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%%%%%%%%%%%%%% Bibliographical database for the book %%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%% TRUST REGION METHODS %%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%% by A.R. Conn, N.I.M. Gould and Ph.L. Toint %%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%% (SIAM, Philadelphia, 2000) %%%%%%%%%%%%%%%%%
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% Copyright: Andrew R. Conn, Nicholas I. M. Gould and Philippe L. Toint
%
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% This version: 18 II 2001
%
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%
% The following works have been cited in the above mentioned book.
% Some are general references to background material, while others
% are central to the development of the trust-region methods we have
% covered. For those references directly relating to trust-region
% methods, we have included a short summary of the work's contents.
% We have deliberately not included any but the most relevant of the
% literally thousands of citations to the Levenberg-Morrison-Marquardt
% method.
%
% We would be delighted to receive any corrections or updates to this list.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% STRING definitions
%
% Journals
@string{AOR = "Annals of Operations Research"}
@string{BIT = "BIT"}
@string{COAP = "Computational Optimization and Applications"}
@string{COMPJ = "Computer Journal"}
@string{CSB = "Chinese Science Bulletin"}
@string{IJNME = "International Journal on Numerical Methods in Engineering"}
@string{IMAJNA = "IMA Journal of Numerical Analysis"}
@string{JIMA = "Journal of the Institute of Mathematics and its Applications"}
@string{JOTA = "Journal of Optimization Theory and Applications"}
@string{JCAM = "Journal of Computational and Applied Mathematics"}
@string{JCM = "Journal of Computational Mathematics"}
@string{LAA = "Linear Algebra and its Applications"}
@string{MC = "Mathematics of Computation"}
@string{MOR = "Mathematics of Operations Research"}
@string{MP = "Mathematical Programming"}
@string{MPA = "Mathematical Programming, Series~A"}
@string{MPB = "Mathematical Programming, Series~B"}
@string{MPS = "Mathematical Programming Studies"}
@string{NUMMATH = "Numerische Mathematik"}
@string{OMS = "Optimization Methods and Software"}
@string{ORSAC = "ORSA Journal on Computing"}
@string{RAIRO-OR = "RAIRO-Recherche Op\'{e}rationnelle---Operations Research"}
@string{RAIRO-MM = "RAIRO-Mathematical Modelling and Numerical
Analysis---Mod\'{e}lisation Math\'{e}matique et Analyse Num\'{e}rique"}
@string{SIADM = "SIAM Journal on Algebraic and Discrete Methods"}
@string{SINUM = "SIAM Journal on Numerical Analysis"}
@string{SICON = "SIAM Journal on Control and Optimization"}
@string{SIOPT = "SIAM Journal on Optimization"}
@string{SIMAA = "SIAM Journal on Matrix Analysis and Applications"}
@string{SISSC = "SIAM Journal on Scientific and Statistical Computing"}
@string{SISC = "SIAM Journal on Scientific Computing"}
@string{SIAPM = "SIAM Journal on Applied Mathematics"}
@string{SIREV = "SIAM Review"}
@string{TOMS = "Transactions of the ACM on Mathematical Software"}
% Places
@string{ANL = "Argonne National Laboratory"}
@string{ANL-ADDRESS = "Argonne, Illinois, USA"}
@string{BELLLABS = "Bell Laboratories"}
@string{BELLLABS-ADDRESS = "Murray Hill, New Jersey, USA"}
@string{CAAM = "Department of Computational and Applied Mathematics,
Rice University"}
@string{CRPC = "Center for Research on Parallel Computers"}
@string{COIMBRA = "Department of Mathematics, University of Coimbra"}
@string{COIMBRA-ADDRESS = "Coimbra, Portugal"}
@string{DAMTP = "Department of Applied Mathematics and Theoretical Physics,
Cambridge University"}
@string{DAMTP-ADDRESS = "Cambridge, England"}
@string{DUNDEE = "Department of Mathematics, University of Dundee"}
@string{DUNDEE-ADDRESS = "Dundee, Scotland"}
@string{ICMSEC = "Institute of Computational Mathematics and
Scientific/Enginering Computing, Chinese Academy of Sciences"}
@string{ICMSEC-ADDRESS = "Beijing, China"}
@string{RICE = "Department of Mathematical Sciences, Rice University"}
@string{RICE-ADDRESS = "Houston, Texas, USA"}
@string{ACRI-CORNELL = "Advanced Computing Research Institute, Cornell Theory
Center"}
@string{CS-CORNELL = "Department of Computer Science, Cornell University"}
@string{CORNELL-ADDRESS = "Ithaca, New York, USA"}
@string{FUNDP = "Department of Mathematics, University of Namur"}
@string{FUNDP-ADDRESS = "Namur, Belgium"}
@string{HAMBURG = "Institute of Applied Mathematics, University of Hamburg"}
@string{HAMBURG-ADDRESS = "Hamburg, Germany"}
@string{HARWELL = "{AERE} {H}arwell Laboratory"}
@string{HARWELL-ADDRESS = "Harwell, Oxfordshire, England"}
@string{HATFIELD = "Numerical Optimization Center, Hatfield Polytechnic"}
@string{IBMWATSON = "T. J. Watson Research Center"}
@string{IBMWATSON-ADDRESS = "Yorktown Heights, NY, USA"}
@string{ICASE = "Institute for Computer Applications in Science and
Engineering"}
@string{ICASE-ADDRESS = "NASA Langley Research Center Hampton, Virginia, USA"}
@string{MADISON = "Computer Sciences Department, University of Wisconsin"}
@string{MADISON-ADDRESS = "Madison, Wisconsin, USA"}
@string{MCCM = "Manchester Centre for Computational Mathematics"}
@string{MCCM-ADDRESS = "Manchester, England"}
@string{NIST = "Applied and Computational Mathematics Division"}
@string{NIST-ADDRESS = "National Institute of Standards and Technology,
Gaithersburg, Maryland, USA"}
@string{NPL = "National Physical Laboratory"}
@string{NPL-ADDRESS = "London, England"}
@string{NWU = "Department of Electrical Engineering and Computer Science,
Northwestern University"}
@string{NWU-ADDRESS = "Evanston, Illinois, USA"}
@string{OTC = "Optimization Technology Center, Argonnne National Laboratory"}
@string{RAL = "Rutherford Appleton Laboratory"}
@string{RAL-ADDRESS = "Chilton, Oxfordshire, England"}
@string{STANFORD = "Department of Operations Research, Stanford University"}
@string{STANFORD-ADDRESS = "Stanford, California, USA"}
@string{MATHWATERLOO = "Faculty of Mathematics, University of Waterloo"}
@string{MATHWATERLOO-ADDRESS = "Waterloo, Ontario, Canada"}
@string{UNICAMP = "Department of Applied Mathematics, IMECC-UNICAMP"}
@string{UNICAMP-ADDRESS = "Campinas, Brasil"}
% Publishers
@string{ADW = "Addison-Wesley Publishing Company"}
@string{ADW-ADDRESS = "Reading, Massachusetts, USA"}
@string{AMS = "American Mathematical Society"}
@string{AMS-ADDRESS = "Providence, Rhode-Island, USA"}
@string{AP = "Academic Press"}
@string{AP-ADDRESS = "London"}
@string{CUP = "Cambridge University Press"}
@string{CUP-ADDRESS = "Cambridge, England"}
@string{FREEMAN = "W. H. Freeman and Company"}, "}
@string{FREEMAN-ADDRESS = "New York and San Francisco"}
@string{KLUWER = "Kluwer Academic Publishers"}
@string{KLUWER-ADDRESS = "Dordrecht, The Netherlands"}
@string{LONGMAN = "Longman Scientific {\&} Technical"}
@string{LONGMAN-ADDRESS = "Harlow, Essex, England"}
@string{MACGH = "McGraw-Hill"}
@string{MACGH-ADDRESS = "New York, USA"}
@string{NH = "North Holland"}
@string{NH-ADDRESS = "Amsterdam, The Netherlands"}
@string{OUP = "Oxford University Press"}
@string{OUP-ADDRESS = "Oxford, England"}
@string{PH = "Prentice-Hall"}
@string{PH-ADDRESS = "Englewood Cliffs, New Jersey, USA"}
@string{WILEY = "J. Wiley and Sons"}
@string{WILEY-ADDRESS = "Chichester, England"}
@string{SIAM = "SIAM"}
@string{SIAM-ADDRESS = "Philadelphia, USA"}
@string{SPRINGER = "Springer Verlag"}
@string{SPRINGER-ADDRESS = "Heidelberg, Berlin, New York"}
@string{WSP = "World Scientific Publishers"}
@string{WSP-ADDRESS = "Singapore"}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% A %%%
@misc{Alex98,
author = {N. M. Alexandrov},
title = {A Trust-Region Algorithm for Bilevel Optimization},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {This paper concerns a trust-region method for solving
nonlinear bilevel optimization problems. No special
assumptions on structure, such as separability or
convexity, are made. Approaches to bilevel optimization
usually fall into one of three categories. One set converts
the bilevel program into a single-level NLP by using the
KKT conditions of the lower-level system as constraints for
the upper-level problem. Another approximates problems of
both levels by series of unconstrained problems resulting
in double-penalty methods. The third category consists of
descent methods on the upper level problem with the use of
some gradient information from the lower-level problem. The
first two approaches suffer from a number of difficulties.
The algorithm presented here is related to those of the
third category and it can be extended to multilevel
optimization. The development of the algorithm is motivated
by applications in multidisciplinary optimization. The
paper presents the algorithm, its analysis, and a number of
numerical examples. It also contrasts the algorithm with a
representative example of the first class---the
collaborative optimization algorithm.},
summary = {A trust-region method for solving nonlinear bilevel
optimization problems is presented without any special
assumptions on structure, such as separability or
convexity. The algorithm is related to descent methods on
the upper level problem which use some gradient information
from the lower-level problem, and can be extended to
multilevel optimization. It is motivated by applications in
multidisciplinary optimization. The algorithm, its
analysis, and a number of numerical examples are discussed
and contrasted with the collaborative optimization method.}}
@techreport{AlexDenn94a,
author = {N. M. Alexandrov and J. E. Dennis},
title = {Multilevel algorithms for nonlinear optimization},
institution = ICASE, address = ICASE-ADDRESS,
number = {94-53}, year = 1994,
abstract = {Multidisciplinary design optimization (MDO) gives rise to
nonlinear optimization problems characterized by a large
number of constraints that naturally occur in blocks. We
propose a class of multilevel optimization methods
motivated by the structure and number of constraints and by
the expense of the derivative computations for MDO. The
algorithms are an extension to the nonlinear programming
problem of the successful class of local
\citebb{Brow69}--\citebb{Bren73} algorithms for nonlinear
equations. Our extensions allow the user to partition
constraints into arbitrary blocks to fit the application,
and they separately process each block and the objective
function, restricted to certain subspaces. The methods use
trust regions as a globalization strategy, and they have
been shown to be globally convergent under reasonable
assumptions. The multilevel algorithms can be applied to
all classes of MDO formulations. Multilevel algorithms for
solving nonlinear systems of equations are a special case
of the multilevel optimization methods. In this case, they
can be viewed as a trust-region globalization of the
Brown-Brent class.},
summary = {Multidisciplinary design optimization (MDO) gives rise to
nonlinear optimization problems with a large number of
constraints that occur in blocks. A class of multilevel
optimization methods motivated by their structure and
number and by the expense of the derivative computations
for MDO is proposed. The algorithms are an extension of the
local \citebb{Brow69}--\citebb{Bren73} algorithms for
nonlinear equations. They allow the user to partition
constraints into arbitrary blocks to fit the application,
and they separately process each block and the objective
function, restricted to certain subspaces. The methods use
trust regions as a globalization strategy, and are globally
convergent.}}
@techreport{AlexDenn94b,
author = {N. M. Alexandrov and J. E. Dennis},
title = {Algorithms for bilevel optimization},
institution = ICASE, address = ICASE-ADDRESS,
number = {94--77}, year = 1994,
abstract = {General multilevel nonlinear optimization problems arise in
design of complex systems and can be used as a means of
regularization for multicriteria optimization problems.
Here for clarity in displaying our ideas, we restrict
ourselves to general bilevel optimization problems, and we
present two solution approaches. Both approaches use a
trust-region globalization strategy, and they can be easily
extended to handle the general multilevel problem. We make
no convexity assumptions, but we do assume that the problem
has a nondegenerate feasible set. We consider necessary
optimality conditions for the bilevel problem formulations
and discuss results that can be extended to obtain
multilevel optimization formulations with constraints at
each level.},
summary = {Two solution approaches are presented for general bilevel
optimization problems. Both use a trust-region
globalization strategy, and can be extended to handle the
general multilevel problem. No convexity assumptions are
made, but it is assumed that the problem has a
non-degenerate feasible set. Necessary optimality conditions
for the bilevel problem formulations are considered and the
results extended to obtain multilevel optimization
formulations with constraints at each level.}}
@techreport{AlexDenn99,
author = {N. M. Alexandrov and J. E. Dennis},
title = {A Class of General Trust-Region Multilevel Algorithms
for Nonlinear Constrained Optimization: Global Convergence
Analysis},
institution = CRPC, address = RICE-ADDRESS,
number = {TR99786-S}, year = 1999,
abstract = {This paper presents a braod class of trust-region multilevel
algorithms for solving large, nonlinear, equality
constrained problems, as well as a global convergence
analysis for the class. The work is motivated by
engineering optimization problems with naturally occurring,
densely or fully-coupled subproblem structure. The
constraints are partitioned into blocks, the number and
composition of which are determined by the application. At
every iteration, a multilevel algorithm minimizes models of
the reduced constraint blocks, followed by a reduced model
of the objective function, in a sequence of subproblems,
each of which yields a substep. The trial step is the sum
of these substeps. The salient feature of the
multilevel class is that there is no prescription on how
the substeps must be computed. Instead, each substep is
required to satisfy mild sufficient decrease and
boundedness conditions on the restricted model that it
minimizes. Within a single trial step computation, all
substeps can be computed by different methods appropriate
to the nature of each subproblem. This feature is
important for the applications of interest in that it allows
for a wide variety of step-choice rules. The trial step is
evaluated via one of two merit functions that take into
account the autonomy of subproblem processing. The
multilevel procedure presented in this work is sequential.
If a problem exhibits full or partial separability, or if
separability is induced by introducing auxiliary variables,
then the multilevel algorithms can easily be stated in
parallel form. However, since this work is devaoted to
analysis, we consider the most general case, that of a
fully coupled problem.},
summary = {A globally convergent class of trust-region multilevel
algorithms for solving equality constrained problems is
presented, motivated by engineering optimization problems
with densely subproblem structure. The constraints are
partitioned into blocks. At every iteration, a multilevel
algorithm minimizes models of the reduced constraint
blocks, followed by a reduced model of the objective
function, each yielding a substep. The trial step is the sum
of these substeps. Each substep is required to satisfy
sufficient decrease and boundedness conditions on its
restricted model and can be computed by a specialized method.
The trial step is evaluated via one of two merit functions
that take into account the autonomy of subproblem
processing.}}
@article{AlexDennLewiTorc98,
author = {N. M. Alexandrov and J. E. Dennis and R. M. Lewis and V.
Torczon},
title = {A Trust Region Framework for Managing the Use of
Approximation Models},
journal = {Structural Optimization},
volume = 15, number = 1, pages = {16--23}, year = 1998,
abstract = {This paper presents an analytically robust, globally
convergent approach to managing the use of approximation
models of various fidelity in optimization. By robust
global behavior we mean the mathematical assurance that the
iterates produced by the optimization algorithm, started at
an arbitrary initial iterate, will converge to a stationary
point or local optimizer for the original problem. The
approach we present is based on the trust region idea from
nonlinear programming and is shown to be provably
convergent to a solution of the original high-fidelity
problem. The proposed method for managing approximations in
engineering optimization suggests ways to decide when the
fidelity, and thus the cost, of the approximations might be
fruitfully increased or decreased in the course of the
optimization iterations. The approach is quite general. We
make no assumptions on the structure of the original
problem, in particular, no assumptions of convexity and
separability, and place only mild requirements on the
approximations. The approximations used in the framework
can be of any nature appropriate to an application; for
instance, they can be represented by analyses, simulations,
or simple algebraic models. This paper introduces the
approach and outlines the convergence analysis.},
summary = {A robust, globally convergent approach to using
approximation models of various fidelity is presented. It
is based on trust regions and converges to a solution
of the original high-fidelity problem. The method suggests
ways to decide when the fidelity, and thus the cost, of the
approximations might be altered in the course of the
iterations. No assumptions on the structure
of the original problem, such as convexity and
separability, are made, and the requirements on the
approximations are mild. These can be of
any nature appropriate to an application; for instance,
they can be represented by analyses, simulations, or simple
algebraic models.}}
@article{Aliz95,
author = {F. Alizadeh},
title = {Interior point methods in semidefinite programming with
applications to combinatorial optimization},
journal = SIOPT,
volume = 5, number = 1, pages = {13--51}, year = 1995}
@inproceedings{Alle95,
author = {D. M. Allen},
title = {Tailoring nonlinear least squares algorithms for the
analysis of compartment models},
booktitle = {Computationally Intensive Statistical Methods.
Proceedings of the 26th Symposium on the Interface,
Fairfax Station, VA, USA},
editor = {J. Sall and A. Lehman},
volume = 26, pages = {533--535}, year = 1995,
abstract = {Compartment models are widely used in
pharmacokinetics. Our objective is to fit compartment
models to data. General numerical optimization
methods frequently perform poorly for this purpose. A
good book on numerical optimization, such as by
J.E. Dennis and R.B. Schnabel, describes multiple
techniques and discusses the advantages and
disadvantages of each. In order to implement these
methods in a software product, one must make a number
of decisions. For example, should a line search
method or a trust region method be used? How should
variables on widely different scales be handled? In
general, these are questions without clear-cut
answers. Compartment models are defined by linear
differential equations. Consequently, compartment
models have a particular structure. We have tailored
general optimization methods to exploit this
structure. Through study and experimentation we have
found workable answers to the questions posed above.},
summary = {Compartment models are defined by linear
differential equations and are widely used in
pharmacokinetics. It is shown how general optimization
methods, including linesearch and trust-region methods
can be tailored to exploit the special structure of
such models.}}
@article{AllgBohmPotrRhei86,
author = {E. L. Allgower and K. B\"{o}hmer and F. A. Potra and W. C.
Rheinboldt},
title = {A mesh-independence principle for operator equations and
their discretizations},
journal = SINUM,
volume = 23, number = 1, pages = {160--169}, year = 1986}
@article{Amay85,
author = {J. Amaya},
title = {On the convergence of curvilinear search algorithms in
unconstrained optimization},
journal = {Operations Research Letters},
volume = 4, number = 1, pages = {31--34}, year = 1985,
abstract = {The purpose of this paper is to unify the conditions under
which curvilinear algorithms for unconstrained optimization
converge. Particularly, two gradient path approximation
algorithms and a trust-region curvilinear algorithm are
examined in this context.},
summary = {The conditions under which curvilinear algorithms for
unconstrained optimization converge are unified. Two
gradient path approximation algorithms and a trust-region
curvilinear algorithm are examined in this light.}}
@book{AndeBaiBiscDemmDongDuCrGreeHammMcKeOstrSore95,
author = {E. Anderson and Z. Bai and C. Bischof and J. Demmel and J.
J. Dongarra and J. DuCroz and A. Greenbaum and S.
Hammarling and A. McKenney and S. Ostrouchov and D. C.
Sorensen},
title = {LAPACK Users' Guide},
publisher = SIAM, address = SIAM-ADDRESS,
edition = {second}, year = 1995}
@inproceedings{AndeGondMeszXu96,
author = {E. D. Andersen and J. Gondzio and C. M\'{e}sz\'{a}ros and
X. Xu},
title = {Implementation of interior point methods for large scale
linear programming},
crossref = {Terl96}, pages = {189--252}}
@article{AndrVice99,
author = {D. A. Andrews and L. N. Vicente},
title = {Characterization of the Smoothness and Curvature of a
Marginal Function for Trust-Region Problem},
journal = MP,
volume = 84, number = 1, pages = {123--137}, year = 1999,
abstract = {This paper studies the smoothness and curvature of a
marginal function for a trust-region problem. In this
problem, a quadratic function is minimized over an
ellipsoid. The marginal function considered is obtained by
perturbing the trust radius, i.e. by changing the size of
the ellipsoid constraint. The values of the marginal
function and of its first and second derivatives are
explicitly calculated in all possible scenarios. A complete
study of the smoothness and curvature of this marginal
function is given. The main motivation for this work arises
from an application in statistics.},
summary = {The smoothness and curvature of a marginal function for a
scaled $\ell_2$ trust-region problem are studied. The
marginal function is obtained by perturbing the trust
radius. The values of the marginal function and of its
first and second derivatives are explicitly calculated in
all possible cases. A complete study of the smoothness and
curvature is given. The work is motivated by an application
in statistics.}}
@techreport{AntoGior96,
author = {F. Antonio and G. Giorgio},
title = {A Bundle type Dual-ascent Approach to Linear
Multi-Commodity Min Cost Flow Problems},
institution = {Pisa University}, address = {Italy},
number = {TR96-01}, year = 1996,
abstract = {We present a cost decomposition approach to Linear
Multicommodity Min Cost Flow problems, based on dualizing
the mutual capacity constraints: the resulting Lagrangian
Dual is solved by means of a new, specialized Bundle
type-dual ascent algorithm. Although decomposition
approaches are generally believed not to be competitive,
even for the solution of large-scale network structured
problems, we present evidence based on extensive
computational comparisons that a careful implementation of
a decomposition algorithm can outperform several other
approaches, especially on problems where the number of
commodities is "large" w.r.t. the size of the graph. Our
specialized Bundle algorithm is characterised by a new
heuristic for the trust region parameter handling, and
embeds a custom, fast Quadratic Programming solver that
permits the implementation of a Lagrangian variables
generation strategy. The Lagrangian Relaxation solver is
capable of exploiting the structural properties of the
single-commodity Min Cost Flow subproblems to avoid using a
"generic" MCF solver whenever it is possible. The proposed
approach can be easily extended to handle extra constraints
or variants such as the Nonsimultaneous Multicommodity
problem. In our computational experience, we also studied
the impact on the relative efficiencies of the different
approaches tested of some characteristics, such as the
number of commodities w.r.t. the total size of the problem.},
summary = {A cost decomposition approach to Linear Multicommodity Min
Cost Flow problems, based on dualizing the mutual capacity
constraints, is presented: the resulting Lagrangian Dual is
solved by means of a specialized Bundle type-dual ascent
algorithm. Extensive computational comparisons show that a
careful implementation of a decomposition algorithm can
outperform several other approaches, especially on problems
where the number of commodities is ``large'' with respect to
the size of the graph. The specialized Bundle algorithm is
characterised by a heuristic for handling the trust region
parameter, and embeds a specialized, efficient quadratic
programming solver that permits the implementation of a
Lagrangian variables generation strategy. Numerical tests
illustrate the relative efficiencies of the different
approaches.}}
@article{AriyLau92,
author = {K. A. Ariyawansa and D. T. M. Lau},
title = {On the Updating Scheme in a Class of Collinear Scaling
Algorithms for Sparse Minimization},
journal = JOTA,
volume = 75, number = 1, pages = {183--193}, year = 1992,
abstract = {Sorensen has proposed a class of algorithms for
sparse unconstrained minimization where the sparsity
pattern of the Cholesky factors of the Hessian is known.
His updates at each iteration depend on the choice of a
vector, and in this reference the question of choosing this
vector is essentially left open. In this note, we propose
a variational problem whose solution may be used to choose
that vector. The major part of the computation of a
solution of this variational problem is similar to the
computation of a trust-region step in unconstrained
optimization. Therefore, well-developed techniques
available for the latter problem can be used to compute
this vector and to perform the updating.},
summary = {The techniques for solving the trust-region subproblem are
used for solving a variational problem arising in the
updating of sparse Hessian approximations for large-scale
unconstrained optimization.}}
@techreport{AshcGrimLewi95,
author = {C. Ashcraft and R. G. Grimes and J. G. Lewis},
title = {Accurate Symmetric Indefinite Linear Equation Solvers},
institution = {Boeing Computer Services},
address = {Seattle, Washington, USA},
year = 1995}
@mastersthesis{Auer93,
author = {G. Auer},
title = {Numerische {B}ehandlung von {T}rust {R}egion {P}roblemen},
school = {Technical University of Graz}, address = {Graz, Austria},
year = 1993}
@article{Axel72,
author = {O. Axelsson},
title = {A generalized {SSOR} method},
journal = BIT,
volume = 12, pages = {443--467}, year = 1972}
@book{Axel96,
author = {O. Axelsson},
title = {Iterative Solution Methods},
publisher = CUP, address = CUP-ADDRESS,
year = 1996}
%%% B %%%
@inproceedings{BandChenMads88,
author = {J. W. Bandler and S. H. Chen and K. Madsen},
title = {An algorithm for one-sided $\ell_1$ optimization with
application to circuit design centering},
booktitle = {Proceedings 1988 IEEE International Symposium on Circuits
and Systems},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 2, pages = {1795--1798}, year = {1988},
abstract = {A highly efficient algorithm for one-sided nonlinear
$\ell_1$ optimization combines a trust region
Gauss-Newton method and a quasi-Newton method. The
proposed method is used as an integral part of an
approach to design centering and yield enhancement.},
summary = {A highly efficient algorithm for one-sided nonlinear
$\ell_1$ optimization combines a trust-region
Gauss-Newton method and a quasi-Newton method. The
proposed method is used as an integral part of an
approach to design centering and yield enhancement.}}
@article{Bann94,
author = {T. Bannert},
title = {A trust region algorithm for nonsmooth optimization},
journal = MP,
volume = 67, number = 2, pages = {247--264}, year = 1994,
abstract = {A trust region algorithm is proposed for minimizing the
nonsmooth composite function $F(x)=h(f(x))$, where $f$ is
smooth and $h$ is convex. The algorithm employs a smoothing
function, which is closely related to \citebb{Flet70b}'s
exact differentiable penalty functions. Global and local
convergence results are given, considering convergence to a
strongly unique minimizer and to a minimizer satisfying
second-order sufficiency conditions.},
summary = {A trust-region algorithm is proposed for minimizing the
non-smooth composite function $F(x)=h(f(x))$, where $f$ is
smooth and $h$ is convex. It uses a smoothing function
closely related to \citebb{Flet70b}'s exact differentiable
penalty function. Global and local convergence to a
strongly unique minimizer and to a minimizer satisfying
second-order sufficiency conditions is considered.}}
@article{BakrBandBiemChenMads98,
author = {M. H. Bakr and J. W. Bandler and R. M. Biemacki and
S. H. Chen and K. Madsen},
title = {A trust region aggressive space mapping algorithm for
{EM} optimization},
journal = {IEEE Transactions on Microwave Theory and Techniques},
volume = 46, number = {12, part~2}, pages = {2412--2425}, year = 1998,
abstract = {A new robust algorithm for EM optimization of
microwave circuits is presented. The algorithm
integrates a trust region methodology with aggressive
space mapping (ASM). A new automated multipoint
parameter extraction process is implemented. EM
optimization of a double-folded stub filter and of an
HTS filter illustrate our new results.},
summary = {A new robust algorithm for EM optimization of
microwave circuits is presented. The algorithm
integrates a trust region methodology with aggressive
space mapping (ASM). A new automated multipoint
parameter extraction process is implemented. EM
optimization of a double-folded stub filter and of an
HTS filter illustrate our new results.}}
@article{BakrBandGeorMads99,
author = {M. H. Bakr and J. W. Bandler and N. Georgieva and K. Madsen},
title = {A hybrid aggressive space mapping algorithm for {EM}
optimization},
journal = {IEEE MTT-S International Microwave Symposium Digest},
volume = 1, pages = {265--268}, year = 1999,
abstract = {We present a novel, hybrid aggressive space mapping (HASM)
optimization algorithm. HASM is a hybrid approach
exploiting both the trust region aggressive space
mapping (TRASM) algorithm and direct optimization. It
does not assume that the final space-mapped design is
the true optimal design and is robust against severe
misalignment between the coarse and the fine
models. The algorithm is based on a novel lemma that
enables smooth switching from the TRASM optimization
to direct optimization and vice versa. The new
algorithm has been tested on several microwave
filters and transformers.},
summary = {A hybrid aggressive space mapping (HASM) optimization
method, combining the trust region aggressive space
mapping (TRASM) algorithm and direct optimization
techiniques, is given. No assumption that the final
space-mapped design is the true optimal design is
made, and the method is robust against severe
misalignment between coarse and fine models. The
algorithm is based on theoretical results that enable
smooth switching from the TRASM optimization to
direct optimization and vice versa. The algorithm is
tested on several microwave filters and transformers.}}
@article{BaraSand92,
author = {R. Barakat and B. H. Sandler},
title = {Determination of the wave-front aberration function from
measured values of the point-spread function---A
2-dimensional phase retrieval problem},
journal = {Journal of the Optical Society of America A-Optics Image
Science and Vision},
volume = 9, number = 10, pages = {1715--1723}, year = 1992,
abstract = {We outline a method for the determination of the unknown
wave-front aberration function of an optical system from
noisy measurements of the corresponding point-spread
function. The problem is cast as a nonlinear unconstrained
minimization problem, and trust region techniques are
employed for its solution in conjunction with analytic
evaluations of the Jacobian and Hessian matrices governing
slope and curvature information. Some illustrative
numerical results are presented and discussed.},
summary = {A method for the determination of the unknown wave-front
aberration function of an optical system from noisy
measurements of the corresponding point-spread function is
considered. The problem is cast as a unconstrained
nonlinear minimization problem. Trust-region techniques are
employed for its solution, using analytic expressions of
the Jacobian and Hessian matrices. Illustrative numerical
results are discussed.}}
@article{BarlTora95,
author = {J. L. Barlow and G. Toraldo},
title = {The effect of diagonal scaling on projected gradient
methods for bound constrained quadratic programming
problems},
journal = OMS,
volume = 5, number = 3, pages = {235--245}, year = 1995}
@inproceedings{BartGoluSaun70,
author = {R. H. Bartels and G. H. Golub and M. A. Saunders},
title = {Numerical techniques in mathematical programming},
crossref = {RoseMangRitt70}, pages = {123--176}}
@article{BazaGood82,
author = {M. S. Bazaraa and J. J. Goode},
title = {Sufficient conditions for a globally exact penalty-function
without convexity},
journal = MPS,
volume = 19, pages = {1--15}, year = 1982}
@incollection{Beal67,
author = {E. M. L. Beale},
title = {Numerical Methods},
booktitle = {Nonlinear programming},
editor = {J. Abadie},
publisher = NH, address = NH-ADDRESS,
pages = {135--205}, year = 1967}
@article{Bell90,
author = {B. M. Bell},
title = {Global convergence of a semi-infinite optimization method},
journal = {Applied Mathematics and Optimization},
volume = 21, pages = {69--88}, year = 1990,
abstract = {A new algorithm for minimizing locally Lipschitz functions
using approximate function values is presented. It yields a
method for minimizing semi-infinite exact penalty functions
that parallels the trust-region methods used in composite
nondifferentiable optimization. A finite method for
approximating a semi-infinite exact penalty function is
developed. A uniform implicit function theorem is
established during this development. An implementation and
test results for the approximate penalty function are
included.},
summary = {An algorithm for minimizing locally Lipschitz functions
using approximate function values is presented. It yields a
method for minimizing semi-infinite exact penalty functions
that parallels the trust-region methods used in composite
non-differentiable optimization. A finite method for
approximating a semi-infinite exact penalty function is
developed. A uniform implicit function theorem is
established during this development. An implementation and
test results for the approximate penalty function are
included.}}
@article{BellRoga95,
author = {M. Bellare and P. Rogaway},
title = {The complexity of approximating a nonlinear program},
journal = MP,
volume = 69, number = 3, pages = {429--441}, year = 1995}
@article{BenTTebo96,
author = {A. Ben{-}Tal and M. Teboulle},
title = {Hidden convexity in some nonconvex quadratically
constrained quadratic-programming},
journal = MP,
volume = 72, number = 1, pages = {51--63}, year = 1996,
abstract = {We consider the problem of minimizing an indefinite
quadratic objective function subject to two-sided
indefinite quadratic constraints. Under a suitable
simultaneous diagonalization assumption (which trivially
holds for a trust-region type problems), we prove that the
original problem is equivalent to a convex minimization
problem with simple linear constraints. We then consider a
special problem of minimizing a concave quadratic function
subject to finitely many convex quadratic constraints,
which is also shown to be equivalent to a minimax convex
problem. In both cases we derive the explicit nonlinear
transformations which allow for recovering the optimal
solution of the nonconvex problems via their equivalent
convex counterparts. Special cases and applications are
also discussed. We outline interior-point polynomial-time
algorithms for the solution of the equivalent convex
programs.},
summary = {The minimization of an indefinite quadratic objective
function subject to two-sided indefinite quadratic
constraints is considered. Under a simultaneous
diagonalization assumption (which trivially holds for a
trust-region type problems), it is shown that the original
problem is equivalent to a convex minimization problem with
simple linear constraints. The special problem of
minimizing a concave quadratic function subject to finitely
many convex quadratic constraints is then considered and
shown to be equivalent to a minmax convex problem. In both
cases, the explicit nonlinear transformations which allow
the recovery of the optimal solution of the non-convex
problems via their equivalent convex counterparts is
derived. Interior-point polynomial-time algorithms for the
solution of the equivalent convex programs are outlined.}}
@article{BenTZibu97,
author = {A. Ben{-}Tal and M. Zibulevsky},
title = {Penalty/Barrier Multiplier Methods for Convex Programming
Problems},
journal = SIOPT,
volume = 7, number = 2, pages = {347--366}, year = 1997}
@article{BenTNemi97,
author = {A. Ben{-}Tal and A. Nemirovskii},
title = {Robust Truss Topology Design via Semidefinite Programming},
journal = SIOPT,
volume = 7, number = 4, pages = {991--1016}, year = 1997}
@article{BereCler97,
author = {Y. Bereaux and J. R. Clermont},
title = {Numerical simulation of two- and three-dimensional complex
flows of viscoelastic fluids using the stream-tube method},
journal = {Mathematics and Computers in Simulation},
volume = 44, number = 4, pages = {387--400}, year = 1997,
abstract = {The present paper examines the stream-tube method in
two-and three-dimensional duct flows. The analysis uses the
concept of stream-tubes in a mapped computational domain of
the physical domain, where streamlines are parallel and
straight. The primary unknown of the problem includes the
transformation between the two domains, together with the
pressure. Mass conservation is automatically verified by
the formulation. Memory-integral constitutive equations may
be considered without the particle-tracking problem. The
method is applied to flows in contractions and a
three-dimensional flow involving a threefold rotational
symmetry. Viscous and elastic liquids involving
memory-integral equations are investigated in the flow
simulations. The discretized schemes for the unknowns are
presented and the relevant equations solved by using
optimization procedures such as the Levenberg-Marquardt and
trust-region methods.},
summary = {The stream-tube method in two-and three-dimensional duct
flows is analyzed using the concept of stream-tubes in a
mapped computational domain of the physical domain, where
streamlines are parallel and straight. The primary unknown
of the problem includes the transformation between the two
domains and the pressure. Mass conservation is
automatically verified by the formulation. Memory-integral
constitutive equations may be considered without the
particle-tracking problem. The method is applied to flows
in contractions and a three-dimensional flow involving a
threefold rotational symmetry. Viscous and elastic liquids
involving memory-integral equations are investigated.. The
discretized schemes are presented and the relevant
equations solved by using optimization procedures such as
the Levenberg-Morrison-Marquardt and trust-region methods.}}
@article{Bert76,
author = {D. P. Bertsekas},
title = {On the {G}oldstein-{L}evitin-{P}oljak gradient projection
method},
journal = {IEEE Transactions on Automatic Control},
volume = {AC-21}, pages = {174--184}, year = 1976}
@article{Bert82a,
author = {D. P. Bertsekas},
title = {Projected {N}ewton Methods for Optimization Problems with
Simple Constraints},
journal = SICON,
volume = 20, number = 2, pages = {221--246}, year = 1982}
@book{Bert82b,
author = {D. P. Bertsekas},
title = {Constrained Optimization and {L}agrange Multiplier Methods},
publisher = AP, address = AP-ADDRESS,
year = 1982}
@article{Best84,
author = {M. J. Best},
title = {Equivalence of some quadratic-programming algorithms},
journal = MP,
volume = 30, number = 1, pages = {71--87}, year = 1984}
@article{BestChak90,
author = {M. J. Best and N. Chakravarti},
title = {Active set algorithms for isotonic regression; a unifying
framework},
journal = MP,
volume = 47, pages = {425--439}, year = 1990}
@techreport{BestRitt76,
author = {M. J. Best and K. Ritter},
title = {An effective algorithm for quadratic minimization problems},
institution = {University of Wisconsin},
address = {Madison, Wisconsin, U.S.A.},
type = {Technical report}, number = 1691, year = 1976}
@article{BiegNoceSchm95,
author = {L. T. Biegler and J. Nocedal and C. Schmid},
title = {A Reduced {H}essian Method for Large-Scale Constrained
Optimization},
journal = SIOPT,
volume = 5, number = 2, pages = {314--347}, year = 1995}
@misc{BielGome98,
author = {R. H. Bielschowsky and F. A. M. Gomes},
title = {Dynamical Control of Infeasibility in Nonlinearly
Constrained Optimization},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {We present a new algorithm for nonconvex nonlinear
programming problems with equality constraints in the form
$\min f(x)$ subject to $h(x)=0$. The algorithm keeps some
characteristics of feasible point methods, but is concerned
with flexibilizing the equality restrictions. Essentially,
the idea is to avoid both forcing the iterates to stay too
near the feasible set and using a merit function. Each
major iteration of the algorithm is divided in two phases.
In the first, we seek a vertical step $d_v$ such that the
infeasibility $x_c=x_k+d_v$ stays controlled, in the sense
that $x_c$ satisfies $\|h(x_c)\|=O(\|g_p(x_c)\|)$, where
$g_p(x_c)$ is the orthogonal projection of $\nabla f(x_c)$
in the tangent space to the restrictions. Frequently, $x_k$
satisfies this condition, so we can make $x_c=x_k$. The aim
of the second phase is to find a horizontal step $d_h$ that
reduces the Lagrangian and stays approximately tangent to
the constraints. This is done by confining $x_c+d_h$ to a
cylinder with radius $r=O(\|g_p(x_c)\|)$ around $h(x)=0$.
The new algorithm is well-suited for large-scale problems.
Implementation details, as well as some preliminary
numerical results based on problems from the CUTE
collection, are presented. We also prove global convergence
results.},
summary = {An algorithm is presented for solving non-convex
problems of the form $\min f(x)$
subject to $h(x)=0$. At each major iteration, one
first seeks a normal step $d_v$ such that $x_c$ satisfies
$\|h(x_c)\|=O(\|g_p(x_c)\|)$, where $g_p(x_c)$ is the
orthogonal projection of $\nabla f(x_c)$ on the tangent
space. A tangential step $d_h$ is then computed that
reduces the Lagrangian and stays approximately tangent to
the constraints. This is done by confining $x_c+d_h$ to a
cylinder with radius $r=O(\|g_p(x_c)\|)$ around $h(x)=0$.
Implementation details, as well as preliminary numerical
results on problems from the {\sf CUTE} collection, are
presented. Global convergence results are also proved.}}
@article{Bier94,
author = {M. Bierlaire},
title = {{HieLoW}: un logiciel d'estimation de mod\`eles Logit
embo\^\i t\'es},
journal = {Cahiers du MET},
volume = 2, pages = {29--43}, month = {Novembre}, year = 1994}
@techreport{Bier95a,
author = {M. Bierlaire},
title = {A robust algorithm for the simultaneous estimation of
hierarchical logit models},
institution = FUNDP, address = FUNDP-ADDRESS,
type = {GRT Report}, number = {95/3}, year = 1995,
abstract = {Estimating simultaneous hierarchical logit models is
conditional to the availability of suitable algorithms.
Powerful mathematical programs are necessary to maximize
the associated non-linear, non-convex, log-likelihood
function. Even if classical methods (e.g. Newton-Raphson)
can be adapted for relatively simple cases, the need of an
efficient and robust algorithm is justified to enable
practioners to consider a wider class of models. The
purpose of this paper is to analyze and to adapt to this
context methodologies available in the optimization
literature. An algorithm is proposed based on two major
concepts from non-linear programming : \emph{a trust-region
method}, that ensures robustness and global convergence,
and \emph{a conjugate gradients iteration}, that can be used
to solve the quadratic subproblems arising in the
estimation process described in this paper. Numerical
experiments are finally presented that indicate the power
of the proposed algorithm and associated software.},
summary = {A trust-region method is proposed for the estimation of
simultaneous hierarchical logit models, where the
subproblem is solved by a truncated conjugate-gradients
technique. Numerical experiments indicate the power of the
proposed algorithm and associated software.}}
@inproceedings{Bier98,
author = {M. Bierlaire},
title = {Discrete Choice Models},
crossref = {LabbLapoTancToin98}, pages = {203--227}}
@article{BierToin95,
author = {M. Bierlaire and Ph. L. Toint},
title = {{MEUSE}: an Origin-Destination Estimator That Exploits
Structure},
journal = {Transportation Research B},
volume = 29, number = 1, pages = {47--60}, year = 1995}
@article{BierToinTuyt91,
author = {M. Bierlaire and Ph. L. Toint and D. Tuyttens},
title = {On iterative algorithms for linear least squares problems
with bound constraints},
journal = LAA,
volume = 143, pages = {111--143}, year = 1991,
abstract = {Three new iterative methods for the solution of the linear
least squares problem with bound constraints are presented
and their performance analyzed. The first is a modification
of a method proposed by \citebb{Lots84}, while the two
others are characterized by a technique allowing for fast
active set changes, resulting in noticeable improvements in
the speed with which constraints active at the solution are
identified. The numerical efficiency of these algorithms is
studied, with particular emphasis on the dependence on the
starting point and the use of preconditioning for
ill-conditioned problems.},
summary = {Three iterative methods for the solution of the linear
least-squares problem with bound constraints are presented
and their performance analyzed. The first is a modification
of a method proposed by \citebb{Lots84}, while the others
two allow fast active set changes. The numerical efficiency
of these algorithms is studied.}}
@inproceedings{Bigg72,
author = {M. C. Biggs},
title = {Constrained Minimization Using Recursive Equality Quadratic
Programming},
crossref = {Loot72}, pages = {411--428}}
@article{Bigg87,
author = {M. C. Bartholomew{-}Biggs},
title = {Recursive quadratic-programming methods based on the
augmented {L}agrangian},
journal = MPS,
volume = 31, pages = {21--41}, year = 1987}
@article{BillFerr97,
author = {S. C. Billups and M. C. Ferris},
title = {{QPCOMP}: A Quadratic Program Based Solver for Mixed
Complementarity Problems},
journal = MP,
volume = 76, number = 3, pages = {533--562}, year = 1997}
@book{Bjor96,
author = {{\AA}. Bj{\"o}rck},
title = {Numerical Methods for Least Squares Problems},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1996}
@article{Bock96,
author = {C. Bockmann},
title = {Curve-fitting and identification of physical spectra},
journal = JCAM,
volume = 70, number = 2, pages = {207--224}, year = 1996,
abstract = {A modification of the trust-region Gauss-Newton method for
the identification of physical spectra is described and
analysed. Local convergence results are presented.},
summary = {A modification of the trust-region Gauss-Newton method for
the identification of physical spectra is described and
analysed. Local convergence results are presented.}}
@inproceedings{BockSchlSchu95,
author = {H. G. Bock and J. P. Schl\"{o}der and V. H. Schulz},
title = {Numerik gro\ss er {D}ifferentiell-{A}lgebraischer
{G}leichungen---{S}imulation und {O}ptimierung},
booktitle = {Proce\ss simulation},
editor = {H. Schuler},
publisher = {VCH Verlaggesellschaft}, address = {Weinheim, Germany},
pages = {35--80}, year = 1995}
@article{Bofi95,
author = {J. M. Bofill},
title = {A Conjugate-Gradient Algorithm with a Trust Region for
Molecular-Geometry Optimization},
journal = {Journal of Molecular Modeling},
volume = 1, number = 1, pages = {11--17}, year = 1995,
abstract = {An algorithm is presented for the optimization of molecular
geometries and general non-quadratic functions using the
nonlinear conjugate gradient method with a restricted step
and a restart procedure. The algorithm only requires the
evaluation of the energy function and its gradient and less
memory storage is needed than for other conjugate gradient
algorithms. Some numerical results are also presented and
the efficiency and behaviour of the algorithm is compared
with the standard conjugate gradient method. On the other
hand we present comparisons of both conjugate gradient and
variable metric methods with and without the trust-region
technique. One of the main conclusions of the present work
is that a trust region always improves the convergence of
an optimization method. A sketch of the algorithm is also
given.},
summary = {An algorithm is presented for the optimization of molecular
geometries and general functions using the
nonlinear conjugate-gradient method with a restricted step
and a restart procedure. The algorithm requires less
memory storage than other conjugate-gradient
algorithms. Numerical results are presented, and the
efficiency of the algorithm is compared with the standard
conjugate-gradient method. A comparison of both
conjugate-gradient and variable-metric methods with and
without the trust-region technique is made. It is concluded
that a trust region always improves the convergence.
A sketch of the algorithm is given.}}
@article{BoggByrdSchn87,
author = {P. T. Boggs and R. H. Byrd and R. B. Schnabel},
title = {A stable and efficient algorithm for nonlinear orthogonal
distance regression},
journal = SISSC,
volume = 8, number = 6, pages = {1052--1078}, year = 1987,
abstract = {One of the most widely used methodologies in scientific and
engineering research is the fitting of equations to data by
least squares. In cases where significant observation
errors exist in the independent variables as well as the
dependent variables, however, the ordinary least squares
(OLS) approach, where all the error are attributed to the
dependent variable, is often inappropriate. An alternate
approach, suggested by several researchers, involves
minimizing the sum of squared orthogonal distances between
each data point and the curve described by the model
equation. We refer to this as orthogonal distance
regression (ODR). This paper describes a method for solving
the orthogonal distance regression problem that is a direct
analog of the trust region Levenberg-Marquardt algorithm.
The number of unknowns involved is the number of model
parameters plus the number of data points, often a very
large number. By exploiting sparsity, however, our
algorithm has a computational effort per step which is of
the same order as required for the Levenberg-Marquardt
method for ordinary least squares. We prove our algorithm
to be globally and locally convergent, and perform
computational tests that illustrate some differences
between ODR and OLS.},
summary = {A method for solving the orthogonal distance regression
problem is described that is an analog of the trust-region
Levenberg-Morrison-Marquardt algorithm. The number of
unknowns involved is the number of model parameters plus
the number of data points, often a very large number. By
exploiting sparsity, the computational effort per step is
of the same order as that required for the
Levenberg-Morrison-Marquardt method for ordinary
least-squares. The algorithm is proved to be globally and
locally convergent. Computational tests
illustrate differences between orthogonal distance
regression and ordinary least-squares.}}
@article{BoggDonaByrdSchn89,
author = {P. T. Boggs and J. R. Donaldson and R. H. Byrd and R. B.
Schnabel},
title = {{ORDPACK} software for weighted orthogonal distance
regression},
journal = TOMS,
volume = 15, number = 4, pages = {348--364}, year = 1989,
abstract = {In this paper, we describe ORDPACK, a software package for
the weighted orthogonal distance regression problem. This
software is an implementation of the algorithm described in
\citebb{BoggByrdSchn87} for finding the parameters that
minimize the sum of the squared weighted orthogonal
distances from a set of observations to a curve or surface
determined by the parameters. It can also be used to solve
the ordinary nonlinear least squares problem. The weighted
orthogonal distance regression procedure implemented is an
efficient and stable trust region (Levenberg-Marquardt)
procedure that exploits the structure of the problem so
that the computational cost per iteration is equal to that
for the same type of algorithm applied to the ordinary
least squares problem. The package allows a general
weighting scheme, provides for finite difference
derivatives, and contains extensive error checking and
report generating facilities.},
summary = {ORDPACK, a software package for the weighted orthogonal
distance regression problem is described. This software is
an implementation of the algorithm described in
\citebb{BoggByrdSchn87} for finding the parameters that
minimize the sum of the squared weighted orthogonal
distances from a set of observations to a curve or surface
determined by the parameters. It can also be used to solve
the ordinary nonlinear least-squares problem. The package
allows a general weighting scheme, provides for finite
difference derivatives, and contains extensive error
checking and report generating facilities.}}
@techreport{BoggDomiRoge95,
author = {P. T. Boggs and P. D. Domich and J. E. Rogers},
title = {An interior point method for general large-scale quadratic
programming problems},
institution = NIST, address = NIST-ADDRESS,
type = {Internal report}, number = {NISTIR 5406}, year = 1995}
@article{BoggToll89,
author = {P. T. Boggs and J. W. Tolle},
title = {A strategy for global convergence in a sequential quadratic
programming algorithm},
journal = SINUM,
volume = 26, number = 3, pages = {600-623}, year = 1989}
@article{BoggToll95,
author = {P. T. Boggs and J. W. Tolle},
title = {Sequential quadratic programming},
journal = {Acta Numerica},
volume = 4, pages = {1--51}, year = 1995}
@techreport{BoggTollKear91,
author = {P. T. Boggs and J. W. Tolle and A. J. Kearsley},
title = {A merit function for inequality constrained nonlinear
programming problems},
institution = NIST, address = NIST-ADDRESS,
type = {Internal report}, number = {NISTIR 4702}, year = 1991}
@techreport{BoggTollKear94,
author = {P. T. Boggs and J. W. Tolle and A. J. Kearsley},
title = {A practical algorithm for general large scale nonlinear
optimization problems},
institution = NIST, address = NIST-ADDRESS,
type = {Internal report}, number = {NISTIR 5407}, year = 1994}
@article{BoggKearToll99,
author = {P. T. Boggs and A. J. Kearsley and J. W. Tolle},
title = {A Practical Algorithm for General Large Scale Nonlinear
Optimization Problems},
journal = SIOPT,
volume = 9, number = 3, pages = {755--778}, year = 1999,
abstract = {We provide an effective and efficient implementation of
a sequential quadratic programming (SQP) algorithm
for the general large scale nonlinear programming
problem. In this algorithm the quadratic programming
subproblems are solved by an interior point method
that can be prematurely halted by a trust region
constraint. Numerous computational enhancements to
improve the numerical performance are
presented. These include a dynamic procedure for
adjusting the merit function parameter and procedures
for adjusting the trust region radius. Numerical
results and comparisons are presented. },
summary = {An effective and efficient implementation of
a sequential quadratic programming (SQP) algorithm
for the general large scale nonlinear programming
problem is given. The quadratic programming
subproblems are solved by an interior point method, and
can be prematurely halted by a trust region
constraint. Numerous computational enhancements to
improve the numerical performance are
presented. These include a dynamic procedure for
adjusting the merit function parameter and procedures
for adjusting the trust region radius. Numerical
results and comparisons are presented.}}
@article{BongConnGoulToin95,
author = {I. Bongartz and A. R. Conn and N. I. M. Gould and Ph. L.
Toint},
title = {{\sf CUTE}: {C}onstrained and {U}nconstrained {T}esting
{E}nvironment},
journal = TOMS,
volume = 21, number = 1, pages = {123--160}, year = 1995}
%abstract = {The purpose of this paper is to discuss the scope and
% functionality of a versatile environment for testing small
% and large-scale nonlinear optimization algorithms.
% Although many of these facilities were originally produced
% by the authors in conjunction with the software package
% {\sf LANCELOT}, we believe that they will be useful in
% their own right and should be available to researchers for
% their development of optimization software. The tools are
% available by anonymous ftp from a number of sources and
% may, in many cases, be installed automatically. The scope
% of a major collection of test problems written in the
% standard input format (SIF) used by the {\sf LANCELOT}
% software package is described. Recognising that most
% software was not written with the SIF in mind, we provide
% tools to assist in building an interface between this
% input format and other optimization packages. These tools
% already provide a link between the SIF and an number of
% existing packages, including MINOS and OSL. In addition,
% as each problem includes a specific classification that is
% designed to be useful in identifying particular classes of
% problems, facilities are provided to build and manage a
% database of this information. There is a UNIX and C-shell
% bias to many of the descriptions in the paper, since, for
% the sake of simplicity, we do not illustrate everything in
% its fullest generality. We trust that the majority of
% potential users are sufficiently familar with UNIX that
% these examples will not lead to undue confusion.}}
@article{BonnBouh95,
author = {J. F. Bonnans and M. Bouhtou},
title = {The trust region affine interior-point algorithm for convex
and nonconvex quadratic-programming},
journal = RAIRO-OR,
volume = 29, number = 2, pages = {195--217}, year = 1995,
abstract = {We study from a theoretical and numerical point of view an
interior point algorithm for quadratic QP using a trust
region idea, formulated by \citebb{YeTse89}. We show that,
under a nondegeneracy hypothesis the algorithm converges
globally in the convex case. For a nonconvex problem, under
a mild additional hypothesis, the sequence of points
converges to a stationary point. We obtain also an
asymptotic linear convergence rate for the cost that
depends only on the dimension of the problem. Then we show
that, provided some modifications are added to the basic
algorithm, the method has a good numerical behaviour.},
summary = {A theoretical and numerical investigation of an interior
point algorithm for quadratic programming using a
trust-region scheme formulated by \citebb{YeTse89} is
performed. Under a non-degeneracy hypothesis, the algorithm
converges globally in the convex case. For a non-convex
problem, the sequence of points converges to a stationary
point under a mild additional assumption. An asymptotic
linear convergence factor that depends only on the
dimension of the problem is given. Provided simple
modifications are made, the method behaves numerically
well.}}
@article{BonnGilbLemaSaga95,
author = {J. F. Bonnans and J. Ch. Gilbert and C. Lemar\'{e}chal and
C. A. Sagastiz\'{a}bal},
title = {A family of variable metric proximal methods},
journal = MPA,
volume = 68, number = 1, pages = {15--47}, year = 1995}
@techreport{BonnLaun92,
author = {J. F. Bonnans and G. Launay},
title = {Implicit trust region algorithm for constrained
optimization},
institution = {Institute Nationale Recherche Informatique et Automation},
address = {Le Chesnay, France},
year = 1992,
abstract = {The authors study the convergence of sequential
quadratic programming algorithms for the nonlinear
programming problems. Assuming only that the
direction is a stationary point of the current
quadratic program they study the local convergence
properties without strict complementarity. They
obtain some global and superlinearly convergent
algorithms. As a particular case they formulate an
extension of Newton's method that is quadratically
convergent to a point satisfying a strong sufficient
second order condition.},
summary = {Convergence of sequential
quadratic programming algorithms for the nonlinear
programming problems is studied without assuming strict
complementarity and global and superlinearly convergence
results are obtained.}}
@article{BonnLaun95,
author = {J. F. Bonnans and G. Launay},
title = {Sequential Quadratic-Programming with Penalization of the
Displacement},
journal = SIOPT,
volume = 5, number = 4, pages = {792--812}, year = 1995,
abstract = {In this paper, we study the convergence of a sequential
quadratic programming algorithm for the nonlinear
programming problem. The Hessian of the quadratic program
is the sum of an approximation of the Lagrangian and of a
multiple of the identity that allows us to penalize the
displacement. Assuming only that direction is a stationary
point of the current quadratic program we study the local
convergence properties without strict complementarity. In
particular, we use a very weak condition on the
approximation of the Hessian to the Lagrangian. We obtain
some global and superlinearly convergent algorithm under
weak hypotheses. As a particular case we formulate an
extension of Newton's method that is quadratically
convergent to a point satisfying a strong sufficient
second-order condition.},
summary = {The convergence of an SQP algorithm for nonlinear
programming is considered. The Hessian of the QP is the sum
of an approximation of the Lagrangian and of a multiple of
the identity that penalizes the displacement. Assuming only
that direction is a stationary point for the current QP,
the algorithm is globally and superlinearly convergent
without strict complementarity. As a particular case, an
extension of Newton's method is given that is quadratically
convergent to a point satisfying a strong sufficient
second-order condition.}}
@article{BonnPaniTitsZhou92,
author = {J. F. Bonnans and E. Panier and A. L. Tits and J. L. Zhou},
title = {Avoiding the {M}aratos effect by means of a nonmonotone
linesearch {II}. Inequality constrained problems---feasible
iterates},
journal = SINUM,
volume = 29, pages = {1187--1202}, year = 1992,
abstract = {When solving inequality constrained optimization problems
via Sequential Quadratic Programming (SQP), it is
potentially advantageous to generate iterates that all
satisfy the constraints: all quadratic programs encountered
are then feasible and there is no need for a surrogate
merit function. (Feasibility of the successive iterates is
in fact required in many contexts such as in real-time
applications or when the objective function is not defined
outside the feasible set.) It has recently been shown that
this, indeed, possible, by means of a suitable perturbation
of the original SQP iteration, without loosing superlinear
convergence. In this context, the well-known
\citebb{Mara78} effect is coumpounded by the possible
infeasibility of the full step of one even close to the
solution. These difficulties have been accomodated by
making use of a suitable modification of a "bending"
technique proposed by \citebb{MaynPola82}, requiring
evaluation of the constraints function at an auxiliary
point at each iteration. In Part I of this two-part paper,
it was shown that, when feasibility of the successive
iterates is not required, the Maratos effect can be avoided
by combining Mayne and Polyak's technique with a
non-monotone line serach proposed by
\citebb{GripLampLuci86} in the context of unconstrained
optimization in such a way that, except possibly at a few
early iterations, function evaluations are no longer
performed at auxiliary points. In this second part, it is
shown that feasibility can be restored without resorting to
additional constraint evaluations, by adaptively estimating
a bound on the second derivatives of the active
constraints. Extension to constrained minimax problems is
briefly discussed.},
summary = {When solving inequality constrained optimization problems
via SQP, it may be advantageous to generate iterates that
always satisfy the constraints. In this context, the
Maratos effect is compounded by the possible
infeasibility of a unit step even close to the solution. It
is shown that feasibility can be restored without resorting
to additional constraint evaluations, by adaptively
estimating a bound on the second derivatives of the active
constraints. The extension to constrained minimax problems
is briefly discussed.}}
@article{BonnPola97,
author = {J. F. Bonnans and C. Pola},
title = {A Trust Region Interior Point Algorithm for Linearly
Constrained Optimization},
journal = SIOPT,
volume = 7, number = 3, pages = {717--731}, year = 1997,
abstract = {We present an extension for nonlinear optimization under
linear constraints of an algorithm for quadratic
programming using a trust region idea, introduced by
\citebb{YeTse89} and extended by \citebb{BonnBouh95}. Due
to the nonlinearity of the cost function, we use a
linesearch in order to reduce the step if necessary. We
prove that, under suitable hypotheses, the algorithm
converges to a point satisfying the first-order optimality
system, and we analyse under which conditions the unit
stepsize will asymptotically be accepted.},
summary = {An extension of the trust-region quadratic programming
algorithm of \citebb{YeTse89} and \citebb{BonnBouh95} to
nonlinear optimization subject to linear constraints is
given. A linesearch is used to reduce the step if
necessary. Under suitable hypotheses, the algorithm
converges to a first-order stationary point. Conditions
under which the unit stepsize is asymptotically accepted
are analysed.}}
@article{BookDennFranSeraTorcTros99,
author = {A. J. Booker and J. E. Dennis and P. D. Frank and D. B.
Serafini and V. Torczon and M. W. Trosset},
title = {A Rigorous Framework for Optimization of Expensive
Functions by Surrogates},
journal = {Structural Optimization},
volume = 17, number = 1, pages = {1--13}, year = 1999}
% abstract = {The goal of the research reported on here is to develop
% rigourous optimization algorithms to apply to some
% engineering design problems for which direct application
% of traditional optimization approaches is not practical.
% This paper presents and analyzes a framework for
% generating and managing a sequence of surrogate objective
% functions to obtain convergence to a minimizer of an
% expensive objective function subject to simple constraints.
% The approach is widely applicable because it does not
% require, or even approximate, derivatives of the objective.
% Numerical results are presented for a 31-variable
% helicopter rotor design example and for a standard
% optimization test example. This is a brief description
% of a portion of the Boeing/IBM/Rice University
% collaboration, whose purpose is to develop effective
% numerical methods for managing the use of approximation
% concepts in design optimization.},
% abstract = {A framework is presented for generating and managing a
% sequence of surrogate objective functions to obtain
% convergence to a minimizer of an expensive objective
% function subject to simple constraints. The approach is
% widely applicable because it does not require, or even
% approximate, derivatives of the objective. This is
% especially useful in the context of engineering design
% problems for which direct application of traditional
% optimization approaches is not practical. Numerical
% results are presented for a 31-variable helicopter
% rotor design example and for a standard optimization
% test example.}}
@book{Boot64,
author = {J. C. Boot},
title = {Quadratic Programming},
publisher = NH, address = NH-ADDRESS,
year = 1964}
@phdthesis{Borg94,
author = {J. Borggaard},
title = {The sensitivity equation method for optimal design},
school = {Department of Mathematics, Virginia Polytechnic Institute
and State University},
address = {Blackburg, Virginia, USA},
year = 1994}
@inproceedings{BorgBurn94,
author = {J. Borggaard and J. Burns},
title = {A sensitivity equation approach to shape optimization in
fluid flows},
booktitle = {Proceedings of the IMA Period of Concentration on Flow
Control},
editor = {M. Gunzburger},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
pages = {49--78}, year = 1997,
abstract = {In this paper we apply a sensitivity equation method to
shape optimization problems. An algorithm is developed and
tested on a problem of designing optimal forebody
simulators for a 2D, inviscid supersonic flow. The
algorithm uses a BFGS/Trust Region optimization scheme with
sensitivities computed by numerically approximating the
linear partial differential equations that determine the
flow sensitivities. Numerical examples are presented to
illustrate the method.},
summary = {A sensitivity equation method is applied to shape
optimization problems. An algorithm is developed and tested
on a problem of designing optimal forebody simulators for a
2D, inviscid supersonic flow. The algorithm uses a
BFGS/trust-region optimization scheme, with sensitivities
computed by numerically approximating the linear partial
differential equations that determine the flow
sensitivities. Numerical examples are presented.}}
@article{BorgBurn97,
author = {J. Borggaard and J. Burns},
title = {A {PDE} sensitivity equation method for optimal aerodynamic
design},
journal = {Journal of Computational Physics},
volume = 136, number = 2, pages = {366--384}, year = 1997,
abstract = {The use of gradient based optimization algorithms in
inverse design is well established as a practical approach
to aerodynamic design. A typical procedure uses a
simulation scheme to evaluate the objective function (from
the approximate states) and its gradient, then passes this
information to an optimization algorithm. Once the
simulation scheme (CFD flow solver) has been selected and
used to provide approximate function evaluations, there are
several possible approaches to the problem of computing
gradients. One popular method is to differentiate the
simulation scheme and compute design sensitivities that are
then used to obtain gradients. Although this black-box
approach has many advantages in shape optimization
problems, one must compute mesh sensitivities in order to
compute the design sensitivity. In this paper, we present
an alternative approach using the PDE sensitivity equation
to develop algorithms for computing gradients. This
approach has the advantage that mesh sensitivities need not
be computed. Moreover, when it is possible to use the CFD
scheme for both the forward problem and the sensitivity
equation, then there are computational advantages. An
apparent disadvantage of this approach is that it does not
always produce consistent derivatives. However, for a
proper combination of discretization schemes, one can show
asymptotic consistency under mesh refinement, which is
often sufficient to guarantee convergence of the optimal
design algorithm. In particular, we show that when
asymptotically consistent schemes are combined with a
trust-region optimization algorithm, the resulting optimal
design method converges. We denote this approach as the
sensitivity equation method. The sensitivity equation
method is presented, convergence results are given and the
approach is illustrated on two optimal design problems
involving shocks.},
summary = {An approach using the PDE sensitivity equation to develop
algorithms for computing gradients in inverse design
problems is considered, in which
that mesh sensitivities need not be computed. Moreover, it is
advantageous when possible to use the CFD scheme for both
the forward problem and the sensitivity equation. For a
proper combination of discretization schemes, asymptotic
consistency under mesh refinement is shown, which is often
sufficient to guarantee convergence of the optimal design
algorithm. In particular, when asymptotically consistent
schemes are combined with a trust-region optimization
algorithm, the resulting optimal design method converges.
Such a method is described, convergence results are given,
and the approach is illustrated on two optimal design
problems involving shocks.}}
@article{Borw82,
author = {J. M. Borwein},
title = {Necessary and sufficient conditions for quadratic
minimality},
journal = {Numerical Functional Analysis and Optimization},
volume = 5, pages = {127--140}, year = 1982}
@article{Boua97,
author = {A. Bouaricha},
title = {Algorithm 765: {STENMIN}: a software package for large,
sparse unconstrained optimization using tensor methods},
journal = TOMS,
volume = 23, number = 1, pages = {81--90}, year = 1997,
abstract = {We describe a new package for minimizing an unconstrained
nonlinear function where the Hessian is large and sparse.
The software allows the user to select between a tensor
method and a standard method based upon a quadratic model.
The tensor method models the objective function by a
fourth-order model, where the third- and fourth-order terms
are chosen such that the extra cost of forming and solving
the model is small. The new contribution of this package
consists of the incorporation of an entirely new way of
minimizing the tensor model that makes it suitable for
solving large, sparse optimization problems efficiently.
The test results indicate that, in general, the tensor
method is often more efficient and more reliable than the
standard Newton method for solving large, sparse
unconstrained optimization problems.},
summary = {A package is presented for minimizing an unconstrained
nonlinear function where the Hessian is large and sparse.
The software allows the user to select between a tensor
method and a standard method based upon a quadratic model.
The tensor method models the objective function by a
fourth-order model, where the third- and fourth-order terms
are chosen such that the extra cost of forming and solving
the model is small. The contribution consists of the
incorporation of a new way of minimizing the tensor model
that is suitable for solving large, sparse problems. The
test results indicate that the tensor method is often more
efficient and more reliable than the standard Newton
method.}}
@article{BouaSchn97,
author = {A. Bouaricha and R. B. Schnabel},
title = {Algorithm 768: {TENSOLVE}: a software package for
solving systems of nonlinear equations and nonlinear
least-squares problems using tensor methods},
journal = TOMS,
volume = 23, number = 2, pages = {174--195}, year = 1997,
abstract = {This article describes a modular software package for
solving systems of nonlinear equations and nonlinear
least-squares problems, using a new class of methods called
tensor methods. It is intended for small- to medium-sized
problems, say with up to 100 equations and unknowns, in
cases where it is reasonable to calculate the Jacobian
matrix or to approximate it by finite differences at each
iteration. The software allows the user to choose between a
tensor method and a standard method based on a linear
model. The tensor method approximates $F(x)$ by a quadratic
model, where the second-order term is chosen so that the
model is hardly more expensive to form, store, or solve
than the standard linear model. Moreover, the software
provides two different global strategies: a line search
approach and a two-dimensional trust region approach. Test
results indicate that, in general, tensor methods are
significantly more efficient and robust than standard
methods on small-and medium-sized problems in iterations
and function evaluations.},
summary = {A modular software package for solving systems of nonlinear
equations and nonlinear least-squares problems, using
tensor methods, is described. It is intended for
small- to medium-sized problems for which it is
reasonable to calculate the Jacobian or to
approximate it by finite differences at each iteration. It
allows the user to choose between a tensor method
and a standard method based on a linear model. The tensor
method approximates $F(x)$ by a quadratic model, where the
second-order term is chosen so that the model is hardly
more expensive to form, store, or solve than the
linear model. The software provides both a linesearch and a
two-dimensional trust-region approach. Test results
indicate that tensor methods are significantly more
efficient and robust than standard methods on small-and
medium-sized problems in iterations and function
evaluations.}}
@article{BouaSchn98,
author = {A. Bouaricha and R. B. Schnabel},
title = {Tensor methods for large sparse systems of nonlinear
equations},
journal = MP,
volume = 82, number = 3, pages = {377--412}, year = 1998,
abstract = {This paper introduces tensor methods for solving large
sparse systems of nonlinear equations. Tensor methods for
nonlinear equations were developed in the context of
solving small to medium-sized dense problems. They base
each iteration on a quadratic model of the nonlinear
equations, where the second-order term is selected so that
the model requires no more derivative or function
information per iteration than standard linear model-based
methods, and hardly more storage or arithmetic operations
per iteration. Computational experiments on small to
medium-sized problems have shown tensor methods to be
considerably more efficient than standard Newton based
methods, with a particularly large advantage on singular
problems. This paper considers the extension of this
approach to solve large sparse problems. The key issue
considered is how to make efficient use of sparsity in
forming and solving the tensor model problem at each
iteration. Accomplishing this turns out to require an
entirely new way of solving the tensor model that
successfully exploits the sparsity of the Jacobian, whether
the Jacobian is nonsingular or singular. We develop such an
approach and, based upon it, an efficient tensor method for
solving large sparse systems of nonlinear equations. Test
results indicate that this tensor method is significantly
more efficient and robust than an efficient sparse
Newton-based method, in terms of iterations, function
evaluations, and execution time.},
summary = {This paper considers the extension of the tensor approach
proposed in \citebb{BouaSchn97}. This requires a new way of
solving the tensor model that exploits the sparsity of the
Jacobian, whether singular or not.}}
@article{BouaSchn99,
author = {A. Bouaricha and R. B. Schnabel},
title = {Tensor methods for large sparse nonlinear
least-squares},
journal = SISC,
volume = 21, number = 4, pages = {1199--1221}, year = 1999}
@book{BoydElGhFeroBala94,
author = {S. Boyd and L. El{-}Ghaoui and E. Feron and V. Balakrishnan},
title = {Linear Matrix Inequalities in Systems and Control Theory},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1994}
@techreport{Bran95,
author = {M. A. Branch},
title = {Inexact reflective {N}ewton methods for large-scale
optimization subject to bound constraints},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR 95-1543}, year = 1995,
abstract = {This thesis addresses the problem of minimizing a
large-scale nonlinear function subject to simple bound
constraints. The most popular methods to handle bound
constrained problems, active-set methods, introduce a
combinatorial aspect to the problem. For these methods, the
number of steps to converge may be related to the number of
constraints. For large problems, this behavior is
particularly detrimental. Reflective Newton methods avoid
this problem by staying strictly within the constrained
region. As a result, these methods have strong theoretical
properties. Moreover, they behave experimentally like an
unconstrained method: the number of steps to a solution is
not strongly correlated with problem size. In this thesis,
we discuss the reflective Newton approach and how it can be
combined with inexact Newton techniques, within a subspace
trust-region method, to efficiently solve large problems.
Two algorithms are presented. The first uses a line search
as its globalizing strategy. The second uses a strictly
trust-region approach to globally converge to a local
minimizer. Global convergence and rate of convergence
results are established for both methods. We present
computational evidence that using inexact Newton steps
preserves the properties of the reflective Newton methods:
the iteration counts are as low as when ``exact'' Newton
steps are used. Also, both the inexact and exact methods
are robust when the starting point is varied. Furthermore,
the inexact reflective Newton methods have fast convergence
when negative curvature is encountered, a trait not always
shared by similar active-set type methods. The role of
negative curvature is further explored by comparing the
subspace trust-region approach to other common
approximations to the full-space trust-region problem. On
problems where only positive curvature is found, these
trust-region methods differ little in the number of
iterations to converge. However, for problems with negative
curvature, the subspace method is more effective in
capturing the negative curvature information, resulting in
faster convergence. Finally a parallel implementation on
the IBM SP2 is described and evaluated; the scalability and
efficiency of this implementation are shown to be as good
as the matrix-vector multiply routine it depends on.},
summary = {The problem of minimizing a large-scale nonlinear function
subject to simple bound constraints using the reflective
Newton approach is addressed, including how it can be
combined with inexact Newton techniques within a subspace
trust-region method. A linesearch and a trust-region
algorithm are presented, that have fast
convergence when negative curvature is encountered. The
subspace trust-region approach is compared to other
approximations to the trust-region subproblem. On problems
where only positive curvature is found, these
methods differ little in efficiency. For problems with
negative curvature, the subspace method is more effective
in capturing the negative curvature information, resulting
in faster convergence. A parallel implementation on
the IBM SP2 is evaluated, whose scalability and efficiency
of are as good as the matrix-vector multiply
routine it depends on.}}
@techreport{BranColeLi95,
author = {M. A. Branch and T. F. Coleman and Y. Li},
title = {A subspace, interior, and conjugate gradient method for
large-scale bound-constrained minimization problems},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR 95-1525}, year = 1995,
abstract = {A subspace adaptation of the \citebb{ColeLi96b} trust
region and interior method is proposed for solving
large-scale bound-constrained minimization problems. This
method can be implemented with either sparse Cholesky
factorization or conjugate gradient computation. Under
reasonable conditions the convergence properties of this
subspace trust region method are as strong as those of its
full-space version. Computational performance on various
large-scale test problems are reported; advantages of our
approach are demonstrated. Our experience indicates our
proposed method represents an efficient way to solve
large-scale bound-constrained minimization problems.},
summary = {A subspace adaptation of the method by \citebb{ColeLi96b}
is proposed for solving large-scale bound-constrained
minimization problems. This method can be implemented with
either sparse Cholesky factorization or
conjugate-gradients. The convergence properties of this
subspace trust-region method are as strong as those of its
full-space version. Computational performance on
large-scale test problems illustrates its advantages.}}
@book{ColeBranGrac99,
author = {T. F. Coleman and M. A. Branch and A. Grace},
title = {Optimization Toolbox for Use with Matlab},
publisher = {The Math Works Inc.}, address = {Natick, Massuchussets, USA},
year = 1999}
@article{Breg67,
author = {L. M. Bregman},
title = {The relaxation method for finding the common points of
convex sets and its applications to the solution of
problems in convex programming},
journal = {USSR Computational Mathematics and Mathematical Physics},
volume = 7, pages = {200--217}, year = 1967}
@inproceedings{BreiShan93,
author = {M. G. Breitfeld and D. F. Shanno},
title = {Preliminary computational experience with modified
log-barrier functions for large-scale nonlinear programming},
crossref = {HageHearPard94}, pages = {45--66}}
@article{BreiShan96,
author = {M. G. Breitfeld and D. F. Shanno},
title = {Computational experience with penalty-barrier methods for
nonlinear programming},
journal = AOR,
volume = 62, pages = {439-463}, year = 1996}
@article{Bren73,
author = {R. P. Brent},
title = {Some efficient algorithms for solving systems of nonlinear
equations},
journal = SINUM,
volume = 10, number = 2, pages = {327--344}, year = 1973}
@article{BrimLove91,
author = {J. Brimberg and R. F. Love},
title = {Estimating travel distances by the weighted
$\ell_p$ norm},
journal = {Naval Research Logistics},
volume = 38, pages = {241-259}, year = 1991 }
@inproceedings{BroyAtti84,
author = {C. G. Broyden and N. F. Attia},
title = {A smooth sequential penalty function method for nonlinear
programming},
crossref = {BalaThom84}, pages = {237--245}}
@article{BroyAtti88,
author = {C. G. Broyden and N. F. Attia},
title = {Penalty functions, {N}ewton's method and quadratic
programming},
journal = JOTA,
volume = 58, number = 3, pages = {377--385}, year = 1988}
@article{Brow69,
author = {K. M. Brown},
title = {A quadratically convergent {N}ewton-like method based on
{G}aussian elimination},
journal = SINUM,
volume = 6, number = 4, pages = {560--569}, year = 1969}
@article{Broy70,
author = {C. G. Broyden},
title = {The Convergence Of A Class Of Double-rank Minimization
Algorithms},
journal = JIMA,
volume = 6, pages = {76--90}, year = 1970}
@article{Buck78a,
author = {A. G. Buckley},
title = {A Combined Conjugate-Gradient Quasi-{N}ewton Minimization
Algorithm},
journal = MP,
volume = 15, pages = {200--210}, year = 1978}
@article{Buck78b,
author = {A. G. Buckley},
title = {Extending the relationship between the conjugate gradient
and the {BFGS} algorithms},
journal = MP,
volume = 15, pages = {343--348}, year = 1978}
@article{BudiLeeSaxeFree96,
author = {D. E. Budil and S. Lee and S. Saxena and J. H. Freed},
title = {Nonlinear-least-squares analysis of slow-motion
{EPR}-spectra in one and 2 dimensions using a modified
{L}evenberg-{M}arquardt algorithm},
journal = {Journal of Magnetic Resonance, Series A},
volume = 120, number = 2, pages = {155--189}, year = 1996,
abstract = {The application of the model trust-region modification of
the Levenberg-Marquardt minimization algorithm to the
analysis of one-dimensional CW EPR and multidimensional
Fourier-transform (FT) EPR spectra especially in the
slow-motion regime is described. The dynamic parameters
describing the slow motion are obtained from least-squares
fitting of model calculations based on the stochastic
Liouville equation (SLE) to experimental spectra, the
trust-region approach is inherently more efficient than the
standard Levenberg-Marquardt algorithm, and the efficiency
of the procedure may be further increased by a
separation-of-variables method in which a subset of fitting
parameters is independently minimized at each iteration,
thus reducing the number of parameters to be fitted by
nonlinear least squares. A particularly useful application
of this method occurs in the fitting of multicomponent
spectra, for which it is possible to obtain the relative
population of each component by the separation-of-variables
method. These advantages, combined with recent improvements
in the computational methods used to solve the SLE, have
led to an order-of-magnitude reduction in computing time,
and have made it possible to carry out interactive,
real-time fitting on a laboratory workstation with a
graphical interface. Examples of fits to experimental data
will be given, including multicomponent CW EPR spectra as
well as two- and three- dimensional FT EPR spectra,
emphasis is placed on the analytic information available
from the partial derivatives utilized in the algorithm, and
how it may be used to estimate the condition and uniqueness
of the fit, as well as to estimate confidence limits for
the parameters in certain cases.},
summary = {The application of the model trust-region modification of
the Levenberg-Morrison-Marquardt algorithm to the analysis
of one-dimensional CW EPR and multidimensional
Fourier-transform (FT) EPR spectra especially in the
slow-motion regime is described. The dynamic parameters
describing the motion are obtained from least-squares
fitting of model calculations based on the stochastic
Liouville equation (SLE) to experimental spectra. The
trust-region approach is more efficient than the
standard Levenberg-Morrison-Marquardt algorithm, and the
efficiency of the procedure may be further increased by a
separation-of-variables method in which a subset of fitting
parameters is independently minimized at each iteration.
An application is the fitting of multicomponent spectra,
for which it is possible to obtain the relative population
of each component by the separation-of-variables method.
These advantages, combined with improvements in the
computational solution of the SLE, have led to an
order-of-magnitude reduction in computing time, and have
made it possible to carry out interactive, real-time
fitting on a laboratory workstation. Examples are given,
including multicomponent CW EPR spectra as well as two- and
three- dimensional FT EPR spectra.}}
@inproceedings{BulsSillSaxe92,
author = {A. B. Bulsari and M. Sillanpaa and H. Saxen},
title = {An expert system for continuous steel casting using neural
networks},
booktitle = {Expert Systems in Mineral and Metal Processing.
Proceedings of the IFAC Workshop},
editor = {J. L. Jamsa-Jounela and A. J. Niemi},
publisher = {Pergamon}, address = {Oxford, England},
pages = {155--159}, year = 1992,
abstract = {Developing an expert system is often time consuming
even after knowledge acquisition. Artificial neural
networks offer an advantageous alternative to coding
such knowledge in an expert system shell or writing a
program for it. This paper illustrates the
feasibility of using a feedforward neural network for
knowledge storage and inferencing for an industrial
problem. The inputs to the network were information
about an incoming ladle of steel, and the output was
about its suitability for successful continuous
casting, giving an indication on whether problems
would be encountered in the beginning and/or at the
end of the casting. A trust-region optimisation
method was used for training the networks, where the
input-output relation of the nodes was given by a
sigmoid function. This training method has been used
successfully for other neural network problems and
was found to be quite reliable and robust. By using a
feedforward neural network as an expert system for
predicting operational problems in the continuous
steel casting process, some inconsistencies in the
knowledge base were also revealed.},
summary = {A feedforward neural network for knowledge storage
and inferencing is studied for an industrial
problem. The inputs to the network receive information
about an incoming ladle of steel, and the output predicts
its suitability for successful continuous
casting. A trust-region optimization
method is used for training the network. This training
method is found to be reliable and robust.}}
@techreport{BultVial83,
author = {J. P. Bulteau and J. P. Vial},
title = {Unconstrained Optimization by Approximation of a Projected
Gradient Path},
institution = {CORE, UCL}, address = {Louvain-la-Neuve, Belgium},
type = {CORE Discussion Paper}, number = 8352, year = 1983,
abstract = {In an earlier paper \citebb{BultVial87} discussed a general
algorithm based on a one-dimensional search over a
curvilinear path according to a trust-region scheme. This
paper proposes a particular implementation of the general
algorithm using as a particular path an approximation of
the projected gradient path on a two dimensional space.
This algorithm is endowed with attractive convergence
properties. Newton and quasi-Newton like variants are
discussed, with corresponding numerical experiments.},
summary = {\citebb{BultVial87} discuss a general algorithm based on a
one-dimensional search over a curvilinear path according to
a trust-region scheme. An implementation using an
approximation of the projected gradient path on a
two-dimensional space is given. This algorithm is endowed
with attractive convergence properties. Newton and
quasi-Newton like variants are discussed, with
corresponding numerical experiments.}}
@article{BultVial85,
author = {J. P. Bulteau and J. P. Vial},
title = {A restricted trust region algorithm for unconstrained
optimization},
journal = JOTA,
volume = 47, number = 4, pages = {413--435}, year = 1985,
abstract = {This paper proposes an efficient implementation of a
trust-region-like algorithm. The trust region is restricted
to an appropriately chosen two-dimensional subspace.
Convergence properties are discussed and numerical results
are reported.},
summary = {An efficient implementation of a trust-region method is
proposed, in which the trust region is restricted to an
appropriately chosen two-dimensional subspace. Convergence
properties are discussed and numerical results are
reported.}}
@article{BultVial87,
author = {J. P. Bulteau and J. P. Vial},
title = {Curvilinear path and trust region in unconstrained
optimization---a convergence analysis},
journal = MPS,
volume = 30, pages = {82--101}, year = 1987,
abstract = {In this paper we propose a general algorithm for solving
unconstrained optimization problems. The basic step of the
algorithm consists in finding a "good" successor point to
the current iterate by choosing it along a curvilinear path
within a trust region. This scheme is due to
\citebb{Powe70c} and has been applied by \citebb{Sore82} to
a particular type of path. We give a series of properties
that an arbitrary path should satisfy in order to achieve
global convergence and fast asymptotical convergence. We
review various paths that have been proposed in the
literature and study the extent to which they satisfy our
properties.},
summary = {A general algorithm for unconstrained optimization is
proposed. Its basic step consists in finding a "good"
successor point to the current iterate by choosing it along
a curvilinear path within a trust region. Properties that
an arbitrary path should satisfy in order to achieve global
convergence and fast asymptotic convergence are given.
Various paths that have been proposed in the literature are
reviewed in this light.}}
@article{Bunc74,
author = {J. R. Bunch},
title = {Partial pivoting strategies for symmetric matrices},
journal = SINUM,
volume = 11, pages = {521--528}, year = 1974}
@article{BuncGayWels93,
author = {D. S. Bunch and D. M. Gay and R. E. Welsch},
title = {Algorithm 717: subroutines for maximum likelihood and
quasi-likelihood estimation of parameters in nonlinear
regression models},
journal = TOMS,
volume = 19, number = 1, pages = {109--130}, year = 1993,
abstract = {The authors present FORTRAN 77 subroutines that solve
statistical parameter estimation problems for general
nonlinear models, e.g., nonlinear least-squares,
maximum likelihood, maximum quasi-likelihood,
generalized nonlinear least-squares, and some robust
fitting problems. The accompanying test examples
include members of the generalized linear model
family, extensions using nonlinear predictors
('nonlinear GLIM'), and probabilistic choice models,
such as linear-in-parameter multinomial probit
models. The basic method, a generalization of the
NL2SOL algorithm for nonlinear least-squares, employs
a model/trust-region scheme for computing trial
steps, exploits special structure by maintaining a
secant approximation to the second-order part of the
Hessian, and adaptively switches between a
Gauss-Newton and an augmented Hessian
approximation. Gauss-Newton steps are computed using
a corrected seminormal equations approach. The
subroutines include variants that handle simple
bounds on the parameters, and that compute
approximate regression diagnostics.},
summary = {FORTRAN 77 subroutines are presented that solve
statistical parameter estimation problems for general
nonlinear models, e.g., nonlinear least-squares,
maximum likelihood, maximum quasi-likelihood,
generalized nonlinear least-squares, and some robust
fitting problems. The basic method, a generalization
of the NL2SOL algorithm for nonlinear least-squares,
employs a model/trust-region scheme for computing trial
steps, maintain a secant approximation to the
second-order part of the Hessian, and adaptively switches
between Gauss-Newton and full Newton
approximations. Gauss-Newton steps are computed using
a corrected seminormal equations approach. The
subroutines include variants that handle simple
bounds on the parameters, and that compute
approximate regression diagnostics.}}
@article{BuncKauf77,
author = {J. R. Bunch and L. C. Kaufman},
title = {Some stable methods for calculating inertia and solving
symmetric linear equations},
journal = MC,
volume = 31, pages = {163--179}, year = 1977}
@article{BuncKauf80,
author = {J. R. Bunch and L. C. Kaufman},
title = {A computational method for the indefinite quadratic
programming problem},
journal = LAA,
volume = 34, pages = {341-370}, year = 1980}
@article{BuncParl71,
author = {J. R. Bunch and B. N. Parlett},
title = {Direct methods for solving symmetric indefinite systems of
linear equations},
journal = SINUM,
volume = 8, number = 4, pages = {639--655}, year = 1971}
@article{Burk90,
author = {J. V. Burke},
title = {On the identification of active constraints {II}: the
nonconvex case},
journal = SINUM,
volume = 27, number = 4, pages = {1081--1102}, year = 1990,
abstract = {In this paper the results of \citebb{BurkMore88} on the
identification of active constraints are extended to the
nonconvex constrained nonlinear programming problem. The
approach is motivated by the geometric structure of certain
polyhedral convex ``linearization'' of the constraint
region at each iteration. As in \citebb{BurkMore88}
questions of constraint identification are couched in
termes of the faces of these polyhedra. The main result
employs a nondegeneracy condition due to \citebb{Dunn87}
and the linear independence condition to obtain a
characterization of those algorithms that identify the
optimal active constraints in a finite number of
iterations. The role of the linear independence condition
is carefully examined and it is argued that it is required
within the context of the
\citebb{Wils63}--\citebb{Han77}--\citebb{Powe78} sequential
quadratic programming algorithm and \citebb{Flet87}'s $QL$
algorithm.},
summary = {The results of \citebb{BurkMore88}, on the identification
of active constraints, are extended to non-convex
constrained programming. The approach is
motivated by the geometric structure of certain polyhedral
convex ``linearization'' of the constraint region at each
iteration. Questions of constraint identification are
couched in termes of the faces of these polyhedra. The main
result employs a non-degeneracy condition due to
\citebb{Dunn87} and the linear independence condition to
obtain a characterization of those algorithms that identify
the optimal active constraints in a finite number of
iterations. It is argued that the linear independence
condition is required for the Wilson--Han--Powell SQP
algorithm and for Fletcher's $QL$ algorithm.}}
@article{Burk92,
author = {J. V. Burke},
title = {A Robust Trust Region Method for Constrained Nonlinear
Programming Problems},
journal = SIOPT,
volume = 2, number = 2, pages = {324--347}, year = 1992,
abstract = {Most of the published work on trust region algorithms for
constrained optimization is derived from the original work
of \citebb{Flet87} on trust region algorithms for
nondifferentiable exact penalty functions. These methods
are restricted to applications where a reasonable estimate
of the magnitude of the optimal Kuhn-Tucker multiplier
vector can be given, More recently an effort has been made
to extend the trust region methodology to the sequential
quadratic programming (SQP) algorithm of \citebb{Wils63},
\citebb{Han77} and \citebb{Powe78}. All of these extensions
to the Wilson--Han--Powell SQP algorithm consider only the
equality-constrained case and require strong global
regularity hypotheses. This paper presents a general
framework for trust region algorithms for constrained
problems that does not require such regularity hypotheses
and allows very general constraints. The approach is
modeled on the one given by Powell for convex composite
optimization problems and is driven by linear subproblems
that yield viable estimates for the value of an exact
penalty parameter. These results are applied to the
Wilson--Han--Powell SQP algorithm and Fletcher's
S$\ell_1$QP algorithm. Local convergence results are also
given.},
summary = {A general framework for trust-region algorithms for
constrained problems is presented, that does not require
strong regularity hypotheses and allows very general
constraints. The approach is modeled on the one given by
Powell for convex composite optimization problems and is
driven by linear subproblems that yield viable estimates
for the value of an exact penalty parameter. These results
are applied to the Wilson-Han-Powell SQP algorithm and
Fletcher's S$\ell_1$QP algorithm. Local convergence results
are given.}}
@article{BurkHan89,
author = {J. V. Burke and S. P. Han},
title = {A robust sequential quadratic-programming method},
journal = MP,
volume = 43, number = 3, pages = {277--303}, year = 1989}
@article{BurkMore88,
author = {J. V. Burke and J. J. Mor\'{e}},
title = {On the identification of active constraints},
journal = SINUM,
volume = 25, number = 5, pages = {1197--1211}, year = 1988,
abstract = {Nondegeneracy conditions that guarantee that the optimal
active constraints are identified in a finite number of
iterations are studied. Results of this type have only been
established for a few algorithms, and then under
restrictive hypothesis. The main result is a
characterization of those algorithms that identify the
optimal constraints in a finite number of iterations. This
result is obtained with a non-degeneracy assumption which is
equivalent, in the standard nonlinear programming problem,
to the assumption that there is a set of strictly
complementary Lagrange multipliers. As an important
consequence of the authors' results the way that this
characterization applies to gradient projection and
sequential quadratic programming algorithms is shown.},
summary = {Non-degeneracy conditions that guarantee that the optimal
active constraints are identified in a finite number of
iterations are studied and a characterization of those
algorithms that identify the optimal constraints in a
finite number of iterations is derived. This result is
obtained with a non-degeneracy assumption which is
equivalent, in the standard nonlinear programming problem,
to the assumption that there is a set of strictly
complementary Lagrange multipliers. As an important
consequence, the way that this characterization applies to
gradient projection and sequential quadratic programming
algorithms is shown.}}
@article{BurkMoreTora90,
author = {J. V. Burke and J. J. Mor\'{e} and G. Toraldo},
title = {Convergence properties of trust region methods for linear
and convex constraints},
journal = MPA,
volume = 47, number = 3, pages = {305--336}, year = 1990,
abstract = {We develop a convergence theory for convex and linearly
constrained trust region methods which only requires that
the step between iterates produce a sufficient reduction in
the trust region subproblem. Global convergence is
established for a general convex minimization problem while
local analysis is for linearly constrained problems. The
main local result establishes that if the sequence
converges to a nondegenerate stationary point then the
active constraints at the solution are identified in a
finite number of iterations. As a consequence of the
identification properties, we develop rate of convergence
results by assuming that the step is a truncated Newton
method. Our development is mainly geometrical; this
approach allows the development of a convergence theory
without any linear independence assumptions.},
summary = {A convergence theory is developed for convex and linearly
constrained trust-region methods which only requires that
the step between iterates produce a sufficient reduction in
the subproblem. Global convergence is established for a
general convex problem while local analysis is for linearly
constrained problems. It is shown that if the sequence
converges to a non-degenerate stationary point then the
active constraints at the solution are identified in a
finite number of iterations. As a consequence, rate of
convergence results are developed by assuming that the step
is a truncated Newton method. This development is mainly
geometrical; such an approach allows the development of a
convergence theory without any linear independence
assumptions.}}
@article{BurkMore95,
author = {J. V. Burke and J. J. Mor\'{e}},
title = {Exposing Constraints},
journal = SIOPT,
volume = 4, number = 3, pages = {573--595}, year = 1994,
abstract = {The development of algorithms and software for the solution
of large-scale optimization problems has been the main
motivation behind the research on the identification
properties of optimization algorithms. The aim of an
identification result for a linearly constrained problem is
to show that if the sequence generated by an optimization
algorithm converges to a stationary point, then there is a
nontrivial face $F$ of the feasible set such that after a
finite number of iterations, the iterates enter and remain
in the face $F$. The paper develops the identification
properties of linearly constrained optimization algorithms
without any nondegeneracy or linear independence
assumptions. The main result shows that the projected
gradient converges to zero if and only if the iterates
enter and remain in the face exposed by the negative
gradient. This result generalizes results of
\citebb{BurkMore88} for nondegenerate cases.},
summary = {The identification properties of linearly constrained
optimization algorithms is developed without any
non-degeneracy or linear independence assumptions. It shown
that the projected gradient converges to zero if and only
if the iterates enter and remain in the face exposed by the
negative gradient, which generalizes results of
\citebb{BurkMore88} for non-degenerate cases.}}
@misc{BurkWeig97,
author = {J. V. Burke and A. Weigmann},
title = {Notes on Limited Memory BFGS Updating in A Trust-Region
Framework},
institution = {Department of Mathematics, University of Washington},
address = {Seattle, Washington, USA},
year = 1997,
abstract = {The limited memory BFGS method pioneered by Jorge Nocedal
is usually implemented as a line search method where the
search direction is computed from a BFGS approximation to
the inverse of the Hessian. the advantage of inverse
updating is that the serach directions are obtained by a
matrix-vector multiplication. Furthermore, experience
shows that when the BFSG approximation is appropriately
re-scaled (or re-sized) at each iteration, the line search
stopping criteria are often satisfied for the first trial
step. In this note it is observed that limited memory
updates to the Hessian approximations can also be applied
in the context of a trust-region algorithm with only modest
increase in the linear algebra costs. This is true even
though in the trust-region framework one maintains
approximations to the Hessian rather than its inverse.
The key to this observation is the compact form of the
limited memory updates derived by Byrd, Nocedal and Schnabel
(1994). Numerical results on a few of the MINPACK-2 test
problems indicate that an implementation that incorporates
re-scaling directly into the trust-region updating
procedure exhibits convergence behavior comparable to a
standard implementation of the algorithm by
\citebb{LiuNoce89}.},
summary = {A limited-memory BFGS method is described that uses
re-scaling at each iteration and a trust-region technique
to ensure convergence. The effects of a non-monotone
technique as well as that of an implicit scheme for
updating the trust-region radius are dicussed. Numerical
experiments are reported.}}
@article{ButcJackMitt97,
author = {J. C. Butcher and Z. Jackiewicz and H. D. Mittelmann},
title = {A nonlinear optimization approach to the construction of
general linear methods of high order},
journal = JCAM,
volume = 81, number = 2, pages = {181--196}, year = 1997,
abstract = {We describe the construction of diagonally implicit
multistage integration methods of order and stage order
$p=q=7$ and $p=q=8$ for ordinary differential equations.
These methods were obtained using state-of-the-art
optimization methods, particularly variable- model
trust-region least-squares algorithms.},
summary = {The construction of diagonally implicit multistage
integration methods of order and stage order $p=q=7$ and
$p=q=8$ for ordinary differential equations is described.
These methods were obtained using variable-model
trust-region least-squares algorithms.}}
@article{Byrd90,
author = {R. H. Byrd},
title = {On the Convergence of Constrained Optimization Methods with
Accurate {H}essian Information on a Subspace},
journal = SINUM,
volume = 27, number = 1, pages = {141--153}, year = 1990}
@misc{Byrd99,
author = {R. H. Byrd},
title = {Step Computation in a Trust Region Interior Point Method},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999,
abstract = {In computing a step on an interior point method for
nonlinear inequality constrained optimization, use of
a trust region provides a unified structure for dealing
with negative curvature and rank deficiency. However, in
an interior point context, exactly solving the complete
trust region subproblem step presents us with an
intractable subproblem. Therefore, most practical methods
use a cheap approximate solution to the trust region
problem. Here, we consider several of these approximate
methods, and point out that methods that are theoretically
adequate can have serious drawbacks. We present some new
approaches to this problem and argue that in many cases
simpler is better. We also consider to what extent these
approximate approaches provide the benefits promised by
trust regions in the cases of negative curvature and rank
deficiency.},
summary = {Approximate methods for computing a trust-region step in
an interior point method for nonlinear inequality constrained
optimization are considered, and some new approaches
proposed. The extent to which they provide the benefits
promised by trust regions in the cases of negative
curvature and rank deficiency are also discussed.}}
@article{ByrdNoce91,
author = {R. H. Byrd and J. Nocedal},
title = {An analysis of reduced {H}essian methods for constrained
optimization},
journal = MP,
volume = 49, number = 3, pages = {285--323}, year = 1991}
@techreport{ByrdGilbNoce96,
author = {R. H. Byrd and J. Ch. Gilbert and J. Nocedal},
title = {A Trust Region Method Based on Interior Point Techniques
for Nonlinear Programming},
institution = {INRIA}, address = {Rocquencourt, France},
number = 2896, year = 1996,
abstract = {An algorithm for minimizing a nonlinear function subject to
nonlinear equality and inequality constraints is described.
It can be seen as an extension of primal interior point
methods to non-convex optimization. The new algorithm
applies sequential quadratic programming techniques to a
sequence of barrier problems, and uses trust regions to
ensure the robustness of the iteration and to allow the
direct use of second order derivatives. An analysis of the
convergence of the new method is presented.},
summary = {An algorithm for minimizing a nonlinear function subject to
nonlinear equality and inequality constraints is described.
It can be seen as an extension of primal interior-point
methods to non-convex optimization. The algorithm applies
SQP techniques to a sequence of barrier problems, and uses
trust regions to ensure the robustness of the iteration and
to allow the direct use of second order derivatives. A
convergence analysis is presented.}}
@article{ByrdHribNoce99,
author = {R. H. Byrd and M. E. Hribar and J. Nocedal},
title = {An Interior Point Algorithm for Large Scale Nonlinear
Programming},
journal = SIOPT,
volume = 9, number = 4, pages = {877-900}, year = 2000,
abstract = {We describe a new algorithm for solving large nonlinear
programming problems. It incorporates within the interior
point method two powerful tools for solving nonlinear
problems: sequential quadratic programming (SQP) and trust
region techniques. SQP ideas are used to efficiently handle
nonlinearities in the constraints. Trust region strategies
allow the algorithm to treat convex and non-convex problems
uniformly, permit the direct use of second derivative
information and provide a safeguard in the presence of
nearly dependent constraint gradients. Both primal and
primal-dual versions of the algorithm are developed, and
their performance is compared with that of LANCELOT on a
set of large and difficult nonlinear problems.},
summary = {An algorithm for solving large nonlinear programming
problems is described. It incorporates
SQP and trust-region techniques within the
interior-point method. SQP ideas are used to efficiently
handle nonlinearities in the constraints. Trust-region
strategies allow the algorithm to treat convex and
non-convex problems uniformly, permit the direct use of
second derivative information and provide a safeguard in
the presence of nearly dependent constraint gradients. Both
primal and primal-dual versions of the algorithm are
developed, and their performance is compared with that of
{\sf LANCELOT} on a set of large and difficult nonlinear
problems.}}
@techreport{ByrdNoceWalt00,
author = {R. H. Byrd and J. Nocedal and R. A. Waltz},
title = {Feasible Interior Methods Using Slacks for Nonlinear
Optimization},
institution = OTC, address = OTC-ADDRESS,
number = 11, year = 2000,
abstract = {A slack-based feasible interior point method is described
which can be derived as a modification of infeasible methods.
The modification is minor for most line search methods, but
trust region methods require special attention. It is shown
how the Cauchy point, which is often computed in trust region
methods, must be modified so that the feasible method is
effective for problems containing both equality and
inequality constraints. The relationship between
slack-based methods and traditional feasible methods is
discussed. Numerical results showing the relative performance
of feasible versus infeasible interior point methods are
presented.},
summary = {A slack-based feasible interior-point method is described
which can be derived as a modification of infeasible methods.
The modification is minor for most line search methods, but
trust-region methods require special attention. It is shown
how the Cauchy point must be modified so that the feasible
trust-region method is effective for problems containing
both equality and inequality constraints. The relationship
between slack-based methods and traditional feasible methods
is discussed. Numerical results showing the relative
performance of feasible versus infeasible interior-point
methods are presented.}}
@article{ByrdKhalSchn96,
author = {R. H. Byrd and H. F. Khalfan and R. B. Schnabel},
title = {Analysis of a symmetric rank-one trust region method},
journal = SIOPT,
volume = 6, number = 4, pages = {1025--1039}, year = 1996,
abstract = {Several recent computational studies have shown that the
symmetric rank-one (SR1) method for unconstrained
optimization and shows that the method has an $n+1$ step
$q$-superlinear rate of convergence. The analysis makes
neither of the assumptions of uniform linear independence
of the iterates nor positive definiteness of the Hessian
approximations that have been made in other recent analysis
of SR1 methods. The trust region method that is analyzed is
fairly standard, except that it includes the feature that
the Hessian approximation is updated after all steps,
including rejected steps. We also present computational
results that show that this feature, safeguarded in a way
that is consistent with the convergence analysis, does not
harm the efficiency of the SR1 trust region method.},
summary = {Computational studies have considered the
symmetric rank-one (SR1) method for unconstrained
optimization and shown that the method has an $n+1$ step
$q$-superlinear rate of convergence. The proposed analysis
makes neither of the assumptions of uniform linear
independence of the iterates nor positive definiteness of
the Hessian approximations that have been made in former
such analyses. The trust-region method is standard, but
requires the Hessian approximation to be updated after
all steps, including rejected ones. Computational results
indicate that this feature, safeguarded in a way that is
consistent with the convergence analysis, does not harm
the efficiency of the SR1 trust-region method.}}
@article{ByrdLuNoceZhu95,
author = {R. H. Byrd and P. Lu and J. Nocedal and C. Zhu},
title = {A limited memory algorithm for bound constrained
optimization},
journal = SISC,
volume = 16, number = 5, pages = {1190--1208}, year = 1995}
@article{ByrdSchn86,
author = {R. H. Byrd and R. B. Schnabel},
title = {Continuity of the null space basis and constrained
optimization},
journal = MP,
volume = 35, number = 1, pages = {32--41}, year = 1986}
@article{ByrdSchnShul87,
author = {R. H. Byrd and R. B. Schnabel and G. A. Shultz},
title = {A trust region algorithm for nonlinearly constrained
optimization},
journal = SINUM,
volume = 24, pages = {1152--1170}, year = 1987,
abstract = {We present a trust region-based method for the general
nonlinearly constrained optimization problem. The method
works by iteratively minimizing a quadratic model of the
Lagrangian subject to a possibly relaxed linearization of
the problem constraints and a trust region constraint. The
model minimization may be done approximately with a
dogleg-type approach. We show that this method is globally
convergent even if singular or indefinite Hessian
approximations are made. A second order correction step
that brings the iterates closer to the feasible set is
described. If sufficiently precise Hessian information is
used, this correction step allows us to prove that the
method is also locally quadratically convergent, and that
the limit satisfies the second order necessary conditions
for constrained optimization. An example is given to show
that, without this correction, a situation similar to the
\citebb{Mara78} effect may occur where the iteration is
unable to move away from a saddle point.},
summary = {A trust-region-based method is given for general
nonlinearly constrained optimization problem. It
iteratively minimizes a quadratic model of the Lagrangian
subject to a possibly relaxed linearization of the problem
constraints and a trust-region constraint. The model
minimization may be done approximately with a dogleg-type
approach. Global convergence is shown. A second order
correction step is also described. If sufficiently precise
Hessian information is used, this step ensures
locally quadratically convergence and
satisfaction of the second order necessary conditions.
An example shows
that, without this correction, a situation similar to the
\citebb{Mara78} effect may occur where the iteration is
unable to move away from a saddle point.}}
@article{ByrdSchnShul88,
author = {R. H. Byrd and R. B. Schnabel and G. A. Shultz},
title = {Approximate solution of the trust region problem by
minimization over two-dimensional subspaces},
journal = MP,
volume = 40, number = 3, pages = {247--263}, year = 1988,
abstract = {The trust region problem, minimization of a quadratic
function subject to a spherical trust region constraint,
occurs in many optimization algorithms. In a previous
paper, the authors introduced an inexpensive approximate
solution technique for this problem that involves the
solution of a two-dimensional trust region problem. They
showed that using this approximation in an unconstrained
optimization algorithm leads to the same theoretical global
and local convergence properties as are obtained using the
exact solution to the trust region problem. This paper
reports computational results showing that the
two-dimensional minimization approach gives nearly optimal
reductions in the $n$-dimension quadratic model over a wide
range of test cases. We also show that there is very little
difference, in efficiency and reliability, between using
the approximate or exact trust region step in solving
standard test problems for unconstrained optimization.
These results may encourage the application of similar
approximate trust region techniques in other contexts.},
summary = {Computational results are given, showing that the
two-dimensional minimization approach of
\citebb{ByrdSchnShul87} gives nearly optimal reductions in
the $n$-dimensional quadratic model over a wide range of
test cases. It is also shown that there is very little
difference, in efficiency and reliability, between using
the approximate or exact trust-region step when solving
standard test problems for unconstrained optimization.}}
@article{ByrdTapiZhan92,
author = {R. H. Byrd and R. A. Tapia and Y. Zhang},
title = {An {SQP} Augmented {L}agrangian {BFGS} Algorithm for
Constrained Optimization},
journal = SIOPT,
volume = 2, number = 2, pages = {210--241}, year = 1992}
%%% C %%%
@article{CalaMore87,
author = {P. H. Calamai and J. J. Mor\'{e}},
title = {Projected gradient methods for linearly constrained
problems},
journal = MP,
volume = 39, pages = {93--116}, year = 1987}
@article{CarpLustMulvShan93,
author = {T. J. Carpenter and I. J. Lustig and J. M. Mulvey
and D. F. Shanno},
title = {Higher-Order Predictor-Corrector Interior Point Methods with
Application to Quadratic Objectives},
journal = SIOPT,
volume = 3, number = 4, pages = {696--725}, year = 1993}
@phdthesis{Carr59,
author = {C. W. Carrol},
title = {An operations research approach to the economic
optimization of a kraft pulping process},
school = {Institute of Paper Chemistry},
address = {Appleton, Wisconsin, USA},
year = 1959}
@article{Carr61,
author = {C. W. Carrol},
title = {The Created Response Surface Technique for Optimizing
Nonlinear Restrained Systems},
journal = {Operations Research},
volume = 9, number = 2, pages = {169--184}, year = 1961}
@techreport{Cart86,
author = {R. G. Carter},
title = {Multi-Model Algorithms for Optimization},
institution = CAAM, address = RICE-ADDRESS,
number = {TR86-3}, year = 1986,
abstract = {A recent approach for the construction of nonlinear
optimization software has been to allow an algorithm to
choose between two possible models to the objective
function at each iteration. The model switching algorithm
NL2SOL of \citebb{DennGayWels81} and the hybrid algorithms
of Al-Baali and Fletcher have proven highly effective in
practice. Although not explicitly formulated as multi-model
methods, many other algorithms implicitly perform a model
switch under certain circumstances (e.g., resetting a
secant model to the exact value of the Hessian). We present
a trust region formulation for multi-model methods which
allows the efficient incorporation of an arbitrary number
of models. Global convergence can be shown for three
classes of algorithms under different assumptions on the
models. First, essentially any multi-model algorithm is
globally convergent if each of the models is sufficiently
well-behaved. Second, algorithms based on the central
feature of the NL2SOL switching system are globally
convergent if one model is well behaved and each other
model obeys a ``sufficient predicted decrease'' condition.
No requirement is made that these alternate models be
quadratic. Third, algorithms of the second type which
directly enforce the ``sufficient predicted decrease''
condition are globally convergent if a single model is
sufficiently well behaved.},
summary = {A trust-region formulation for multi-model methods is
presented which allows the efficient incorporation of an
arbitrary number of models. Global convergence is
established for three classes of algorithms under different
assumptions on the models. Firstly, essentially any
multi-model algorithm is globally convergent if each of the
models is sufficiently well-behaved. Secondly, algorithms
based on the central feature of the NL2SOL switching system
are globally convergent if one model is well behaved and
each other model obeys a ``sufficient predicted decrease''
condition. No requirement is made that these alternate
models be quadratic. Finally, algorithms of the second type
which directly enforce the ``sufficient predicted
decrease'' condition are globally convergent if a single
model is sufficiently well behaved.}}
@techreport{Cart87,
author = {R. G. Carter},
title = {Safeguarding {H}essian Approximations in Trust Region
Algorithms},
institution = CAAM, address = RICE-ADDRESS,
number = {TR87-12}, year = 1987,
abstract = {In establishing global convergence results for trust region
algorithms applied to unconstrained optimization, it is
customary to assume either a uniform upper bound on the
sequence of Hessian approximations or an upper bound linear
in the iteration count. The former property has not been
established for most commonly used secant updates, and the
latter has only been established for some updates under the
highly restrictive assumption of convexity. One purpose of
the uniform upper bound assumption is to establish a
technical condition we refer to as the \emph{uniform
predicted decrease condition}. We show that this condition
can also be obtained by milder assumptions, the simplest of
which is a uniform upper bound on the sequence of Rayleigh
quotients of the Hessian approximations \emph{in the
gradient directions}. This in turn suggests both a simple
procedure for detecting questionable Hessian
approximations, and several natural procedures for {\em
correcting} them when detected. In numerical testing, one
of these procedures increased the reliability of the
popular BFGS method by a factor of two (i.e., the procedure
halved the number of test cases to fail to converge to a
critical point in a reasonable number of iterations).
Further, for those problems where both methods were
successful, this safeguarding procedure actually improved
the average efficiency of the BFGS by ten to twenty
percent.},
summary = {It is shown that the assumptions on the Hessian
approximations in a trust-region method for unconstrained
optimization can be replaced by a uniform upper bound on
the sequence of Rayleigh quotients of the Hessian
approximations in the gradient directions. This suggests
both a simple procedure for detecting questionable
approximations, and several natural procedures for
correcting them when detected. In numerical tests, one of
these procedures increased the reliability of the BFGS
method by a factor of two. For those problems where both
the safeguarded and original methods were successful, this
safeguarding procedure improved the average efficiency of
the BFGS by ten to twenty percent.}}
@article{Cart91,
author = {R. G. Carter},
title = {On the global convergence of trust region methods using
inexact gradient information},
journal = SINUM,
volume = 28, number = 1, pages = {251--265}, year = 1991,
abstract = {Trust region algorithms are an important class of methods
that can be used to solve unconstrained optimization
problems. Strong global convergence results are
demonstrated for a class of methods where the gradient
values are approximated rather than computed exactly,
provided they obey a simple relative error condition. No
requirement is made that gradients be recomputed to
successively greater accuracy after unsuccessful
iterations.},
summary = {Strong global convergence results are demonstrated for
trust-region methods for unconstrained minimization where
the gradient values are approximated rather than computed
exactly, provided they obey a simple relative error
condition. No requirement is made that gradients be
recomputed to successively greater accuracy after
unsuccessful iterations.}}
@article{Cart93,
author = {R. G. Carter},
title = {Numerical Experience with a class of Algorithms for
nonlinear Optimization using inexact function and gradient
information},
journal = SISSC,
volume = 14, number = 2, pages = {368--388}, year = 1993,
abstract = {For optimization problems associated with engineering
design, parameter estimation, image reconstruction,
and other optimization/simulation applications, low
accuracy function and gradient values are frequently
much less expensive to obtain than high accuracy
values. The computational performance of trust region
methods for nonlinear optimization is investigated
for cases when high accuracy evaluations are
unavailable or prohibitively expensive, and earlier
theoretical predictions that such methods are
convergent even with relative gradient errors of 0.5
or more is confirmed. The proper choice of the amount
of accuracy to use in function and gradient
evaluations can result in orders-of-magnitude savings
in computational cost.},
summary = {The computational performance of trust-region
methods for nonlinear optimization is investigated
for cases when high accuracy evaluations of function and
gradient are unavailable or prohibitively expensive, and
theoretical predictions that such methods are
convergent even with relative gradient errors of 0.5
or more is confirmed. The proper choice of the amount
of accuracy to use in function and gradient
evaluations can result in orders-of-magnitude savings
in computational cost.}}
@phdthesis{Case97,
author = {L. Case},
title = {An $\ell_1$ penalty function approach to the nonlinear
bilevel programming problem},
school = {University of Waterloo}, address = {Waterloo, Canada},
year = 1997,
summary = {The nonlinear bilevel problem is a difficult constrained
optimization problem where the variables are partitioned
into two sets, $z$ and $y$.
The feasibility conditions require that y is the solution
of a separate optimization problem.
The approach of this thesis replaces the original problem
by stating the necessary conditions for a solution and
determining a one level programming problem by using
an exact penalty function to attempt to satisfy these
conditions. The resulting non-convex, non-smooth problems
are solved by a trust-region approach and specialized
techniques are used to overcome difficulties arising from
the non-differentiability. A unique method is developed
to handle degeneracy. Proof of convergence to a minimum
of the penalty function is given. Test results and an
analysis of the solutions are included.
}}
@article{Cauc47,
author = {A. Cauchy},
title = {M\'{e}thode g\'{e}n\'{e}rale pour la r\'{e}solution des
syst\`{e}mes d'\'{e}quations simultan\'{e}es},
journal = {Comptes Rendus de l'Acad\'{e}mie des Sciences},
pages = {536--538}, year = 1847}
@techreport{Celi85,
author = {M. R. Celis},
title = {A Trust Region Strategy for Nonlinear Equality Constrained
Optimization.},
institution = CAAM, address = RICE-ADDRESS,
number = {TR85-4}, year = 1985,
abstract = {Many current algorithms for nonlinear constrained
optimization problems determine a search direction by
solving a quadratic programming subproblem. The global
convergence properties are addressed by using a line search
technique and a merit function to modify the length of the
step obtained from the quadratic program. In unconstrained
optimization, trust region strategies have been very
successful. In this thesis we present a new approach for
equality constrained optimization problems based on a trust
region strategy. The direction selected is not necessarily
the solution of the standard quadratic programming
subproblem.},
summary = {An approach for equality constrained optimization problems
based on a trust-region strategy is presented. The
direction selected is not necessarily the solution of the
standard quadratic programming subproblem.}}
@inproceedings{CeliDennTapi85,
author = {M. R. Celis and J. E. Dennis and R. A. Tapia},
title = {A trust region strategy for nonlinear equality constrained
optimization},
crossref = {BoggByrdSchn85}, pages = {71--82},
abstract = {Many current algorithms for nonlinear constrained
optimization problems determine a direction by solving a
quadratic programming subproblem. The global convergence
properties are addressed by using a line search technique
and a merit function to modify the length of the step
obtained from the quadratic program. In unconstrained
optimization, trust regions strategies have been very
successful. In this paper, we present a new approach for
equality constrained optimization problems based on a trust
region strategy. The direction selected is not necessarily
the solution of the standard quadratic programming
subproblem.},
summary = {As for \citebb{Celi85}.}}
@article{CesaAgreHelgJorgJens91,
author = {A. Cesar and H. Agren and T. Helgaker and P. Jorgensen and
H. J. A. Jensen},
title = {Excited-State Structures and Vibronic Spectra of
{H}$_2${CO}$_+$, {HDCO}$_+$, and {D}$_2${CO}$_+$ Using
Molecular Gradient and {H}essian Techniques},
journal = {Journal of Chemical Physics},
volume = 95, number = 8, pages = {5906--5917}, year = 1991,
abstract = {We choose H$_2$CO$_+$ and its deuterated species to
demonstrate the potential for using second-order
multiconfigurational self-consistent field theory to
optimize structures and calculate properties of ionized and
excited states. We focus on the calculation of
multidimensional vibronic spectra using only the local
information of the potential hypersurface, viz. the
molecular energy, gradient, and Hessian. Second-order
multiconfigurational self-consistent field optimization on
lowest excited states using the trust radius algorithm is
found to give the same stable convergence as for neutral
ground states, while for higher lying states, the problem
of multidimensional potential crossings renders the
calculations more difficult.},
summary = {H$_2$CO$_+$ and its deuterated species are chosen to
demonstrate the potential for using second-order
multiconfigurational self-consistent field theory to
optimize structures and calculate properties of ionized and
excited states. The focus is on calculation of
multidimensional vibronic spectra using only the
molecular energy, gradient, and Hessian of the potential
hypersurface. Second-order multiconfigurational
self-consistent field optimization on lowest excited states
using the trust-region algorithm is found to give the same
stable convergence as for neutral ground states. For higher
lying states, the problem of multidimensional potential
crossings renders the calculations more difficult.}}
@article{ChabCrou84,
author = {Y. Chabrillac and J.-P. Crouzeix},
title = {Definiteness and semidefiniteness of quadratic forms
revisited},
journal = LAA,
volume = 63, pages = {283--292}, year = 1984}
@article{ChamPoweLemaPede82,
author = {R. M. Chamberlain and M. J. D. Powell and C. Lemar\'{e}chal
and H. C. Pedersen},
title = {The watchdog technique for forcing convergence in
algorithms for constrained optimization},
journal = MPS,
volume = 16, number = {MAR}, pages = {1--17}, year = 1982}
% abstract = {Han proves that a line search objective function, which is
% of a form that occurs in many algorithms for constrained
% optimization, can be used to force convergence to a
% Kuhn-Tucker point. We give an example, however, to show
% that this line search objective function can prevent a
% superlinear rate of convergence. If this situation occurs,
% we find that it is advantageous to replace the line search
% objective function by an estimate of the Lagrangian
% function. Therefore a technique is proposed, which chooses
% automatically between Han's line search function and the
% Lagrangian function, in a way that gives superlinear
% convergence. We call it the ``watchdog technique'', because
% the conditions on the step-length of a line search are
% restricted on some iterations by a monitor, in order to
% retain global convergence to a Kuhn-Tucker point from a
% poor initial estimate of the solution.},
% summary = {\citebb{Han77} proved that a line search objective
% function, which is of a form that occurs in many algorithms
% for constrained optimization, can be used to force
% convergence to a Kuhn-Tucker point. An example is given
% which shows that this linesearch objective function can
% prevent a superlinear rate of convergence. If this
% situation occurs, it is advantageous to replace the line
% search objective function by an estimate of the Lagrangian
% function. Therefore a technique is proposed, which chooses
% automatically between Han's linesearch function and the
% Lagrangian function, in a way that gives superlinear
% convergence. This technique is known as the ``watchdog
% technique'', because the conditions on the step-length of a
% linesearch are restricted on some iterations by a monitor,
% in order to retain global convergence to a Kuhn-Tucker
% point from a poor initial estimate of the solution.}}
@article{ChanOlkiCool92,
author = {T. F. Chan and J. A. Olkin and D. W. Cooley},
title = {Solving quadratically constrained least squares using
black box solvers},
journal = BIT,
volume = 32, pages = {481-495}, year = 1992}
@phdthesis{Chan78,
author = {R. Chandra},
title = {Conjugate gradient methods for partial differential
equations},
school = {Yale University}, address = {New Haven, USA},
year = 1978}
@article{ChanCott80,
author = {Y. Y. Chang and R. W. Cottle},
title = {Least-index resolution of degeneracy in quadratic
programming},
journal = MP,
volume = 18, number = 2, pages = {127--137}, year = 1980}
@article{Char78,
author = {C. Charalambous},
title = {A lower bound for the controlling parameter of the exact
penalty functions},
journal = MP,
volume = 15, number = 3, pages = {278--290}, year = 1978}
@article{Char79,
author = {C. Charalambous},
title = {Acceleration of the least $p$-th algorithm for minimax
optimization with engineering applications},
journal = MP,
volume = 17, number = 1, pages = {270--297}, year = 1979}
@article{CharConn75,
author = {A. R. Conn and C. Charalambous},
title = {Optimization of Microwave Networks},
journal = {IEEE Transactions on Microwave Theory and Techniques},
volume = 23, number = 10, pages = {834--838}, year = 1975}
@inproceedings{Chen95,
author = {Z. Chen},
title = {A new trust region algorithm for optimization with
simple bounds},
booktitle = {Operations Research and Its Applications. Proceedings of the
First International Symposium, ISORA '95},
editor = {D. Z. Du and X. S. Zhang and K. Cheng},
publisher = {Beijing World Publishing}, address = {Beijing, China},
pages = {49--58}, year = 1995,
abstract = {We present an algorithm of trust region type for
minimizing a differentiable function of many
variables with simple bounds. Under milder
conditions, we prove the global convergence of the
main algorithm. It is also proved that the correct
active set is identified in a finite number of
iterations with a strict complementarity condition,
and so the proposed algorithm reduces to an
unconstrained minimization method in a finite number
of iterations, allowing a fast asymptotic rate of
convergence.},
summary = {A globally convergent trust-region algorithm is presented for
minimizing a differentiable function of many
variables with simple bounds.It is proved that the correct
active set is identified in a finite number of
iterations under a strict complementarity condition,
allowing a fast asymptotic rate of convergence.}}
@incollection{Chen96,
author = {Z. Chen},
title = {Some algorithms for a class of CDT subproblems},
booktitle = {Lecture Notes in Operations Research},
editor = {D. Du and X. Zhang and W. Wang},
publisher = {Beijing World Publishing}, address = {Beijing, China},
pages = {108--114}, year = 1996}
@inproceedings{ChenDengZhan95,
author = {L. Chen and N. Deng and J. Zhang},
title = {A trust region method with partial-update technique for
unary optimization},
booktitle = {Operations Research and Its Applications. Proceedings of the
First International Symposium, ISORA '95},
editor = {D. Z. Du and X. S. Zhang and K. Cheng},
publisher = {Beijing World Publishing}, address = {Beijing, China},
pages = {40--46}, year = 1995,
abstract = {We propose a modified partial-update algorithm for
solving unconstrained unary optimization problems
based on trust region stabilization via indefinite
dogleg curves. This algorithm only partially updates
an approximation to the Hessian matrix in each
iteration by utilizing limited times of rank-one
updating of Bunch-Parlett factorization. In contrast
with the original algorithms in \citebb{GoldWang93},
the algorithm not only converges globally,
but also possesses a locally quadratic convergence
rate. Furthermore, our numerical experiments show
that the new algorithm outperforms the trust region
method which uses the partial update criteria
suggested in the above paper.},
summary = {A modified partial-update algorithm for
solving unconstrained unary optimization problems
is proposed, based on trust-region stabilization via
indefinite dogleg curves. This algorithm only partially
updates an approximation to the Hessian matrix in each
iteration by applying a limited number of rank-one
updates to its Bunch-Parlett factorization. In contrast
with the original algorithms proposed by \citebb{GoldWang93},
the algorithm not only converges globally,
but also possesses a locally quadratic convergence
rate. Furthermore, numerical experiments show
improved performance.}}
@techreport{ChenChenKanz97,
author = {B. Chen and X. Chen and Ch. Kanzow},
title = {A penalized {F}ischer-{B}urmeister {NCP}-Function:
theoretical Investigation and Numerical Results },
institution = HAMBURG, address = HAMBURG-ADDRESS,
number = {A-126}, year = 1997}
@article{ChenDengZhan98,
author = {L. Chen and N. Deng and J. Zhang},
title = {Modified partial-update {N}ewton-type algorithms for unary
optimization},
journal = JOTA,
volume = 97, number = 2, pages = {385--406}, year = 1998,
abstract = {In this paper, we propose two modified partial-update
algorithms for solving unconstrained unary optimization
problems based on trust-region stabilization via indefinite
dogleg curves. The two algorithms partially update an
approximation to the Hessian matrix in each iteration by
utilizing a number of times the rank-one updating of the
Bunch-Parlett factorization. In contrast with earlier
algorithms, the two algorithms not only converge
globally, but possess also a locally quadratic or
superlinear convergence rate. Furthermore, our numerical
experiments show that the new algorithms outperform the
trust-region method which uses the partial update criteria
suggested in Ref. 1.},
summary = {Two modified partial-update algorithms for solving
unconstrained unary optimization problems based on
trust-region stabilization via indefinite dogleg curves are
proposed. They both partially update an approximation to
the Hessian matrix in each iteration by using
the SR1 updating of the Bunch-Parlett factorization.
They converge globally with a locally quadratic or
superlinear convergence rate. Numerical experiments
indicate that they outperform the trust-region method which
uses some other partial update criteria.}}
@article{ChenHigh98,
author = {S. H. Cheng and N. J. Higham},
title = {A Modified {Cholesky} Algorithm Based on a Symmetric
Indefinite Factorization},
journal = SIMAA,
volume = 19, number = 4, pages = {1097--1110}, year = 1998}
institution = MCCM, address = MCCM-ADDRESS,
type = {Numerical Analysis Report}, number = {No. 289},
year = 1996}
@article{ChenHan96,
author = {Z. W. Chen and J. Y. Han},
title = {A trust region algorithm for optimization with nonlinear
equality and linear inequality constraints},
journal = {Science in China Series A --- Mathematics Physics Astronomy},
volume = 39, number = 8, pages = {799--806}, year = 1996,
abstract = {A new algorithm of trust region type is presented to
minimize a differentiable function of many variables with
nonlinear equality and linear inequality constraints. Under
the milder conditions, the global convergence of the main
algorithm is proved. Moreover, since any nonlinear
inequality constraint can be converted into an equation by
introducing a slack variable, the trust region method can
be used in solving general nonlinear programming problems.},
summary = {A globally convergent trust-region algorithm is presented
to minimize a smooth function of many variables with
nonlinear equality and linear inequality constraints.}}
@article{ChenMang96,
author = {C. Chen and O. L. Mangasarian},
title = {A class of smoothing functions for nonlinear and mixed
complementarity problems},
journal = COAP,
volume = 5, number = 2, pages = {97--138}, year = 1996}
@article{ChenSyku96,
author = {Y. B. Cheng and J. K. Sykulski},
title = {Automated design and optimization of electromechanical
actuators},
journal = {International Journal of Numerical Modelling-Electronic
Networks Devices and Fields},
volume = 9, number = {1--2}, pages = {59--69}, year = 1996,
abstract = {The paper investigates various optmization techniques and
their suitability for the magnetic design of
electromechanical actuators. Selected algorithms, including
Gauss-Newton, \citebb{Leve44}--\citebb{Marq63} and Trust
region, are examined and compared using 18 test functions.
The Levenberg-Marquardt method is chosen for its robustness
and fast convergence, and incorporated into an automated
CAD optimization system (EAMON), which interfaces an
external optimizer to a general purpose finite element
package; The EAMON program, which is user friendly with
pull-down menus, searches for constrained shape design
variables that fulfill prescribed performance criteria. The
electromagnetic field analysis forms part of the
optimization iterative cycle. Finally, two application
examples are described. First, a DC solenoid actuator with
truncated cone pole face is optimized to produce a user
specified force-displacement characteristic. Secondly, an
actuator solenoid is optimized to produce maximum energy
per stroke.},
summary = {The suitability of various optimization techniques for the
magnetic design of electromechanical actuators is examined.
The Gauss-Newton, Levenberg-Morrison-Marquardt and
trust-region algorithms are compared using 18 test
functions. The Levenberg-Morrison-Marquardt method is
chosen for its robustness and fast convergence, and
incorporated into an automated CAD optimization system
(EAMON), which interfaces an external optimizer to a
general purpose finite element package. The user-friendly
EAMON program searches for constrained shape design
variables that fulfill prescribed performance criteria. The
electromagnetic field analysis forms part of the
optimization iterative cycle. As examples, a DC solenoid
actuator with truncated cone pole face is optimized to
produce a user-specified force-displacement characteristic,
and an actuator solenoid is optimized to produce maximum
energy per stroke.}}
@article{ChenYuan99,
author = {X. D. Chen and Y. Yuan},
title = {On local solutions of the CDT subproblem},
journal = SIOPT,
note = {To appear.}, year = 1999,
abstract = {In this paper, we discuss the distribution of the local
solutions of the CDT subproblem which appears in some
trust region algorithms for nonlinear optimization. We
also give some examples to show the differences between the
CDT subproblem and the single-ball constraint subproblem.
These results show that the complexity of the CDT subproblem
does not depend on the complexity of the structure of the
dual plane, thus they provide the possibility to search the
global minimizer in the dual plane.},
summary = {The distribution of the local solutions of the CDT
subproblem which appears in some trust region algorithms for
nonlinear optimization is discussed. Examples illustrate the
differences between the CDT subproblem and the single-ball
constraint subproblem. The complexity of the CDT subproblem
is shown not to depend on the complexity of the structure
of the dual plane, which opens the possibility of searching
the global minimizer in this plane.}}
@techreport{ChinFlet99,
author = {C. M. Chin and R. Fletcher},
title = {Convergence Properties of SLP-filter Algorithms that
use EQP steps},
institution = DUNDEE, address = DUNDEE-ADDRESS,
type = {Numerical Analysis Report},
number = {(in preparation)}, year = 1999}
@inproceedings{ChowChen94,
author = {T. T. Chow and P. K. Chen},
title = {A new trust region global strategy for unconstrained
optimization},
booktitle = {1994 International Computer Symposium Conference Proceedings.
National Chiao Tung University, Hsinchu, Taiwan},
volume = 1, pages = {394--401}, year = 1994,
abstract = {This paper introduces a new global strategy, the
tensor dogleg method, for solving unconstrained
optimization problems, especially using tensor
methods. Tensor methods for unconstrained
optimization were first introduced by R.B. Schnabel
and T.T. Chow (1991). They adopted line search method
and two trust region methods as global strategies,
but these trust region methods were either
inefficient or too complicated. Therefore, the
software package, TENMIN, developed by T.T. Chow et
al. employed only the line search method as the
global strategy for the tensor methods. We tested
several different versions of our tensor dogleg
algorithm. Although the performance of each version
of the algorithm differed slightly, most of them
performed better than TENMIN. The final version of
the tensor dogleg algorithm comprises eleven
states. During our tests we found that the candidate
steps generated by our algorithm were in the tensor
step directions more often than in other
directions. The test results indicate that our tensor
dogleg algorithm performs better than not only the
standard double dogleg algorithm with Newton steps
but also the conventional line search method using
tensor steps.},
summary = {This paper introduces the tensor dogleg method, a
trust-region technique for solving unconstrained
optimization problems, which seems to outperform the
linesearch based TENMIN package.}}
@book{Chva83,
author = {V. Chv\'{a}tal},
title = {Linear Programming},
publisher = FREEMAN, address = FREEMAN-ADDRESS,
year = 1983}
@book{Clar83,
author = {F. H. Clarke},
title = {Optimization and Nonsmooth Analysis},
publisher = WILEY, address = WILEY-ADDRESS,
series = {Canadian Mathematical Society series in mathematics},
year = 1983,
note = {Reprinted as \emph{Classics in Applied Mathematics 5}, SIAM,
Philadelphia, USA, 1990}}
@article{ClerDelaPhamYass91,
author = {J. R. Clermont and M. E. Delalande and Pham Dinh, T. and A.
Yassine},
title = {Analysis of Plane and Axisymmetrical Flows of
Incompressible Fluids with the Steam Tube
Method---Numerical-Simulation by Trust-Region Optimization
Algorithm},
journal = {International Journal for Numerical Methods in Fluids},
volume = 13, number = 3, pages = {371--399}, year = 1991,
abstract = {New concepts for the study of incompressible plane or
axisymmetric flows are analysed by the stream tube method.
Flows without eddies and pure vortex flows are considered
in a transformed domain where the mapped streamlines are
rectilinear or circular. The transformation between the
physical domain and the computational domain is an unknown
of the problem. In order to solve the non-linear set of
relevant equations, we present a new algorithm based on a
trust region technique which is effective for non-convex
optimization problems. Experimental results show that the
new algorithm is more robust compared to the Newton-Raphson
method.},
summary = {Concepts for the study of incompressible plane or
axisymmetric flows are analysed by the stream tube method.
Flows without eddies and pure vortex flows are considered
in a transformed domain where the mapped streamlines are
rectilinear or circular. The transformation between the
physical domain and the computational domain is an unknown
of the problem. A trust-region algorithm is given for
solving the relevant nonlinear set of equations,
Experimental results show that it is more robust than the
Newton-Raphson method.}}
@inproceedings{ClinConnVanL82,
author = {A. K. Cline and A. R. Conn and Van Loan, C. F.},
title = {Generalizing the {LINPACK} condition estimator},
crossref = {Henn82}, pages = {73--83}}
@article{ClinMoleStewWilk79,
author = {A. K. Cline and C. B. Moler and G. W. Stewart and J. H.
Wilkinson},
title = {An estimate for the condition number of a matrix},
journal = SINUM,
volume = 16, pages = {368-375}, year = 1979}
@inproceedings{Cole94,
author = {T. F. Coleman},
title = {Linearly Constrained Optimization and Projected
Preconditioned Conjugate Gradients},
booktitle = {Proceedings of the Fifth SIAM Conference on Applied Linear
Algebra},
editor = {J. Lewis},
publisher = SIAM, address = SIAM-ADDRESS,
pages = {118--122}, year = 1994}
@article{ColeConn80,
author = {T. F. Coleman and A. R. Conn},
title = {Second-order conditions for an exact penalty function},
journal = MP,
volume = 19, number = 2, pages = {178--185}, year = 1980}
@article{ColeConn82,
author = {T. F. Coleman and A. R. Conn},
title = {Non-linear programming via an exact penalty-function:
asymptotic analysis},
journal = MP,
volume = 24, number = 2, pages = {123--136}, year = 1982}
@article{ColeConn82b,
author = {T. F. Coleman and A. R. Conn},
title = {Non-linear programming via an exact penalty-function:
global analysis},
journal = MP,
volume = 24, number = 2, pages = {137--161}, year = 1982}
@article{ColeFeny92,
author = {T. F. Coleman and P. A. Fenyes},
title = {Partitioned quasi-{N}ewton methods for nonlinear equality
constrained optimization},
journal = MP,
volume = 53, number = 1, pages = {17--44}, year = 1992}
@article{ColeHemp90,
author = {T. F. Coleman and C. Hempel},
title = {Computing a trust region step for a penalty function},
journal = SISSC,
volume = 11, number = 1, pages = {180--201}, year = 1990,
abstract = {The problem of minimizing a quadratic function subject to
an ellipsoidal constraint when the matrix involved is the
Hessian of a penalty function (i.e., a function of the form
$p(x)=f(x)+(1/2\mu)c(x)^Tc(x)$) is considered. Most
applications of penalty functions require $p(x)$ to be
minimized for values of $\mu$ decreasing to zero. In
general, as $\mu$ tends to zero the nature of finite
precision arithmetic causes a considerable loss of
information about the null space of the constraint
gradients when $\nabla^2p(x)$ is formed. This loss of
information renders ordinary trust region Newton's method
unstable and degrades the accuracy of the solution to the
trust region problem. The algorithm of \citebb{MoreSore83}
is modified so as to be more stable and less sensitive to
the nature of finite precision arithmetic in this
situation. Numerical experiments clearly demonstrate the
stability of the proposed algorithm.},
summary = {The minimization of a quadratic function subject to
an ellipsoidal constraint is considered in the case when
the matrix involved is the Hessian of a penalty function
$p(x)=f(x)+(1/2\mu)c(x)^Tc(x)$. Most applications
require $p(x)$ to be minimized for values
of $\mu$ decreasing to zero. The algorithm of
\citebb{MoreSore83} is modified so as to be less sensitive
to the nature of finite precision arithmetic in this
situation. Numerical experiments illustrate the stability
of the modified algorithm.}}
@article{ColeHulb89,
author = {T. F. Coleman and L. A. Hulbert},
title = {A direct active set algorithm for large sparse quadratic
programs with simple bounds},
journal = MPB,
volume = 45, number = 3, pages = {373--406}, year = 1989}
@article{ColeHulb93,
author = {T. F. Coleman and L. A. Hulbert},
title = {A Globally and Superlinearly Convergent Algorithm for
Convex Quadratic Programs with Simple Bounds},
journal = SIOPT,
volume = 3, number = 2, pages = {298--321}, year = 1993}
abstract = {We present a globally and superlinearly convergent
algorithm for solving convex quadratic programs with simple
bounds. We develop our algorithm using a new formulation of
the problem: the minimization of an unconstrained piecewise
quadratic function that has the same optimality conditions
as the original problem. The major work at each iteration
is the Cholesky factorization of a positive definite matrix
with the size and structure of the Hessian of the
quadratic. Hence our algorithm is suitable for solving
large-scale problems and for implementation on parallel
computers. We implemented our algorithm and tested it on a
sequential computer on a variety of dense problems, and we
present numerical results which show that our algorithm
solves many problems quickly.},
summary = {A globally and superlinearly convergent algorithm for
solving convex quadratic programs with simple bounds is
given. It uses the minimization
of an unconstrained piecewise quadratic function that has
the same optimality conditions as the original problem.
The major work at each iteration is the Cholesky
factorization
of a positive definite matrix with the size and structure of
the Hessian of the quadratic. Hence the algorithm is suitable
for solving large-scale problems and for implementation on
parallel computers. An implementation of the algorithm is
tested on a sequential computer using a variety of dense
problems. Numerical results are presented.}}
@article{ColeLi94,
author = {T. F. Coleman and Y. Li},
title = {On the convergence of interior-reflective {N}ewton methods
for nonlinear minimization subject to bounds},
journal = MP,
volume = 67, number = 2, pages = {189--224}, year = 1994}
@article{ColeLi96,
author = {T. F. Coleman and Y. Li},
title = {A Reflective {N}ewton Method for Minimizing a Quadratic
Function Subject to Bounds on Some of the Variables},
journal = SIOPT,
volume = 6, number = 4, pages = {1040--1058}, year = 1996,
abstract = {We propose a new algorithm, a reflective Newton method, for
the minimization of a quadratic function of many variables
subject to upper and lower bounds on some of the variables.
The method applies to a general (indefinite) quadratic
function for which a local minimum subject to bounds is
required and is particularly suitable for the large-scale
problem. Our new method exhibits strong convergence
properties and global and second-order convergence and
appears to have significant practical potential. Strictly
feasible points are generated. We provide experimental
results on moderately large and sparse problems based on
both sparse Cholesky and preconditioned conjugate gradient
linear solvers.},
summary = {A reflective Newton method is proposed for the minimization
of a quadratic function of many variables subject to upper
and lower bounds on some of the variables. It applies to a
general (indefinite) quadratic function for which a local
minimum subject to bounds is required and is particularly
suitable for the large-scale problem. The method is
globally and asymptotically quadratically convergent.
generates strictly feasible points. Experimental results
are presented for moderately large and sparse problems
based on both sparse Cholesky and preconditioned
conjugate-gradient linear solvers.}}
@article{ColeLi96b,
author = {T. F. Coleman and Y. Li},
title = {An Interior Trust Region Approach for Nonlinear
Minimization Subject to Bounds},
journal = SIOPT,
volume = 6, number = 2, pages = {418--445}, year = 1996,
abstract = {We propose a new trust region approach for minimizing a
nonlinear function subject to simple bounds. Unlike most
existing methods, our proposed method does not require that
a quadratic programming subproblem, with inequality
constraints, be solved each iteration. Instead, a solution
to a trust region subproblem is defined by minimizing a
quadratic function subject only to an ellipsoidal
constraint. The iterates generated are strictly feasible.
Our proposed method reduces to a standard trust region
approach for the unconstrained problem when there are no
upper or lower bounds on the variables. Global and local
quadratic convergence is established. Preliminary numerical
experiments are reported indicating the practical viability
of this approach.},
summary = {A trust-region approach for minimizing a nonlinear function
subject to simple bounds is proposed, that does not require
that a quadratic programming subproblem with inequality
constraints be solved every iteration. Instead, a solution
to a trust-region subproblem is sought. The iterates
generated are strictly feasible. The proposed method reduces
to a standard trust-region approach for the unconstrained
problem. Global and locally quadratic convergence is
established. Preliminary numerical experiments are reported.}}
@techreport{ColeLi97b,
author = {T. F. Coleman and Y. Li},
title = {A trust region and affine scaling interior point method for
nonconvex minimization with linear inequality constraints},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR 97-1642}, year = 1997,
abstract = {A trust region and affine scaling interior point method
(TRAM) is proposed for a general nonlinear minimization
with linear inequality constraints by \citebb{ColeLi98}. In
the proposed approach, a Newton step is derived from the
complementarity conditions. Based on this Newton step, a
trust region subproblem is formed, and the original
objective function is monotonically decreased. Explicit
sufficient decrease conditions are proposed for satisfying
complementarity, dual feasibility and second order
optimality. The objective of this paper is to establish
global and local convergence properties of the proposed
trust region and affine scaling interior point method. It
is shown that the proposed decrease conditions are
sufficient for achieving complementarity, dual feasibility
and second order optimality respectively. It is also
established that a trust region solution is asymptotically
in the interior of the proposed trust region subproblem and
a damped trust region step can achieve quadratic
convergence.},
summary = {Global and local convergence properties of the trust-region
and affine-scaling interior-point method (TRAM) by
\citebb{ColeLi98} are established. It is shown that a
trust-region solution is asymptotically in the interior of
the trust region subproblem and a damped trust-region step
can achieve quadratic convergence.}}
@inproceedings{ColeLi98,
author = {T. F. Coleman and Y. Li},
title = {Combining trust region and affine scaling for linearly
constrained nonconvex minimization},
crossref = {Yuan98}, pages = {219--250},
abstract = {An interior point method is proposed for a general
nonlinear (non-convex) minimization with linear inequality
constraints. This method is a combination of the trust
region idea for nonlinearity and affine scaling technique
for constraints. Using this method, the original objective
function is monotonically decreased. In the proposed
approach, a Newton step is derived directly from the
complementarity conditions. A trust region subproblem is
formed which yields an approximate Newton step as its
solution asymptotically. The objective function of the
trust region subproblem is the quadratic approximation to
the original objective function plus an augmented quadratic
convex term. Similar to an augmented Lagrangian function,
this augmentation adds positive curvature in the range
space of the constraint normals. The global convergence is
achieved by possibly using trust regions with different
shapes. A reflection technique, which accelerates
convergence, is described. Explicit sufficient decrease
conditions are proposed. Computational results of a
two-dimensional trust region implementation are reported
for large-scale problems. Preliminary experiments suggest
that this method can be effective; a relatively small
number of function evaluations are required for some medium
and large test problems.},
summary = {An interior-point method is proposed for
non-convex minimization with linear inequality
constraints. It combines the trust-region idea for
nonlinearity and affine-scaling technique for constraints,
and ensures that the original objective function is
monotonically decreased. A subproblem is formed which
asymptotically yields an approximate Newton step, directly
derived from the complementarity conditions. Global
convergence is achieved by possibly using trust regions
with different shapes. A reflection technique accelerates
convergence. Explicit sufficient decrease conditions are
proposed. Computational results of a two-dimensional
implementation are reported for large-scale problems.}}
@misc{ColeLi98b,
author = {T. F. Coleman and Y. Li},
title = {A primal-dual Trust Region Algorithm for Nonconvex
Programming using a $\ell_1$ Penalty Function},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {A primal and dual algorithm is proposed for nonconvex
programming. Primal and dual steps are derived directly
from the complementarity conditions. A primal step is used
to yield decrease for the $\ell_1$ penalty function. In
addition, a dual step yields decrease for an appropriate
function of dual variables. Reflection procedures are used
to accelerate convergence and preliminary computational
results are reported.},
summary = {A primal-dual algorithm is proposed for non-convex
programming. Primal and dual steps are derived directly
from the complementarity conditions. A primal trust-region
step is used to yield decrease for the $\ell_1$ penalty
function, and a dual constrained least-squares step yields
decrease for an appropriate function of dual variables.
Reflection procedures are used to accelerate convergence
and preliminary computational results are reported.}}
@article{ColeLiao95,
author = {T. F. Coleman and A. Liao},
title = {An efficient trust region method for unconstrained
discrete-time optimal control problem},
journal = COAP,
volume = 4, number = 1, pages = {47--66}, year = 1995,
abstract = {Discrete-time optimal control (DTOC) problems are
large-scale optimization problems with a dynamic structure.
In previous work this structure has been exploited to
provide very fast and efficient local procedures. Two
examples are the differential dynamic programming algorithm
(DDP) and the stagewise Newton procedure---both require
only $O(N)$ operations, where $N$ is the number of
timesteps. Both exhibit a quadratic convergence rate.
However, most algorithms in this category do not have a
satisfactory global convergence strategy. The most popular
global strategy is shifting: this sometimes works poorly
due to the lack of automatic adjustment to the shifting
element. In this paper we propose a method that
incorporates the trust region idea with the local stagewise
Newton's method. This method possesses advantages of both
the trust region idea and the stagewise Newton's method,
i.e., our proposed method has strong global and local
convergence properties yet remains economical. Preliminary
numerical results are presented to illustrate the behavior
of the proposed algorithm. We also collect in the Appendix
some DTOC problems that have appeared in the literature.},
summary = {A method is proposed that incorporates the trust-region
idea with the local stagewise Newton's method for
discrete-time optimal control (DTOC) problems. This method
has strong global and local convergence properties yet
remains economical. Preliminary numerical results
illustrate the behaviour of the algorithm. Some DTOC
problems that have appeared in the literature are collected
in appendix.}}
@article{ColeLiu99,
author = {T. F. Coleman and J. Liu},
title = {An interior {N}ewton method for quadratic programming},
journal = MPA,
volume = 85, number = 3, pages = {491--524}, year = 1999,
abstract = {We propose a new (interior) approach for the general
quadratic programming problem. We establish that the
new method has strong convergence properties: the
generated sequence converges globally to a point
satisfying the second-order necessary optimality
conditions, and the rate of convergence is 2-step
quadratic if the limit point is a strong minimizer.
Published alternative interior approaches do not share
such strong convergence properties for the nonconvex case.
We also report on the results of preliminary numerical
experiments: the results indicate that the proposed method
has considerable practical potential.},
summary = {An interior point method is proposed for the general
quadratic programming problem. The method converges
globally to a point satisfying the second-order necessary
optimality conditions, and the rate of convergence is
2-step quadratic if the limit point is a strong minimizer.
Preliminary numerical experiments indicate that the method
has practical potential.}}
@article{ColeMore83,
author = {T. F. Coleman and J. J. Mor\'{e}},
title = {Estimation of sparse {J}acobian matrices and graph coloring
problems},
journal = SINUM,
volume = 20, pages = {187--209}, year = 1983}
@article{ColeMore84,
author = {T. F. Coleman and J. J. Mor\'{e}},
title = {Estimation of sparse {H}essian matrices and graph coloring
problems},
journal = MP,
volume = 28, pages = {243--270}, year = 1984}
@article{ColePlas92,
author = {T. F. Coleman and P. E. Plassman},
title = {A Parallel Nonlinear Least-Squares Solver--Theoretical
Analysis and Numerical Results},
journal = SISSC,
volume = 13, number = 3, pages = {771--793}, year = 1992,
abstract = {The authors (1989) proposed a parallel algorithm,
based on the sequential Levenberg-Marquardt method
for the nonlinear least-squares problem. The
algorithm is suitable for message-passing
multiprocessor computers. A parallel efficiency
analysis is provided and computational results are
reported. The experiments were performed on an Intel
iPSC/2 multiprocessor with 32 nodes: the paper
presents experimental results comparing the given
parallel algorithm with sequential MINPACK code
executed on a single processor. These experimental
results show that essentially full efficiency is
obtained for problems where the row size is
sufficiently larger than the number of processors.},
summary = {The paper presents experimental results comparing a
parallel version of the Levenberg-Morrison-Marquardt
algorithm on an Intel iPSC/2 multiprocessor with 32
nodes with sequential MINPACK code executed on a single
processor. These experimental results show that essentially
full efficiency is obtained for problems where the row
size is sufficiently larger than the number of processors.}}
@article{ColePoth86a,
author = {T. F. Coleman and A. Pothen},
title = {The Null Space Problem {I}: complexity},
journal = SIADM,
volume = 7, number = 4, pages = {527-537}, year = 1986}
@article{ColePoth87,
author = {T. F. Coleman and A. Pothen},
title = {The Null Space Problem {II}: algorithms},
journal = SIADM,
volume = 8, number = 4, pages = {544-563}, year = 1987}
@article{ColeSore84,
author = {T. F. Coleman and D. C. Sorensen},
title = {A note on the computation of an orthonormal basis for the
null space of a matrix},
journal = MP,
volume = 29, number = 2, pages = {234--242}, year = 1984}
@techreport{ColeYuan95,
author = {T. F. Coleman and W. Yuan},
title = {A New Trust Region Algorithm for Equality Constrained
Optimization},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR95-1477}, year = 1995,
abstract = {We present a new trust region algorithm for solving
nonlinear equality constrained optimization problems. At
each iterate a change of variables is performed to improve
the ability of the algorithm to follow the constraint level
sets. The algorithm employs $L_2$ penalty functions for
obtaining global convergence. Under certain assumptions we
prove that this algorithm globally converges to a point
satisfying the second order necessary conditions; the local
convergence rate is quadratic. Results of preliminary
numerical experiments are presented.},
summary = {A trust-region algorithm for solving nonlinear equality
constrained problems is presented. At each
iterate a change of variables improves the
ability of the algorithm to follow the constraint level
sets. The algorithm employs quadratic penalty functions to
obtain global convergence. It converges globally and
Q-quadratically to a point satisfying second-order
necessary optimality conditions. Preliminary numerical
experiments are presented.}}
@book{Coll66,
author = {L. Collatz},
title = {Functional Analysis and Numerical Mathematics},
publisher = AP, address = AP-ADDRESS,
year = 1966}
@mastersthesis{Cols99,
author = {B. Colson},
title = {Mathematical Programs with Equilibrium Constraints and
Nonlinear Bilevel Programming Problems},
school = FUNDP, address = FUNDP-ADDRESS,
year = 1999}
@inproceedings{ConcGoluOLea76,
author = {P. Concus and G. H. Golub and D. P. O'Leary},
title = {Numerical Solution of Nonlinear Elliptic Partial
Differential Equations by a Generalized Conjugate Gradient
Method},
booktitle = {Sparse Matrix Computations},
editor = {J. Bunch and D. Rose},
publisher = AP, address = AP-ADDRESS,
pages = {309--332}, year = 1976}
@article{Conn73,
author = {A. R. Conn},
title = {Constrained optimization via a nondifferentiable penalty
function},
journal = SINUM,
volume = 10, number = 4, pages = {760--779}, year = 1973}
@inproceedings{ConnCoulHariMoriVisw96,
author = {A. R. Conn and P. K. Coulman and R. A. Haring and G. L.
Morrill and C. Visweswariah},
title = {Optimization of custom {MOS} circuits by transistor sizing},
booktitle = {IEEE/ACM International Conference on Computer-Aided Design.
Digest of Technical Papers (Cat. No.96CB35991)},
publisher = {IEEE},
address = {IEEE Comput. Soc. Press, Los Alamitos, USA},
pages = {p.174--180}, year = 1996,
abstract = {Optimization of a circuit by transistor sizing is often a
slow, tedious and iterative manual process which relies on
designer intuition. Circuit simulation is carried out in
the inner loop of this tuning procedure. Automating the
transistor sizing process is an important step towards
being able to rapidly design high-performance, custom
circuits. JiffyTune is a new circuit optimization tool that
automates the tuning task. Delay, rise/fall time, area and
power targets are accommodated. Each (weighted) target can
be either a constraint or an objective function. Minimax
optimization is supported. Transistors can be ratioed and
similar structures grouped to ensure regular layouts.
Bounds on transistor widths are supported. JiffyTune uses
{\sf LANCELOT}, a large-scale nonlinear optimization
package with an augmented Lagrangian formulation. Simple
bounds are handled explicitly and trust region methods are
applied to minimize a composite objective function. In the
inner loop of the optimization, the fast circuit simulator
SPECS is used to evaluate the circuit. SPECS is unique in
its ability to efficiently provide time-domain
sensitivities, thereby enabling gradient-based
optimization. Both the adjoint and direct methods of
sensitivity computation have been implemented in SPECS. To
assist the user, interfaces in the Cadence and SLED design
systems have been constructed.},
summary = {JiffyTune is a circuit optimization tool that automates the
tuning task. Delay, rise/fall time, area and power targets
are accommodated. Each (weighted) target can be either a
constraint or an objective function. Minimax optimization
is supported. Transistors can be ratioed and similar
structures grouped to ensure regular layouts. Bounds on
transistor widths are supported. JiffyTune uses {\sf
LANCELOT}. In the inner loop of the optimization, the fast
circuit simulator SPECS is used to evaluate the circuit.
SPECS is unique in its ability to provide time-domain
sensitivities, thereby enabling gradient-based
optimization. Both the adjoint and direct methods of
sensitivity computation have been implemented in SPECS.
Interfaces in the Cadence and SLED design systems have been
constructed.}}
@inproceedings{ConnGoulLescToin94,
author = {A. R. Conn and N. I. M. Gould and M. Lescrenier and Ph. L.
Toint},
title = {Performance of a multifrontal scheme for partially
separable optimization},
crossref = {GomeHenn94}, pages = {79--96}}
%abstract = {We consider the solution of partially separable
% minimization problems subject to simple bounds
% constraints. At each iteration, a quadratic model is used
% to approximate the objective function within a trust
% region. To minimize this model, the iterative method of
% conjugate gradients has usually been used. The aim of this
% paper is to compare the performance of a direct method, a
% multifrontal scheme, with the conjugate gradient method
% (with and without preconditioning). To assess our
% conclusions, a set of numerical experiments, including
% large-dimensional problems, is presented.}
@article{ConnGoulToin88a,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Global convergence of a class of trust region algorithms
for optimization with simple bounds},
journal = SINUM,
volume = 25, number = 182, pages = {433--460}, year = 1988,
note = {See also same journal 26:764--767, 1989.},
abstract = {This paper extends the known excellent global convergence
properties of trust-region algorithms for unconstrained
optimization to the case where bounds on the variables are
present. Weak conditions on the accuracy of the Hessian
approximations are considered. It is also shown that, when
the strict complementarity condition holds, the proposed
algorithms reduce to an unconstrained calculation after
finitely many iterations, allowing a fast rate of
convergence.},
summary = {The global convergence properties of trust-region
algorithms for unconstrained optimization are extended to
the case where bounds on the variables are present. Weak
conditions on the accuracy of the Hessian approximations
are considered. When the strict complementarity condition
holds, the proposed algorithms reduce to an unconstrained
calculation after finitely many iterations, allowing fast
convergence.}}
@article{ConnGoulToin88b,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Testing a class of methods for solving minimization
problems with simple bounds on the variables},
journal = MC,
volume = 50, pages = {399--430}, year = 1988,
abstract = {We describe the results of a series of tests upon a class
of new methods of trust region type for solving the simple
bound constrained minimization problem. The results are
encouraging and lead us to believe that the method will
prove useful in solving large problems.},
summary = {The results of tests on the trust-region methods proposed by
\citebb{ConnGoulToin88a} for solving the bound constrained
minimization problem are discussed.}}
@inproceedings{ConnGoulToin89b,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {An introduction to the structure of large scale nonlinear
optimization problems and the {{\sf LANCELOT}} project},
booktitle = {Computing Methods in Applied Sciences and Engineering},
editor = {R. Glowinski and A. Lichnewsky},
publisher = SIAM, address = SIAM-ADDRESS,
pages = {42--51}, year = 1990}
%abstract = {This paper presents the authors' personal views on two
% fundamental aspects amongst the recent developments in the
% growing field of large-scale nonlinear mathematical
% programming. Important concepts for the description of
% problem structure are discussed in detail. A systematic
% approach to software development for this class of
% problems is also presented. The approach incorporates both
% suitable numerical algorithms and user oriented standard
% format for problem specification in a modular and coherent
% system.}
@article{ConnGoulToin91,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {A Globally Convergent Augmented {L}agrangian Algorithm for
Optimization with General Constraints and Simple Bounds},
journal = SINUM,
volume = 28, number = 2, pages = {545--572}, year = 1991}
@article{ConnGoulToin91a,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Convergence of quasi-{N}ewton matrices generated by the
Symmetric Rank One update},
journal = MP,
volume = 50, number = 2, pages = {177--196}, year = 1991}
@inproceedings{ConnGoulToin91e,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {On the number of inner iterations per outer iteration of a
globally convergent algorithm for optimization with general
nonlinear equality constraints and simple bounds},
crossref = {GrifWats92}, pages = {49--68}}
@book{ConnGoulToin92,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {{\sf LANCELOT}: a {F}ortran package for Large-scale
Nonlinear Optimization ({R}elease {A})},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
series = {Springer Series in Computational Mathematics}, year = 1992}
@techreport{ConnGoulToin92g,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Intensive numerical tests with {{\sf LANCELOT}} ({R}elease
{A}): the complete results},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {92/15}, year = 1992,
note = {Also issued as Research Report RC 18750, IBM T.J. Watson
Center, Yorktown Heights, USA, and as Research Report
92-069, RAL, Chilton, Oxfordshire, England},
abstract = {This report contains the detailed results of the numerical
experiments on the {\sf LANCELOT} package for nonlinear
optimization (Release A). These results constitute the
basis of the discussion and analysis presented by the
authors in \citebb{ConnGoulToin96a}.},
summary = {The detailed results for the tests reported by
\citebb{ConnGoulToin96a} are presented.}}
@article{ConnGoulToin94a,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {A note on using alternative second-order models for the
subproblems arising in barrier function methods for
minimization},
journal = NUMMATH,
volume = 68, pages = {17--33}, year = 1994}
@article{ConnGoulToin96a,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Numerical experiments with the {{\sf LANCELOT}} package
({R}elease {A}) for large-scale nonlinear optimization},
journal = MPA,
volume = 73, number = 1, pages = {73--110}, year = 1996,
abstract = {In this paper, we describe the algorithmic options of
Release A of {\sf LANCELOT}, a Fortran package for
large-scale nonlinear optimization. We then present the
results of intensive numerical tests and discuss the
relative merits of the options. The experiments described
involve both academic and applied problems. Finally, we
propose conclusions, both specific to {\sf LANCELOT} and of
more general scope.},
summary = {The algorithmic options available within Release A of {\sf
LANCELOT}, a Fortran package for large-scale nonlinear
optimization, are presented. The results of intensive
numerical tests are described, and the relative merits of
the options discussed. The experiments described involve
both academic and applied problems. Conclusions specific to
{\sf LANCELOT} and of more general scope are made.}}
@inproceedings{ConnGoulToin97,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {Methods for Nonlinear Constraints in Optimization
Calculations},
crossref = {DuffWats97}, pages = {363--390}}
%abstract = {Ten years ago, the broad consensus among researchers in
% constrained optimization was that sequential quadratic
% programming (SQP) methods were the methods of choice.
% While, in the long term, this position may be justified,
% the past ten years have exposed a number of difficulties
% with the SQP approach. Moreover, alternative methods have
% shown themselves capable of solving large-scale problems.
% In this paper, we shall outline the defects with SQP
% methods, and discuss the alternatives. In particular, we
% shall indicate how our understanding of the subproblems
% which inevitably arise in constrained optimization
% calculations has improved. We shall also consider the
% impact of interior-point methods for inequality
% constrained problems, described elsewhere in this volume,
% and argue that these methods likely provide a more useful
% Newton model for such problems than do traditional SQP
% methods. Finally, we shall consider trust-region methods
% for constrained problems, and the impact of automatic
% differentiation on algorithm design. }
@article{ConnGoulToin97a,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {A globally convergent {L}agrangian barrier algorithm for
optimization with general inequality constraints and simple
bounds},
journal = MC,
volume = 66, pages = {261--288}, year = 1997}
@article{ConnGoulToin97z,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {On the number of inner iterations per outer iteration of a
globally convergent algorithm for optimization with general
nonlinear inequality constraints and simple bounds},
journal = COAP,
volume = 7, number = 1, pages = {41--70}, year = 1997}
@inproceedings{ConnGoulToin99,
author = {A. R. Conn and N. I. M. Gould and Ph. L. Toint},
title = {A Primal-Dual Algorithm for Minimizing a Nonconvex Function
Subject to Bound and Linear Equality Constraints},
crossref = {DiPiGian99}, pages = {15--30}}
@article{ConnGoulSartToin93,
author = {A. R. Conn and N. I. M. Gould and A. Sartenaer and Ph. L.
Toint},
title = {Global convergence of a class of trust region algorithms
for optimization using inexact projections on convex
constraints},
journal = SIOPT,
volume = 3, number = 1, pages = {164--221}, year = 1993,
abstract = {A class of trust region based algorithms is presented for
the solution of nonlinear optimization problems with a
convex feasible set. At variance with previously published
analysis of this type, the theory presented allows for the
use of general norms. Furthermore, the proposed algorithms
do not require the explicit computation of the projected
gradient, and can therefore be adapted to cases where the
projection onto the feasible domain may be expensive to
calculate. Strong global convergence results are derived
for the class. It is also shown that the set of linear and
nonlinear constraints that are binding at the solution are
identified by the algorithms of the class in a finite
number of iterations.},
summary = {Trust-region algorithms for the solution
of nonlinear optimization problems with a convex feasible
set are presented. The theory given allows for the use of
general norms. Furthermore, the proposed algorithms do not
require the explicit computation of the projected gradient,
and can therefore be adapted to cases where the projection
onto the feasible domain may be expensive to calculate.
Strong global convergence results are derived. The set of
linear and nonlinear constraints that are binding at the
solution are identified by the algorithms in a finite
number of iterations.}}
@article{ConnGoulSartToin96,
author = {A. R. Conn and N. I. M. Gould and A. Sartenaer and Ph. L.
Toint},
title = {Convergence properties of an Augmented {L}agrangian
Algorithm for Optimization with a Combination of General
Equality and Linear Constraints},
journal = SIOPT,
volume = 6, number = 3, pages = {674--703}, year = 1996}
@article{ConnGoulSartToin96a,
author = {A. R. Conn and N. I. M. Gould and A. Sartenaer and Ph. L.
Toint},
title = {Convergence properties of minimization algorithms for
convex constraints using a structured trust region},
journal = SIOPT,
volume = 6, number = 4, pages = {1059--1086}, year = 1996,
abstract = {In this paper, we present a class of trust region
algorithms for minimization problems within convex feasible
regions in which the structure of the problem is explicitly
used in the definition of the trust region. This
development is intended to reflect the possibility that
some parts of the problem may be more accurately modelled
than others, a common occurrence in large-scale nonlinear
applications. After describing the structured trust region
mechanism, we prove global convergence for all algorithms
in our class.},
summary = {A class of structured trust-region algorithms is presented
for minimization problems within convex feasible regions,
in which the structure of the problem is explicitly used in
the definition of the trust region. Global convergence is
established.}}
@techreport{ConnGoulOrbaToin99,
author = {A. R. Conn and N. I. M. Gould and D. Orban and Ph. L. Toint},
title = {A primal-dual trust-region algorithm for minimizing a
non-convex function subject to bound and linear equality
constraints},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {TR99-04}, year = 1999,
abstract = {A new primal-dual algorithm is proposed for the
minimization of non-convex objective functions subject to
simple bounds and linear equality constraints. The method
uses a primal-dual trust-region model to ensure descent on
a suitable merit function. Convergence of a well-defined
subsequence of iterates is proved to a second-order
critical point from arbitrary starting points. Algorithmic
variants are discussed and preliminary numerical results
presented.},
summary = {A primal-dual algorithm is proposed for the minimization of
non-convex objective functions subject to simple bounds and
linear equality constraints. The method uses a primal-dual
trust-region model to ensure descent on a suitable merit
function. Convergence of a well-defined subsequence of
iterates is proved to a second-order critical point from
arbitrary starting points. Algorithmic variants are
discussed and preliminary numerical results presented.}}
@article{ConnLi92,
author = {A. R. Conn and Y. Li},
title = {A structure-exploiting algorithm for nonlinear minimax
problems},
journal = SIOPT,
volume = 2, number = 2, pages = {242-263}, year = 1992}
@techreport{ConnSinc75,
author = {A. R. Conn and J. W. Sinclair},
title = {Quadratic programming via a non-differentiable
penalty function},
institution = {Faculty of Mathematics, University of Waterloo},
number = {CORR 75/15}, year = 1975}
@inproceedings{ConnToin96,
author = {A. R. Conn and Ph. L. Toint},
title = {An algorithm using quadratic interpolation for
unconstrained derivative free optimization},
crossref = {DiPiGian96}, pages = {27--47},
abstract = {This paper explores the use of multivariate interpolation
techniques in the context of method for unconstrained
optimization that do not require derivatives of the
objective function. A new algorithm is proposed that uses
quadratic models in a trust region framework. The algorithm
is constructed to require few evaluations of the objective
function and to be relatively insensitive to noise in the
objective function values. Its performance is analyzed on a
set of 20 examples, both with and without noise.},
summary = {The use of multivariate interpolation techniques is
explored in the context of method for unconstrained
optimization that do not require derivatives. An algorithm
is proposed that uses quadratic models in a trust-region
framework. It requires few evaluations of
the objective function and is relatively
insensitive to noise in the objective function values. Its
performance is analyzed on a set of 20 examples, both with
and without noise.}}
@article{ConnPiet77,
author = {A. R. Conn and T. Pietrzykowski},
title = {A penalty function method converging directly to a
constrained optimum},
journal = SINUM,
volume = 14, number = 2, pages = {348--375}, year = 1977}
@inproceedings{ConnScheToin97,
author = {A. R. Conn and K. Scheinberg and Ph. L. Toint},
title = {On the convergence of derivative-free methods for
unconstrained optimization},
booktitle = {Approximation Theory and Optimization: Tributes to M. J. D.
Powell},
editor = {A. Iserles and M. Buhmann},
publisher = CUP, address = CUP-ADDRESS,
pages = {83--108}, year = 1997,
abstract = {The purpose of this paper is to examine a broad class of
derivative-free trust-region methods for unconstrained
optimization inspired by the proposals of \citebb{Powe94b}
and to derive a general framework in which reasonable
global convergence results can be obtained. The
developments make extensive use of an interpolation error
bound derived by \citebb{SaueXu95} in the context of
multivariate polynomial interpolation.},
summary = {Derivative-free trust-region methods for
unconstrained optimization, inspired by \citebb{Powe94b},
are discussed and global convergence results obtained. The
developments make extensive use of an interpolation error
bound derived by \citebb{SaueXu95} in the context of
multivariate polynomial interpolation.}}
@article{ConnScheToin97b,
author = {A. R. Conn and K. Scheinberg and Ph. L. Toint},
title = {Recent progress in unconstrained nonlinear optimization
without derivatives},
journal = MPB,
volume = 79, number = 3, pages = {397--414}, year = 1997,
abstract = {We present an introduction to a new class of derivative
free methods for unconstrained optimization. We start by
discussing the motivation for such methods and why they are
in high demand by practitioners. We then review the past
developments in this field, before introducing the features
that characterize the newer algorithms. In the context of a
trust region framework, we focus on techniques that ensure
a suitable ``geometric quality'' of the considered models.
We then outline the class of algorithms based on these
techniques, as well as their respective merits. We finally
conclude the paper with a discussion of open questions and
perspectives.},
summary = {Derivative-free trust-region methods for unconstrained
optimization are introduced. Motivation is given, and past
developments in the field reviewed.
Techniques that ensure a suitable ``geometric quality'' of
the models are considered. A discussion of open
questions and perspectives is given.}}
@techreport{ConnScheToin98,
author = {A. R. Conn and K. Scheinberg and Ph. L. Toint},
title = {A Derivative Free Optimization Algorithm in Practice},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {TR98/11}, year = 1998,
abstract = {We consider an algorithm for optimizing, at first without
constraints, a nonlinear function whose first-order
derivatives exist but are unavailable. The algorithm is
based on approximating the objective function by a
quadratic polynomial interpolation model and using this
model within a trust-region framework. We study some
practical properties of the algorithm and show how it can
be extended to solve certain constrained optimization
problems. We present computational results for analytical
and for real-life problems from the aeronautical industry.},
summary = {An algorithm is presented for optimizing, at first without
constraints, a nonlinear function whose first-order
derivatives exist but are unavailable. It is based on
approximating the objective function by a quadratic
polynomial interpolation model and using this model within
a trust-region framework. Some practical properties of the
algorithm are studied, including how it can be extended to
solve certain constrained optimization problems.
Computational results are presented for analytical and for
real-life problems from the aeronautical industry.}}
@article{ConnViceVisw99,
author = {A. R. Conn and L. N. Vicente and C. Visweswariah},
title = {Two-step algorithms for nonlinear optimization with
structured applications},
journal = SIOPT,
volume = 9, number = 4, pages = {924--947}, year = 1999,
abstract = {In this paper we propose extensions to trust-region and
line-search algorithms in which the classical step is
augmented with a second step that is required to yield a
decrease in the value of the objective function. The
classical convergence theory for trust-region and
line-search algorithms is adapted to this class of two-step
algorithms. It is shown that the algorithms are globally
convergent to a stationary point under the ``classical''
assumptions. The algorithms can be applied to any problem
with variable(s) whose contribution to the objective
function is a known functional form. In the nonlinear
programming package {\sf LANCELOT}, they have been applied
to update slack variables and variables introduced to solve
minimax problems, leading to enhanced optimization
efficiency. Numerical results are presented to show the
effectiveness of these techniques.},
summary = {Extensions to trust-region and line-search algorithms are
proposed in which the classical step is augmented with a
second step that yields a decrease of the
objective function. The convergence theory for
trust-region and linesearch algorithms is adapted to this
class of two-step algorithms. It is shown that the
algorithms are globally convergent. The
algorithms can be applied to any problem with variable(s)
whose contribution to the objective function is a known
functional form. In the nonlinear programming package {\sf
LANCELOT}, they have been applied to update slack variables
and variables introduced to solve minimax problems, leading
to enhanced optimization efficiency. Numerical results are
presented.}}
@article{ContTapi93,
author = {M. Contreras and R. A. Tapia},
title = {Sizing the {BFGS} and {DFP} updates: numerical study},
journal = JOTA,
volume = 78, number = 1, pages = {93--108}, year = 1993,
abstract = {The authors develop and test a strategy for
selectively sizing (multiplying by an appropriate
scalar) the approximate Hessian matrix before it is
updated in the BFGS and DFP trust-region methods for
unconstrained optimization. The numerical results
imply that, for use with the DFP update, the
Oren-Luenberger sizing factor is completely
satisfactory and selective sizing is vastly superior
to the alternatives of never sizing or
first-iteration sizing and is slightly better than
the alternative of always sizing. Numerical
experimentation showed that the Oren-Luenberger
sizing factor is not a satisfactory sizing factor for
use with the BFGS update. Therefore, based on the
newly acquired understanding of the situation, the
authors propose a centered Oren-Luenberger sizing
factor to be used with the BFGS update. The numerical
experimentation implies that selectively sizing the
BFGS update with the centered Oren-Luenberger sizing
factor is superior to the alternatives. These results
contradict the folk axiom that sizing should be done
only at the first iteration. They also show that,
without sufficient sizing, DFP is vastly inferior to
BFGS and however, when selectively sized, DFP is
competitive with BFGS.},
summary = {A strategy is developed for selectively sizing the
approximate Hessian matrix before it is
updated in the BFGS and DFP trust-region methods for
unconstrained optimization. The numerical results
suggest that sizing should be done not
only at the first iteration. They also show that,
without sufficient sizing, DFP is vastly inferior to
BFGS and however, when selectively sized, DFP is
competitive with BFGS.}}
@article{Cont80,
author = {L. B. Contesse},
title = {Une caract\'{e}risation compl\`{e}te des minima locaux en
programmation quadratique},
journal = NUMMATH,
volume = 34, pages = {315--332}, year = 1980}
@article{Corr97,
author = {G. Corradi},
title = {A trust region algorithm for unconstrained optimization},
journal = {International Journal of Computer Mathematics},
volume = 65, number = {1-2}, pages = {109--119}, year = 1997,
abstract = {In this paper a trust region method, based on approximation
of $f(\cdot)$ and $f'(\cdot)$ of higher order, is
presented. A convergence analysis for the method is
considered too. Numerical results are reported.},
summary = {A trust region method is proposed which is based on
approximation of $f(\cdot)$ and $f'(\cdot)$ of higher
order. A convergence analysis for the method is also
presented. Numerical results are reported.}}
@inproceedings{CostStanSnym96,
author = {J. E. Coster and N. Stander and J. A. Snyman},
title = {Trust region augmented {L}agrangian methods with secant
{H}essian updating applied to structural optimization},
booktitle = {Proceedings of the ASME Design Engineering Technical
Conference and Computers in Engineering Conference, August
18-22, 1996, Irvine, California},
year = 1996,
abstract = {The problem of determining the optimal sizing design of
truss structures is considered. An augmented Lagrangian
optimization algorithm which uses a quadratic penalty term
is formulated. The implementation uses a first-order
Lagrange multiplier update and a strategy for progressively
increasing the accuracy with which the bound constrained
minimizations are performed. The allowed constraint
violation is also progressively decreased but at a slower
rate so as to prevent ill-conditioning due to large penalty
values. Individual constraint penalties are used and only
the penalties of the worst violated constraints are
increased. The scheme is globally convergent. The bound
constrained minimizations are performed using the {\em
SBMIN} algorithm where a sophisticated trust-region
strategy is employed. The Hessian of the augmented
Lagrangian function is approximated using partitioned
secant updating. Each function contributing to the
Lagrangian is individually approximated by a secant update
and the augmented Lagrangian Hessian is formed by
appropriate accumulation. The performance of the algorithm
is evaluated for a number of different secant updates on
standard explicit and truss sizing optimization problems.
The results show the formulation to be superior to other
implementations of augmented Lagrangian methods reported in
the literature and that, under certain conditions, the
method approaches the performance of the state-of-the-art
SQP and SAM methods. Of the secant updates, the symmetric
rank one update is superior to the other updates including
the BFGS scheme. It is suggested that the individual
function, secant updating employed may be usefully applied
in contexts where structural analysis and optimization are
performed simultaneously, as in the simultaneous analysis
and design method. In such cases the functions are
partially separable and the associated Hessians are of low
rank.},
summary = {An augmented Lagrangian algorithm is formulated for the
optimal sizing design of
truss structures. The scheme is globally convergent. The
bound constrained minimizations are performed using the
{\sf SBMIN} trust-region algorithm. The Hessian of the
augmented Lagrangian is approximated using partitioned
secant updating. The performance of the algorithm is
evaluated for different secant updates on standard explicit
and truss sizing optimization problems. The results show
the formulation to be superior to other implementations of
augmented Lagrangian methods and that the method may
approach the performance of the state-of-the-art SQP and
SAM methods. Of the secant updates, the SR1 update is
superior to the other updates including the BFGS scheme.
Secant updating may be usefully applied in contexts where
structural analysis and optimization are performed
simultaneously, as in the simultaneous analysis and design
method. In such cases the functions are partially
separable.}}
@book{CottPangSton92,
author = {R. W. Cottle and J.-S. Pang and R. E. Stone},
title = {The Linear Complementarity Problem},
publisher = AP, address = AP-ADDRESS,
year = 1992}
@article{Cour43,
author = {R. Courant},
title = {Variational methods for the solution of problems of
equilibrium and vibrations},
journal = {Bulletin of the American Mathematical Society},
volume = 49, pages = {1--23}, year = 1943}
@manual{CPLEX98,
author = {{CPLEX 6.0}},
title = {High-performance linear, integer and quadratic programming
software},
organization = {ILOG SA}, address = {Gentilly, France},
year = 1998}
@book{Crye82,
author = {C. W. Cryer},
title = {Numerical Functional Analysis},
publisher = OUP, address = OUP-ADDRESS,
year = 1982}
@article{CullWill80,
author = {J. Cullum and R. A. Willoughby},
title = {The {L}anczos phenomenon---an interpretation based upon
conjugate gradient optimization},
journal = LAA,
volume = 29, pages = {63--90}, year = 1980}
% abstract = {The equivalence in exact arithmetic of the Lanczos
% tridiagonalization procedure and the conjugate gradient
% optimization procedure for solving Ax=b, where A is a real
% symmetric, positive definite matrix, is well known.
% We demonstrate that a relaxed equivalence is valid in the
% presence of errors. Specifically we demonstrate that local
% $\epsilon$-orthonormality of the Lanczos vectors guarantees
% local $\epsilon$-A-conjugacy of the direction vectors in the
% associated conjugate gradient procedure. Moreover we
% demonstrate that all the conjugate gradient relationships
% are satisfied approximately. Therefore, any statements valid
% for the conjugate gradient optimization procedure, which we
% show converges under very weak conditions, apply directly to
% the Lanczos procedures. We then use this equivalence to
% obtain an explanation of the Lanczos phenomenon: the
% empirically observed "convergence" of Lanczos eigenvalue
% procedures despite total loss of the global orthogonality of
% the Lanczos vectors.},
% summary = {It is demonstrated that there is a one sided
% $\epsilon$-equivalence relationship between the practical
% Lanczos tridiagonalizarion procedure and the conjugate
% gradient optimization procedure for solving Ax=b. This
% equivalence allows the exploitation of properties of the
% optimization procedure to provide a plausible mechanism for
% explaining the observed Lanczos phenomenon. The arguments
% required an assumption on the variation in the ratios of the
% norms of the residuals that seems to occur in practice,
% although no proof that it occurs in general was presented.
% This equivalence was also used to provide, for positive
% definite matrices, a convergence argument for the SYMMLQ
% Lanczos procedure for solving systems of equations developed
% in \citebb{PaigSaun75} and for the related procedure for
% computing elements of A-1 used in \citebb{KaplGray76}}
@article{CurtPoweReid74,
author = {A. R. Curtis and M. J. D. Powell and J. K. Reid},
title = {On The Estimation of Sparse {J}acobian Matrices},
journal = JIMA,
volume = 13, pages = {117--119}, year = 1974}
%%% D %%%
@article{Dafe80,
author = {S. Dafermos},
title = {Traffic equilibrium and variational inequalities},
journal = {Transportation Science},
volume = 14, pages = {42--54}, year = 1980}
@article{Dani67a,
author = {J. W. Daniel},
title = {The conjugate gradient method for linear and nonlinear
operator equations},
journal = SINUM,
volume = 4, pages = {10--25}, year = 1967}
@article{Dani67b,
author = {J. W. Daniel},
title = {Convergence of the conjugate gradient method with
computationally convenient modifications},
journal = NUMMATH,
volume = 10, pages = {125--131}, year = 1967}
@book{Dant63,
author = {G. B. Dantzig},
title = {Linear Programming and Extensions},
publisher = {Princeton Uiversity Press}, address = {Princeton, USA},
year = 1963}
@techreport{Das96,
author = {I. Das},
title = {An interior point algorithm for the general nonlinear
programming problem with trust region globalization},
institution = ICASE, address = ICASE-ADDRESS,
number = {96--61}, year = 1996,
abstract = {This paper presents an SQP-based interior point technique
for solving the general nonlinear programming problem using
trust region globalization and the \citebb{ColeLi96b}
scaling. The SQP subproblem is decomposed into a normal and
a reduced tangential subproblem in the tradition of
numerous works on equality constrained optimization, and
strict feasibility is maintained with respect to the
bounds. This is intended to be an extension of previous
work by \citebb{ColeLi96b} and \citebb{Vice96}. Though no
theoretical proofs of convergence are provided, some
computational results are presented which indicate that
this algorithm holds promise. The computational experiments
have been geared towards improving the semi-local
convergence of the algorithm; in particular high
sensitivity of the speed of convergence with respect to the
fraction of the trust region radius allowed for the normal
step and with respect to the initial trust region radius
are observed. The chief advantages of this algorithm over
primal-dual interior point algorithms are better handling
of the `sticking problem' and a reduction in the number of
variables by elimination of the multipliers of bound
constraints.},
summary = {An SQP-based interior-point technique is presented for
solving the general nonlinear programming problem using
trust-region globalization and the \citebb{ColeLi96b}
scaling. The SQP subproblem is decomposed into a normal
and a reduced tangential subproblem, and strict feasibility
is maintained with respect to the bounds. Computational
experiments have been geared towards improving the
semi-local convergence of the algorithm; in particular high
sensitivity of the speed of convergence with respect to the
fraction of the trust region radius allowed for the normal
step and with respect to the initial trust region radius
are observed. The chief advantages of this algorithm over
primal-dual interior-point algorithms are better handling
of the `sticking problem' and a reduction in the number of
variables by elimination of the multipliers of bound
constraints.}}
@article{Davi68,
author = {W. C. Davidon},
title = {Variance algorithms for minimization},
journal = COMPJ,
volume = 10, pages = {406--410}, year = 1968}
@article{Davi75,
author = {W. C. Davidon},
title = {Optimally Conditioned Optimization Algorithms Without Line
Searches},
journal = MP,
volume = 9, number = 1, pages = {1--30}, year = 1975}
@article{Dean92,
author = {E. J. Dean},
title = {A Model Trust-Region Modification of {N}ewton Method for
Nonlinear 2-Point Boundary-Value-Problems},
journal = {Journal of Optimization Theory and Applications},
volume = 75, number = 2, pages = {297--312}, year = 1992,
abstract = {The method of quasilinearization for nonlinear two-point
boundary-value problems is Newton's method for a nonlinear
differential operator equation. A model trust-region
approach to globalizing the quasilinearization algorithm is
presented. A double-dogleg implementation yields a globally
convergent algorithm that is robust in solving difficult
problems.},
summary = {The method of quasilinearization for nonlinear two-point
boundary- value problems is Newton's method for a nonlinear
differential operator equation. A model trust-region
approach to globalizing the quasilinearization algorithm is
presented. A double-dogleg implementation yields a globally
convergent algorithm that is robust in solving difficult
problems.}}
@article{DeBoRon92,
author = {De Boor, C. and A. Ron},
title = {Computational aspects of polynomial interpolation in
several variables},
journal = MC,
volume = 58, number = 198, pages = {705--727}, year = 1992}
@techreport{DeLuFaccKanz95,
author = {De Luca, T. and F. Facchinei and Ch. Kanzow},
title = {A semismooth approach to the solution of nonlinear
complementarity problems},
institution = HAMBURG, address = HAMBURG-ADDRESS,
number = 93, year = 1995}
@article{DembEiseStei82,
author = {R. S. Dembo and S. C. Eisenstat and T. Steihaug},
title = {Inexact-{N}ewton Methods},
journal = SINUM,
volume = 19, number = 2, pages = {400--408}, year = 1982}
@article{DembStei83,
author = {R. S. Dembo and T. Steihaug},
title = {Truncated-{N}ewton Algorithms for Large-Scale Unconstrained
Optimization},
journal = MP,
volume = 26, pages = {190--212}, year = 1983}
@techreport{DembTulo83,
author = {R. S. Dembo and U. Tulowitzki},
title = {On the minimization of quadratic functions subject to box
constraints},
institution = {Yale University}, address = {Yale, USA},
type = {School of Organization and Management Working paper },
number = {Series B no. 71}, year = 1983}
@book{DemyMalo74,
author = {V. F. Dem'yanov and V. N. Malozemov},
title = {Introduction to Minmax},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1974}
@article{DengXiaoZhou93,
author = {N. Deng and Y. Xiao and F. Zhou},
title = {Nonmonotonic Trust Region Algorithms},
journal = JOTA,
volume = 76, number = 2, pages = {259--285}, year = 1993,
abstract = {A non-monotonic trust region method for unconstrained
optimization problems is presented. Although the method
allows the sequence of values of the objective function to
be non-monotonic, convergence properties similar to those
for the usual trust region method are proved under certain
conditions, including conditions on the approximate
solutions to the subproblem. To make the solution satisfy
these conditions, an algorithm to solve the subproblem is
also established. Finally, some numerical results are
reported which show that the non-monotonic trust region
method is superior to the usual trust region method
according to both the number of gradient evaluations and
the number of function evaluations.},
summary = {A non-monotonic trust-region method for unconstrained
optimization is presented, whose convergence
properties are similar to those for the usual trust-region
method under conditions including requirements
on the approximate solutions to the subproblem. An
algorithm to solve the subproblem is also presented and
numerical results discussed.}}
@inproceedings{Denn78,
author = {J. E. Dennis},
title = {A brief introduction to quasi-{N}ewton methods},
booktitle = {Numerical Analysis},
editor = {G. H. Golub and J. Oliger},
publisher = AMS, address = AMS-ADDRESS,
series = {Proceedings of Symposia in Applied Mathematics},
number = 22, pages = {19--52}, year = 1978}
@article{DennElAlMaci97,
author = {J. E. Dennis and M. El{-}Alem and M. C. Maciel},
title = {A global convergence theory for general Trust-Region based
algorithms for equality constrained optimization},
journal = SIOPT,
volume = 7, number = 1, pages = {177--207}, year = 1997,
abstract = {A class of algorithms based on the successive quadratic
programming method for solving the general nonlinear
programming problem is presented. The objective function
and the constraints of the problem are only required to be
differentiable and their gradients to satisfy a Lipschitz
condition. The strategy for obtaining global convergence is
based on the trust-region approach. The merit function is
an augmented Lagrangian. A new updating technique is
introduced for the penalty parameter, by means of which
monotone increase is not necessary. Global convergence
results are proved and numerical experiments are presented.},
summary = {A class of algorithms based on the SQP method for general
nonlinear programming is presented. The objective function
and the constraints of the problem are only required to be
differentiable and their gradients to satisfy a Lipschitz
condition. Global convergence is obtained by using a
trust-region approach. The merit function is an augmented
Lagrangian. A possibly non-monotone updating technique is
introduced for the penalty parameter. Global convergence
results are proved and numerical experiments presented.}}
@article{DennElAlWill99,
author = {J. E. Dennis and M. El{-}Alem and K. A. Williamson},
title = {A trust-region approach to nonlinear systems of equalities
and inequalities},
journal = SIOPT,
volume = 9, number = 2, pages = {291--315}, year = 1999,
abstract = {In this paper, two new trust-region algorithms for the
numerical solution of systems of nonlinear equalities and
inequalities are introduced. The formulation is free of
arbitrary parameters and possesses sufficient smoothness to
exploit the robustness of the trust-region approach. The
proposed algorithms are one-sided least-squares
trust-region algorithms. The first algorithm is a single
model algorithm, and the second one is a multi-model
algorithm where the Cauchy point computation is a model
selection procedure. Global convergence analysis for the
two algorithms are presented. Our analysis generalizes to
nonlinear systems of equalities and inequalities the
well-developed theory for nonlinear least-squares problems.
Numerical experiments on the two algorithms are also
presented. The performance of the two algorithm are
reported. The numerical results validate the effectiveness
of our approach.},
summary = {Two one-sided trust-region algorithms for the numerical
solution of systems of nonlinear equalities and
inequalities are introduced. The first is a single model
algorithm, while the second uses multiple models with
the Cauchy point computation being used a model selection
procedure. Global convergence analysis is presented for
both algorithms, and numerical experiments show their
effectiveness.}}
@article{DennEcheGuarMartScolVacc91,
author = {J. E. Dennis and N. Echebest and M. T. Guardarucci and
J. M. Mart\'{\i}nez and H. D. Scolnik and C. Vacchino},
title = {A Curvilinear Search using Tridiagonal Secant Updates for
Unconstrained Optimization},
journal = SIOPT,
volume = 1, number = 3, pages = {333--357}, year = 1991}
@article{DennGayWels81,
author = {J. E. Dennis and D. M. Gay and R. E. Welsh},
title = {An adaptive nonlinear least squares algorithm},
journal = TOMS,
volume = 7, number = 3, pages = {348--368}, year = 1981}
abstract = {NL2SOL is a modular program for solving nonlinear
least-squares problems that incorporates a number of
novel features. It maintains a secant approximation S
to the second-order part of the least-squares Hessian
and adaptively decides when to use this
approximation. S is 'sized' before updating,
something that is similar to Oren-Luenberger
scaling. The step choice algorithm is based on
minimizing a local quadratic model of the sum of
squares function constrained to an elliptical trust
region centered at the current approximate
minimizer. This is accomplished using ideas discussed
by More (1978) together with a special module for
assessing the quality of the step thus
computed. These and other ideas behind NL2SOL are
discussed, and its evolution and current
implementation are also described briefly.},
summary = {NL2SOL is a modular program for solving nonlinear
least-squares problems. It maintains a sized
secant approximation to the second-order part of
the least-squares Hessian and adaptively decides when
to use this approximation. The step choice algorithm
is based on minimizing a local quadratic model of the
sum of squares function constrained to an elliptical trust
region.}}
@article{DennHeinVice98,
author = {J. E. Dennis and M. Heinkenschloss and L. N. Vicente},
title = {Trust-region interior-point {SQP} algorithms for a class of
nonlinear programming problems},
journal = SICON,
volume = 36, number = 5, pages = {1750--1794}, year = 1998,
abstract = {In this paper, a family of trust-region interior-point
sequential quadratic programming (SQP) algorithms for the
solution of a class of minimization problems with nonlinear
equality constraints and simple bounds on some of the
variables is described and analyzed. Such nonlinear
programs arise, e.g., from the discretization of optimal
control problems. The algorithms treat states and controls
as independent variables. They are designed to take
advantage of the structure of the problem. In particular
they do not rely on matrix factorizations of the linearized
constraints but use solutions of the linearized state
equation and the adjoint equation. They are well suited for
large scale problems arising from optimal control problems
governed by partial differential equations. The algorithms
keep strict feasibility with respect to the bound
constraints by using an affine scaling method proposed, for
a different class of problems, by \citebb{ColeLi96b} and
they exploit trust-region techniques for
equality-constrained optimization. Thus, they allow the
computation of the steps using a variety of methods,
including many iterative techniques. Global convergence of
these algorithms to a first-order Karush- Kuhn-Tucker (KKT)
limit point is proved under very mild conditions on the
trial steps. Under reasonable, but more stringent,
conditions on the quadratic model and on the trial steps,
the sequence of iterates generated by the algorithms is
shown to have a limit point satisfying the second-order
necessary KKT conditions. The local rate of convergence to
a nondegenerate strict local minimizer is q-quadratic. The
results given here include, as special cases, current
results for only equality constraints and for only simple
bounds. Numerical results for the solution of an optimal
control problem governed by a nonlinear heat equation are
reported.},
summary = {Trust-region interior-point SQP algorithms are presented
for solving minimization problems with nonlinear equality
constraints and simple bounds. The algorithms treat states
and controls as independent variables and take advantage of
the structure of the problem. In particular they do not rely
on matrix factorizations of the linearized constraints but
use solutions of the linearized state equation and the
adjoint equation. They are suited for large-scale problems
arising from optimal control problems governed by partial
differential equations. They keep strict feasibility with
respect to the bound constraints by using an affine-scaling
method inspired by \citebb{ColeLi96b} and they exploit
trust-region techniques for equality-constrained
optimization. They allow the computation of the steps using
a variety of methods, including many iterative techniques.
Global convergence is proved under very mild conditions on
the trial steps. Under more stringent
conditions on the quadratic model and on the trial steps,
the iterates converge Q-quadratically to a
limit point satisfying the second-order necessary
conditions. Numerical results are reported for an optimal
control problem governed by a nonlinear heat equation.}}
@article{DennLiTapi95,
author = {J. E. Dennis and S. B. B. Li and R. A. Tapia},
title = {A unified approach to global convergence of trust region
methods for nonsmooth optimization},
journal = MP,
volume = 68, number = 3, pages = {319--346}, year = 1995,
abstract = {This paper investigates the global convergence of
trust-region (TR) mthods for solving nonsmooth minimization
problems. For a class of nonsmooth objective functions
called regular functions, conditions are found on the TR
local models that imply three fundamental convergence
properties. These conditions are shown to be satisfied by
appropriate forms of \citebb{Flet87}'s TR method for
solving constrained optimization problems, \citebb{Powe83}
and \citebb{Yuan83}'s TR method for solving nonlinear
fitting problems, \citebb{DuffNoceReid87}'s TR method for
solving systems of nonlinear equations, and
\citebb{ElHaTapi93}'s TR method for solving systems of
nonlinear equations. Thus our results can be viewed as a
unified convergence theory for TR methods for nonsmooth
problems.},
summary = {The global convergence of trust-region methods for
non-smooth minimization is investigated. Conditions are
found on the local models that imply three convergence
properties for regular problems. These conditions are
satisfied by appropriate forms of \citebb{Flet87}'s method
for constrained optimization, \citebb{Powe83} and
\citebb{Yuan83}'s method for solving nonlinear fitting
problems, and \citebb{DuffNoceReid87}'s and
\citebb{ElHaTapi93}'s methods for solving systems of
nonlinear equations. The results may thus be viewed as a
unified convergence theory for trust-region methods for
non-smooth problems.}}
@article{DennMei79,
author = {J. E. Dennis and H. H. W. Mei},
title = {Two New Unconstrained Optimization Algorithms Which Use
Function and Gradient Values},
journal = JOTA,
volume = 28, number = 4, pages = {453--482}, year = 1979,
abstract = {Two new methods for unconstrained optimization are
presented. Both methods employ a hybrid direction strategy
which is a modification of \citebb{Powe70c}'s dogleg
strategy. They also employ a projection technique
introduced by \citebb{Davi75} which uses projection images
of $\Delta x$ and $\Delta g$ in updating the approximate
Hessian. The first method uses Davidon's optimally
conditioned update formula, while the second uses only the
BFGS update. Both methods performed well without Powell's
special iterations and singularity safeguards, and the
numerical results are very promising.},
summary = {Two methods for unconstrained optimization are presented.
They employ a hybrid direction strategy, which is a
modification of \citebb{Powe70c}'s dogleg strategy, and a
projection technique introduced by \citebb{Davi75} which
uses projection images of $\Delta x$ and $\Delta g$ in
updating the approximate Hessian. The first method uses
Davidon's optimally conditioned update formula, while the
second uses only the BFGS update. Both methods performed
well without Powell's special iterations and singularity
safeguards.}}
@book{DennSchn83,
author = {J. E. Dennis and R. B. Schnabel},
title = {Numerical Methods for Unconstrained Optimization and
Nonlinear Equations},
publisher = PH, address = PH-ADDRESS,
year = 1983,
note = {Reprinted as \emph{Classics in Applied Mathematics 16}, SIAM,
Philadelphia, USA, 1996}}
@article{DennTorc91,
author = {J. E. Dennis and V. Torczon},
title = {Direct Search Methods on Parallel Machines},
journal = SIOPT,
volume = 1, number = 4, pages = {448--474}, year = 1991}
@inproceedings{DennTorc97,
author = {J. E. Dennis and V. Torczon},
title = {Managing Approximation Models in Optimization},
booktitle = {Multidisciplinary Design Optimization},
editor = {N. M. Alexandrov and M. Y. Hussaini},
publisher = SIAM, address = SIAM-ADDRESS,
pages = {330--347}, year = 1997,
abstract = {It is standard engineering practice to use approximation
models in place of expensive simulations to drive an
optimal design process based on nonlinear programming
algorithms. This paper uses well-established notions from
the literature on trust-region methods and a powerful
global convergence theory for pattern search methods to
manage the interplay between optimization and the fidelity
of the approximation models to insure that the process
converges to a reasonable solution of the original design
problem. We present a specific example from the class of
algorithms outlined here, but many other interesting
options exist that we will explore in later work. The
algorithm we present as an example of the management
strategies we propose is based on a family of pattern
search algorithms developed by the authors. Pattern search
methods can be successfully applied when only ranking
(ordinal) information is available and when derivatives are
either unavailable or unreliable. Since we are interested
here in using approximations to provide arguments for the
objective function, our choice seems relevant. This work is
in support of the Rice effort in collaboration with Boeing
and IBM to look at the problem of designing helicopter
rotor blades.},
summary = {It is standard engineering practice to use approximation
models in place of expensive simulations to drive an
optimal design process based on nonlinear programming
algorithms. Known notions on trust-region methods and a
global convergence theory for pattern search methods are
used to manage the interplay between optimization and the
fidelity of the approximation models to insure that the
process converges to a reasonable solution of the original
problem. The algorithm given as an example is based on the
family of pattern search algorithms by \citebb{DennTorc91},
which can be successfully applied when only ranking (ordinal)
information is available and when derivatives are either
unavailable or unreliable.}}
@inproceedings{DennVice96,
author = {J. E. Dennis and L. N. Vicente},
title = {Trust Region Interior-Point Algorithms for minimization
problems with simple bounds},
booktitle = {Applied Mathematics and Parallel Computing,
Festschrift for Klaus Ritter},
editor = {H. Fisher and B. Riedm\u{u}ller and S. Sch\u{a}ffler},
publisher = {Physica-Verlag, Springer-Verlag},
address = SPRINGER-ADDRESS,
pages = {97--107}, year = 1996,
abstract = {Two trust-region interior-point algorithms for the solution
of minimization problems with simple bounds are presented.
The algorithms scale the local model in a way proposed by
\citebb{ColeLi96b}, but they are new otherwise. The first
algorithm is more usual in that the trust region and the
local quadratic model are consistently scaled. The second
algorithm proposed here uses an unscaled trust region. A
first-order convergence result for these algorithms is
given and dogleg and conjugate-gradient algorithms to
compute trial steps are introduced. Some numerical examples
that show the advantages of the second algorithm are
presented.},
summary = {Two trust-region interior-point algorithms for the solution
of minimization problems with simple bounds are presented,
that scale the local model as proposed by
\citebb{ColeLi96b}. In the first, the
trust region and the local quadratic model are consistently
scaled. The second uses an unscaled trust region. A
first-order convergence result is
given and dogleg and conjugate-gradient algorithms to
compute trial steps introduced. Numerical examples
illustrate the advantages of the second algorithm.}}
%also
%institution = CAAM, address = RICE-ADDRESS,
%number = {TR94-42}, year = 1994,
@article{DennVice97,
author = {J. E. Dennis and L. N. Vicente},
title = {On the Convergence Theory of Trust-Region Based Algorithms
for Equality-Constrained Optimization},
journal = SIOPT,
volume = 7, number = 4, pages = {927--950}, year = 1997,
abstract = {In a recent paper, \citebb{DennElAlMaci97} developed a
global first-order convergence theory for a general
trust-region-based algorithm for equality-constrained
optimization. This general algorithm is based on
appropriate choices of trust-region subproblems and seems
particularly suitable for large problems. This paper
presents the global second-order convergence theory for the
same general trust-region-based algorithm. The results
given here can be seen as a generalization of the
second-order convergence results for trust-region methods
for unconstrained optimization obtained by
\citebb{MoreSore83}. The behavior of the trust region
radius and the local rate of convergence are analyzed. Some
interesting facts concerning the trust-region subproblem
for the linearized constraints, the quasi-normal component
of the step and the hard case are presented. It is shown
how these results can be applied to a class of discretized
optimal control problems.},
summary = {A global second-order convergence theory is given
for the algorithm of \citebb{DennElAlMaci97}, that
generalizes second-order
convergence results of \citebb{MoreSore83}. The behaviour
of the trust-region radius and the local rate of
convergence are analyzed. Some results concerning
the trust-region subproblem for the linearized constraints,
the quasi-normal component of the step and the hard case
are presented. It is shown how these results can be applied
to some discretized optimal control problems.}}
%also
%institution = CAAM, address = RICE-ADDRESS,
%number = {TR94-36}, year = 1994,
@inproceedings{DennWill88,
author = {J. E. Dennis and K. A. Williamson},
title = {A new parallel optimization algorithm for parameter
identification in ordinary differential equations},
booktitle = {Proceedings of the 27th IEEE Conference on Decision and
Control},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 3, pages = {1836--1840}, year = 1988,
abstract = {Standard approaches to the solution of the parameter
identification problem in systems of ordinary
differential equations are reviewed. An algorithm
that is based on the \citebb{CeliDennTapi85}
trust region algorithm for
equality-constrained optimization problems is
described. This algorithm should be both more
efficient and more stable than standard solution
techniques, and it also provides a flexible framework
for introducing parallelism into the parameter
identification problem.},
summary = {A variant of the \citebb{CeliDennTapi85} trust-region
algorithm for equality-constrained optimization problems is
described in the context of parameter identification
in ordinary differential equations}}
@article{deSaYuanSun97,
author = {de Sampaio, R. J. B. and J. Yuan and W. Sun},
title = {Trust region algorithm for nonsmooth optimization},
journal = {Applied Mathematics and Computation},
volume = 85, number = {2-3}, pages = {109--116}, year = 1997,
abstract = {Minimization of a composite function $h(f(x))$ is
considered here, where $f: \Re^n \leftarrow \Re^m$ is a
locally Lipschitzian function, and $h: \Re^m \leftarrow
\Re$ is a continuously differentiable convex function.
Theory of trust region algorithm for nonsmooth optimization
given by \citebb{Flet87}, and \citebb{PoweYuan90} is
extended to this case. Trust region algorithm and its
global convergence are studied. Finally, some applications
on nonlinear and nonsmooth least squares problems are also
given.},
summary = {Minimization of a composite function $h(f(x))$ is
considered, where $f:\Re^n \leftarrow \Re^m$ is a locally
Lipschitzian function, and $h: \Re^m \leftarrow \Re$ is a
continuously differentiable convex function. The theory of
trust region algorithms for non-smooth optimization given
by \citebb{Flet87}, and \citebb{PoweYuan90} is extended to
this case. A trust-region algorithm and its global
convergence are studied. Some applications to nonlinear and
non-smooth least-squares problems are given.}}
@inproceedings{DeScDeMo97,
author = {De Schutter, B. and De Moor, B.},
title = {The Extended Linear Complementarity Problem and Its
Applications in the Max-Plus Algebra},
crossref = {FerrPang97}, pages = {22--39}}
@article{DeufPotr92,
author = {P. Deuflhard and F. A. Potra},
title = {Asymptotic Mesh Independence for {N}ewton--{G}alerkin
Methods via a Refined {M}ysovskii Theorem},
journal = SINUM,
volume = 29, number = 5, pages = {1395--1412}, year = 1992}
@article{DiSun96,
author = {S. Di and W. Sun},
title = {A trust region method for conic model to solve
unconstrained optimization},
journal = OMS,
volume = 6, number = 4, pages = {237--263}, year = 1996,
abstract = {A trust region method for conic models to solve
unconstrained optimization problems is proposed. We analyze
the trust region approach for conic models and present
necessary and sufficient conditions for the solution of the
associated trust region subproblems. A corresponding
numerical algorithm is developed and has been tested for 19
standard test functions in unconstrained optimization. The
numerical results show that this method is superior to some
advanced methods in the current software libraries.
Finally, we prove that the proposed method has global
convergence and Q-superlinear convergence properties.},
summary = {A trust-region method using conic models is proposed for
solving unconstrained optimization problems. Necessary and
sufficient conditions for the solution of the associated
subproblems are given. The method is globally
and Q-superlinearly convergent. Numerical experiments are
reported.}}
@techreport{DiniGomeSant98,
author = {M. A. Diniz{-}Ehrhardt and M. A. Gomes{-}Ruggiero and S. A.
Santos},
title = {Numerical analysis of leaving-face parameters in
bound-constrained quadratic minimization},
institution = UNICAMP, address = UNICAMP-ADDRESS,
number = {52/98}, year = 1998,
abstract = {In this work, we focus our attention on the quadratic
subproblem of trust-region algorithms for large-scale
bound-constrained minimization. An approach that combines a
mild active set strategy with gradient projection
techniques is employed in the solution of large-scale
bound-constrained quadratic problems. To fill in some gaps
that have appeared in previous work, we propose, test and
analyze heuristics which dynamically choose the parameters
in charge of the decision of leaving or not the current
face of the feasible set. The numerical analysis is based
on problems from CUTE collection and randomly generated
convex problems with controlled conditioning and
degeneracy. The practical consequences of an appropriate
decision of such parameters are shown to be crucial,
particularly when dual degenerate and ill-conditioned
problems are solved.},
summary = {The problem of leaving the current face of the feasible
domain in a combined active-set and gradient-projection
method for large-scale bound-constrained quadratic problems
is considered. Heuristics which dynamically choose the
parameters in charge of the decision are proposed, tested
and analyzed. The practical consequences of an appropriate
choice of such parameters are crucial for
dual-degenerate and ill-conditioned problems.}}
@article{DiniGomeSant98b,
author = {M. A. Diniz{-}Ehrhardt and M. A. Gomes{-}Ruggiero and S. A.
Santos},
title = {Comparing the numerical performance of two trust-region
algorithms for large-scale bound-constrained minimization},
journal = {Investigaci\'{o}n Operativa},
note = {To appear.}, year = 1998,
abstract = {In this work we compare the numerical performance of the
software BOX-QUACAN with the package LANCELOT. We put
BOX-QUACAN in a context by means of solving an extensive
set of problems, so that specific features of both
approaches are compared. Through the computational tests,
conclusions are made about the classes of problems for
which each algorithm suits better and ideas for future
research are devised.},
summary = {The numerical performance of the BOX-QUACAN and {\sf
LANCELOT} software packages are compared on an extensive set
of problems. Conclusions are drawn about the classes of
problems for which each package performs better.}}
@article{DiPiFaccGrip92,
author = {Di Pillo, G. and F. Facchinei and L. Grippo},
title = {An {RQP} algorithm using a differentiable exact penalty
function for inequality constrained problems},
journal = MP,
volume = 55, number = 1, pages = {49--68}, year = 1992}
@article{DipiGrip85,
author = {Di Pillo, G. and L. Grippo},
title = {A continuously differentiable exact penalty-function method
for nonlinear programming with inequality constraints},
journal = SICON,
volume = 23, number = 1, pages = {72--84}, year = 1985}
@article{DipiGrip86,
author = {Di Pillo, G. and L. Grippo},
title = {An exact penalty-function method with global convergence
properties for nonlinear programming problems},
journal = MP,
volume = 36, number = 1, pages = {1--18}, year = 1986}
@article{Diki67,
author = {I. I. Dikin},
title = {Iterative solution of problems of linear and quadratic
programming},
journal = {Doklady Akademiia Nauk USSR},
volume = 174, pages = {747--748}, year = 1967}
@book{DikiZork80,
author = {I. I. Dikin and V. I. Zorkaltsev},
title = {Iterative Solutions of Mathematical Programming Problems},
publisher = {Nauka}, address = {Novosibirsk},
year = 1980}
@article{DirkFerr95,
author = {S. P. Dirkse and M. C. Ferris},
title = {The {PATH} Solver: a Non-Monotone Stabilization Scheme
for Mixed Complementarity Problems},
journal = OMS,
volume = 5, number = 2, pages = {123--156}, year = 1995}
@phdthesis{Djan79,
author = {A. Djang},
title = {Algorithmic equivalence in quadratic programming},
school = STANFORD, address = STANFORD-ADDRESS,
year = 1979}
@book{DongBuncMoleStew79,
author = {J. J. Dongarra and J. R. Bunch and C. B. Moler and G. W.
Stewart},
title = {{LINPACK} Users's guide},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1979}
@book{DongDuffSoreVors98,
author = {J. J. Dongarra and I. S. Duff and D. C. Sorensen and van
der Vorst, H. A.},
title = {Numerical Linear Algebra for High-Performance Computers},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1998}
@article{Dost97,
author = {Z. Dost\'{a}l},
title = {Box Constrained Quadratic Programming with Proportioning
and Projections },
journal = SIOPT,
volume = 7, number = 3, pages = {871--887}, year = 1997}
@article{DrusGreeKniz98,
author = {V. Druskin and A. Greenbaum and L. Knizherman},
title = {Using nonorthogonal {L}anczos vectors in the computation of
matrix functions},
journal = SISC,
volume = 19, number = 1, pages = {38--54}, year = 1998}
@book{DuffErisReid86,
author = {I. S. Duff and A. M. Erisman and J. K. Reid},
title = {Direct Methods for Sparse Matrices},
publisher = OUP, address = OUP-ADDRESS,
year = 1986}
@article{DuffNoceReid87,
author = {I. S. Duff and J. Nocedal and J. K. Reid},
title = {The use of linear programming for the solution of sparse
sets of nonlinear equations},
journal = SISSC,
volume = 8, number = 2, pages = {99--108}, year = 1987,
abstract = {In this paper, we propose a trust region algorithm for
solving sparse sets of nonlinear equations. It is based on
minimizing the $\ell_1$-norm of the linearized residual
within an $\ell_{\infty}$-norm trust region, thereby
permitting linear programming techniques to be easily
applied. The new algorithm has sparsity advantages over the
\citebb{Leve44}--\citebb{Marq63} algorithm},
summary = {A trust-region algorithm for solving sparse sets of
nonlinear equations is proposed. It is based on minimizing
the $\ell_1$-norm of the linearized residual within an
$\ell_{\infty}$-norm trust region, thereby permitting
linear programming techniques to be applied. The algorithm
has sparsity advantages over the
Levenberg-Morrison-Marquardt algorithm}}
@inproceedings{Duff97,
author = {I. S. Duff},
title = {Sparse numerical linear algebra: direct methods and
preconditioning},
crossref = {DuffWats97}, pages = {27--62}}
@article{DuffReid83b,
author = {I. S. Duff and J. K. Reid},
title = {The multifrontal solution of indefinite sparse symmetric
linear equations},
journal = TOMS,
volume = 9, number = 3, pages = {302--325}, year = 1983}
@article{DuffReid96b,
author = {I. S. Duff and J. K. Reid},
title = {Exploiting zeros on the diagonal in the direct solution of
indefinite sparse symmetric linear systems},
journal = TOMS,
volume = 22, number = 2, pages = {227--257}, year = 1996}
@article{DuffReidMunkNeil79,
author = {I. S. Duff and J. K. Reid and N. Munksgaard and H. B.
Neilsen},
title = {Direct solution of sets of linear equations whose matrix is
sparse, symmetric and indefinite},
journal = JIMA,
volume = 23, pages = {235--250}, year = 1979}
@article{Dunn80,
author = {J. C. Dunn},
title = {{N}ewton's method and the {G}oldstein step-length rule for
constrained minimization problems},
journal = SICON,
volume = 6, pages = {659--674}, year = 1980}
@article{Dunn87,
author = {J. C. Dunn},
title = {On the convergence of projected gradient processes to
singular critical points},
journal = JOTA,
volume = 55, pages = {203--216}, year = 1987}
@article{DussFerlLema86,
author = {J. P. Dussault and J. A. Ferland and B. Lemaire},
title = {Convex quadratic programming with one constraint and
bounded variables},
journal = MP,
volume = 36, number = 1, pages = {90--104}, year = 1986}
%%% E %%%
@article{EckeNiem75,
author = {J. G. Ecker and R. D. Niemi},
title = {A dual method for quadratic programs with quadratic
constraints},
journal = SIAPM,
volume = 28, number = 3, pages = {568--576}, year = 1975,
abstract = {A dual method is developed for minimizing a convex
quadratic function of several variables subject to
inequality constraints on the same type of function. The
dual program is a concave maximization problem with
constraints that are essentially linear.
However, the dual objective function is not differentiable
over the dual constraint region. The numerical
difficulties associated with this nondifferentiability are
circumvented by considering a sequence of dual programs via
a modified penalty function technique that does not
eliminate the dual constraints but does insure that they
will all be active at optimality. A numerical example is
included.},
summary = {A dual method is developed for minimizing a convex
quadratic function of several variables subject to
inequality constraints on the same type of function. The
method solves a sequence of dual programs via
a modified penalty function technique that does not
eliminate the dual constraints but ensures that they
will be active at optimality. A numerical example is
included.}}
@article{Ecks93,
author = {J. Eckstein},
title = {Nonlinear Proximal Point Algorithms Using {B}regman
Functions with Applications to Convex Programming},
journal = MOR,
volume = 18, number = 1, pages = {202--226}, year = 1993}
@article{Edlu97,
author = {O. Edlund},
title = {Linear {M}-estimation with bounded variables},
journal = {BIT},
volume = 37, number = 1, pages = {13--23}, year = 1997,
abstract = {A subproblem in the trust region algorithm for non-linear
M-estimation by \citebb{EkblMads89} is to find the
restricted step. It is found by calculating the M-estimator
of the linearized model, subject to an $\ell_2$-norm bound
on the variables. In this paper it is shown that this
subproblem can be solved by applying \citebb{Hebd73}
iterations to the minimizer of the Lagrangian function. The
new method is compared with an Augmented Lagrange
implementation.},
summary = {A subproblem in the trust-region algorithm for nonlinear
M-estimation by \citebb{EkblMads89} is to find the
restricted step, by calculating the M-estimator of the
linearized model, subject to an $\ell_2$-norm bound on the
variables. It is shown that this subproblem can be solved
by applying \citebb{Hebd73} iterations to the minimizer of
the Lagrangian function. The method is compared with an
Augmented Lagrange implementation.}}
@article{EdluEkblMads97,
author = {O. Edlund and H. Ekblom and K. Madsen},
title = {Algorithms for non-linear {M}-estimation},
journal = {Computational Statistics},
volume = 12, number = 3, pages = {373--383}, year = 1997,
abstract = {In non-linear regression, the least squares method is most
often used. Since this estimator is highly sensitive to
outliers in the data, alternatives have become increasingly
popular during the last decades. We present algorithms for
non-linear M-estimation. A trust region approach is used,
where a sequence of estimation problems for linearized
models is solved. In the testing we apply four estimators
to ten non-linear data fitting problems. The test problems
are also solved by the Generalized
\citebb{Leve44}--\citebb{Marq63} method and standard
optimization BFGS method. It turns out that the new method
is in general more reliable and efficient.},
summary = {Algorithms for nonlinear M-estimation are presented. A
trust-region approach is used, where a sequence of
estimation problems for linearized models is solved.
Numerical tests involving four estimators and ten
non-linear data fitting problems are performed.}}
@article{EdsbWedi95,
author = {L. Edsberg and P. A. Wedin},
title = {Numerical tools for parameter-estimation in {ODE} systems},
journal = {Optimization Methods and Software},
volume = 6, number = 3, pages = {193--217}, year = 1995,
abstract = {The numerical problem of estimating unknown parameters in
systems of ordinary differential equations from complete or
incomplete data is treated. A new numerical method for the
optimization part, based on the Gauss-Newton method with a
trust region approach to subspace minimization for the
weighted nonlinear least squares problem, is presented. The
method is implemented in the framework of a toolbox (called
diffpar) in Matlab and several test problems from
applications, giving non-stiff and stiff ODE-systems, are
treated.},
summary = {The numerical problem of estimating unknown parameters in
systems of ordinary differential equations from complete or
incomplete data is treated. A numerical method for the
optimization part is presented, based on the Gauss-Newton
method with a trust-region approach to subspace
minimization for the weighted nonlinear least-squares
problem. The method is implemented in Matlab and several
test problems from applications, giving non-stiff and stiff
ODE-systems, are treated.}}
@misc{EinaMads98,
author = {H. Einarsson and K. Madsen},
title = {Cutting planes and trust-regions for nondifferentiable
optimization},
howpublished = {Presentation at the International Conference on Nonlinear
Programming and Variational Inequalities, Hong Kong},
year = 1998,
abstract = {We discuss the problem of minimizing a nonsmooth function
$f:\Re^n \righarrow \Re$. $f$ is assumed to be continuous
and piecewise smooth, and the number of sooth pieces is
assumed to be finite. The bundle trust region method of
\citebb{SchrZowe92}, is discussed. it is shown that in
the neighbourhood of a minimizer the general function $f$
may be considered as a minimax function, and the relation
between the cutting plane methods of \citebb{SchrZowe92}
and the minimax trust region method of \citebb{Mads75} is
discussed. Based on these ideas an iterative method for
minimizing $f$ is proposed. The basic principles are the
following. At each iteration $f$ is approximated by a
piecewise linear minimax function which is intended to
model the set of generalized gradients in the neighbourhood
of the current iterate. Two trust regions are used: an
inner $R_i$ in which the piecewise linear approximation to
$f$ is found, and an outer $R_o$ in which the next tentative
step is calculated. Initially the two trust regions are
equal, but if the iterate is close to a kink (i.e.\ the
intersection between two or more smooth pieces) then the
inner trust region radius may be smaller. The method has
been tested on a number of convex as well as nonconvex test
problems, and the results are compared with those of
\citebb{SchrZowe92}. Some of the test problems are of the
minimax or $L_1$ type, and in these cases the new algorithm
is compared with the dedicated methods of \citebb{Mads75}
and Hald and Madsen (1985).},
summary = {A bundle trust-region method is proposed for the
minimization of piecewise smooth functions. At each
iteration $f$ is approximated by a piecewise linear minimax
function that models the set of generalized gradients in
the neighbourhood of the current iterate. Two trust regions
are used: an inner one in which the piecewise linear
approximation to the objective function is found, and an
outer one in which the trial step is calculated. Initially
the two radii are equal, but the inner radius may be smaller
if the iterate is close to a the intersection between
smooth pieces. The method is tested on convex and non-convex
test problems}}
@techreport{EiseWalk94,
author = {S. C. Eisenstat and H. F. Walker},
title = {Choosing the forcing terms in an inexact {N}ewton method},
institution = {Dept of Mathematics and Statistics, Utah State University},
address = {Logan, USA},
number = {6/94/75}, year = 1994}
@book{Eispack76,
author = {B. T. Smith and J. M. Boyle and J. J. Dongarra and B. S.
Garbow and Y. Ikebe and V. C. Klema and C. B. Moler},
title = {Matrix Eigensystem Routine---EISPACK Guide},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
year = 1976}
@article{EkblMads89,
author = {H. Ekblom and K. Madsen},
title = {Algorithms for non-linear {H}uber estimation},
journal = BIT,
volume = 29, number = 1, pages = {60--76}, year = 1989,
abstract = {The Huber criterion for data fitting is a combination
of the $\ell_1$ and the $\ell_2$ criteria which is
robust in the sense that the influence of ``wild'' data
points can be reduced. The authors present a trust
region and a Marquardt algorithm for Huber estimation
in the case where the functions used in the fit are
nonlinear. It is demonstrated that the algorithms
converge under the usual conditions.},
summary = {A converfent Levenberg-Morrison-Marquardt method for
nonlinear Huber estimation is presented.}}
@techreport{ElAl88,
author = {M. El{-}Alem},
title = {A Global Convergence Theory for a Class of Trust Region
Algorithms for Constrained Optimization},
institution = CAAM, address = RICE-ADDRESS,
number = {TR88-5}, year = 1988,
abstract = {In this research, we present a trust region algorithm for
solving the equality constrained optimization problem. This
algorithm is a variant of the \citebb{CeliDennTapi85}
algorithm. The augmented Lagrangian function is used as a
merit function. A scheme for updating the penalty parameter
is presented. The behavior of the penalty parameter is
discussed. We present a global and local convergence
analysis for this algorithm. We also show that under mild
assumptions, in a neighborhood of the minimizer, the
algorithm will reduce to the standard SQP algorithm; hence
the local rate of convergence of SQP is maintained. Our
global convergence theory is sufficiently general that it
holds for any algorithm that generates steps that give at
least a fraction of Cauchy decrease in the quadratic model
of constraints.},
summary = {A variant of the \citebb{CeliDennTapi85} trust-region
algorithm for equality constrained optimization is given.
An augmented Lagrangian merit function is used, and a
scheme for updating the penalty parameter presented.
A global and local convergence
analysis is given, showing that
the algorithm reduces to the standard SQP algorithm in a
neighborhood of the minimizer. The global convergence
theory is sufficiently general that it holds for any
algorithm that generates steps giving at least a
fraction of Cauchy decrease in the quadratic model of
constraints.}}
@article{ElAl91,
author = {M. El{-}Alem},
title = {A global convergence theory for the
{D}ennis-{C}elis-{T}apia trust-region algorithm for
constrained optimization},
journal = SINUM,
volume = 28, number = 1, pages = {266--290}, year = 1991,
abstract = {A global convergence theory for a class of trust-region
algorithms for solving the equality constrained
optimization problem is presented. This theory is
sufficiently general that it holds for any algorithm that
generates steps giving at least a fraction of Cauchy
decrease in the quadratic model of the constraints, and
that uses the augmented Lagrangian as a merit function.
This theory is used to establish global convergence of the
\citebb{CeliDennTapi85} algorithm with a different scheme
for updating the penalty parameter. The behaviour of the
penalty parameter is also discussed.},
summary = {A global convergence theory for a class of trust-region
algorithms for equality constrained optimization is
presented, that holds for any algorithm that generates
steps giving at least a fraction of Cauchy decrease in the
quadratic model of the constraints, and that uses the
augmented Lagrangian as a merit function. This theory is
used to establish global convergence of the
\citebb{CeliDennTapi85} algorithm with a different scheme
for updating the penalty parameter. The behaviour of the
penalty parameter is also discussed.}}
@article{ElAl95,
author = {M. El{-}Alem},
title = {A Robust Trust-Region Algorithm with a Nonmonotonic Penalty
Parameter Scheme for Constrained Optimization},
journal = SIOPT,
volume = 5, number = 2, pages = {348--378}, year = 1995,
abstract = {An algorithm for solving the problem of minimizing a
non-linear function subject to equality constraints is
introduced. This algorithm is a trust-region algorithm. In
computing the trial step, a projected-Hessian technique is
used that converts the trust-region subproblem to a one
similar to that of the unconstrained case. To force global
convergence, the augmented Lagrangian is empoyed as a merit
function. One of the main advantages of this algorithm is
the way that the penalty parameter is updated. We introduce
an updating scheme that allows (for the first time to the
best of our knowledge) the penalty parameter to be
decreased whenever it is warranted. The behaviour of this
penalty parameter is studied. A convergence theory for this
algorithm is presented. It is shown that this algorithm is
globally convergent and that the globalization strategy
will not disrupt fast local convergence. The local rate of
convergence is also discussed. This theory is sufficiently
general that it holds for any algorithm that generates
steps whose normal component give at least a fraction of
the Cauchy decrease in the quadratic model of the
constraints and uses \citebb{Flet70b}'s exact penalty
function as a merit function.},
summary = {A trust-region algorithm for nonlinear optimization subject
to equality constraints is introduced. In computing the
trial step, a projected-Hessian technique converts the
trust-region subproblem to one similar to that of the
unconstrained case. To force global convergence, the
augmented Lagrangian is employed as a merit function. An
updating scheme that allows the penalty parameter to be
decreased whenever it is warranted is proposed, and the its
behaviour is studied. It is shown that this algorithm is
globally convergent and that the globalization strategy
does not disrupt fast local convergence. The local rate of
convergence is also discussed. This theory is sufficiently
general that it holds for any algorithm that generates
steps whose normal component give at least a fraction of
the Cauchy decrease in the quadratic model of the
constraints and uses \citebb{Flet70b}'s exact penalty
function as a merit function.}}
@article{ElAl95b,
author = {M. El{-}Alem},
title = {Global convergence without the assumption of linear
independence for a trust-region algorithm for constrained
optimization},
journal = JOTA,
volume = 87, number = 3, pages = {563--577}, year = 1995,
abstract = {A trust-region algorithm for solving the equality
constrained optimization problem is presented. This
algorithm uses the Byrd and \citebb{Omoj89} way of
computing the trial steps, but it differs from the Byrd and
Omojokun algorithm in the way steps are evaluated. A global
convergence theory for this new algorithm is presented. The
main feature of this theory is that the linear independence
assumption on the gradients of the constraints is not
assumed.},
summary = {A trust-region algorithm for solving the equality
constrained optimization problem is presented. This
algorithm uses the Byrd and \citebb{Omoj89} mechanism for
computing the trial steps, but it differs from this
algorithm in the way steps are evaluated. Global
convergence is proved without assuming linear independence
of the constraints' gradients.}}
@article{ElAl96,
author = {M. El{-}Alem},
title = {Convergence to a 2nd order point of a trust-region
algorithm with nonmonotonic penatly parameter for
constrained optimization},
journal = JOTA,
volume = 91, number = 1, pages = {61--79}, year = 1996,
abstract = {In a recent paper, the author proposed a trust-region
algorithm for solving the problem of minimizing a nonlinear
function subject to a set of equality constraints. The main
feature of the algorithm is that the penalty parameter in
the merit function can be decreased whenever it is
warranted. He studied the behavior of the penalty parameter
and proved several global and local convergence results.
One of these results is that there exists a subsequence of
the iterates generated by the algorithm that converges to a
point that satisfies the first-order necessary conditions.
In the current paper, we show that, for this algorithm,
there exists a subsequence of iterates that converges to a
point that satisfies both the first-order and the
second-order necessary conditions.},
summary = {It is shown that a subsequence of iterates produced by the
trust-region algorithm of \citebb{ElAl95} converges to a
point that satisfies both the first- and second-order
necessary conditions.}}
%also
%institution = CRPC, address = RICE-ADDRESS,
%number = {CRPC-TR96654}, year = 1996,
@article{ElAl99,
author = {M. El{-}Alem},
title = {A Global Convergence Theory for a General Class of
Trust-Region-Based Algorithms for Constrained Optimization
Without Assuming Regularity},
journal = SIOPT,
volume = 9, number = 4, pages = {965--990}, year = 1999,
abstract = {This work presents a convergence theory for a general class
of trust-region-based algorithms for solving the smooth
nonlinear programming problem with equality constraints.
The results are proved under very mild conditions on the
quasi-normal and tangential components of the trial steps.
The Lagrange multiplier estimates and the Hessian estimates
are assumed to be bounded. In addition, the regularity
assumption is not made. In particular, the linear
independence of the gradients of the constraints is not
assumed. The theory proves global convergence to one of
four different types of Mayer-Bliss stationary points. The
theory holds for any algorithm that uses the augmented
Lagrangian as a merit function, the \citebb{ElAl95} scheme
for updating the penalty parameter, and bounded multiplier
and Hessian estimates.},
summary = {A convergence theory is presented for a general class of
trust-region algorithms for solving the smooth nonlinear
programming problem with equality constraints. The results
are proved under very mild conditions on the quasi-normal
and tangential components of the trial steps. The Lagrange
multiplier estimates and the Hessian estimates are assumed
to be bounded. In addition, no regularity assumption, such
as linear independence of the constraints' gradients, is
made. The theory proves global convergence to one of four
different types of Mayer-Bliss stationary points, and holds
for any algorithm that uses the augmented Lagrangian as a
merit function, the \citebb{ElAl95} scheme for updating the
penalty parameter, and bounded multiplier and Hessian
estimates.}}
%also
%institution = CRPC, address = RICE-ADDRESS,
%number = {CRPC-TR96655}, year = 1996,
@techreport{ElAl96c,
author = {M. El{-}Alem},
title = {A Strong Global Convergence Result for {D}ennis,
{E}l-{A}lem, and {M}aciel's Class of Trust Region
Algorithms},
institution = CAAM, address = RICE-ADDRESS,
number = {TR96-15}, year = 1996,
abstract = {In a recent paper, \citebb{DennElAlMaci97} suggested a
class of trust-region-based algorithms for solving the
equality constrained optimization problem. They established
a global convergence result that is analogous to
\citebb{Powe75}'s result for the unconstrained optimization
problem. In this paper, a global convergence theory for
\citebbs{DennElAlMaci97} class of algorithms is presented.
The theory is analogous to \citebb{Thom75}'s result for
unconstrained optimization. In particular, it proves that
every accumulation point of the sequence of iterates
generated by any member of \citebbs{DennElAlMaci97} class
of algorithms is a first-order point. In other words, the
sequence of iterates converges to the set of first-order
points of the problem. To the best of our knowledge, the
global convergence result presented in this paper
generalizes all existing global convergence theories for
trust region algorithms that are suggested for solving the
equality constrained optimization problem and use the
augmented Lagrangian as a merit function.},
summary = {A global convergence theory for \citebb{DennElAlMaci97}'s
class of algorithms is presented, which is analogous to
\citebb{Thom75}'s result for unconstrained optimization. In
particular, every accumulation point of the sequence of
iterates is a first-order stationary point. This
result generalize all current global convergence theories
for trust-region algorithms that have been suggested for
solving the equality constrained optimization problem and
use the augmented Lagrangian as a merit function.}}
%also
%institution = CRPC, address = RICE-ADDRESS,
%number = {CRPC-TR96656}, year = 1996,
@article{ElAlTapi95,
author = {M. El{-}Alem and R. A. Tapia},
title = {Numerical Experience with a polyhedral-norm {CDT}
trust-region algorithm},
journal = JOTA,
volume = 85, number = 3, pages = {575--591}, year = 1995,
abstract = {In this paper, we study a modification of the
\citebb{CeliDennTapi85} trust-region subproblem, which is
obtained by replacing the $l_2$-norm with a polyhedral
norm. The polyhedral norm Celis-Dennis-Tapia (CDT)
subproblem can be solved using a standard quadratic
programming code. We include computational results which
compare the performance of the polyhedral-norm CDT
trust-region algorithm with the performance of existing
codes. The numerical results validate the effectiveness of
the approach. These results show that there is not much
loss of robustness or speed and suggest that the
polyhedral-norm CDT algorithm may be a viable alternative.
The topic merits further investigation.},
summary = {A modification of the \citebb{CeliDennTapi85} (CDT)
trust-region subproblem, which is obtained by replacing the
$l_2$-norm with a polyhedral norm, is studied. The
polyhedral norm CDT subproblem can be solved using a
standard quadratic programming code. Computational results
which compare the performance of the polyhedral-norm CDT
trust-region algorithm with the performance of existing
codes are given.}}
@article{ElBa98,
author = {A. S. El{-}Bakry},
title = {Convergence rate of primal-dual reciprocal barrier {N}ewton
interior-point methods},
journal = OMS,
volume = 9, number = {1--3}, pages = {37--44}, year = 1998}
@techreport{ElHa87,
author = {M. El{-}Hallabi},
title = {A Global Convergence Theory for Arbitrary Norm Trust Region
Methods for Nonlinear Equations},
institution = CAAM, address = RICE-ADDRESS,
number = {TR87-5}, year = 1987,
abstract = {In this research we extend the
\citebb{Leve44}--\citebb{Marq63} algorithm for
approximating zeros of the nonlinear system $F(x)=0$, where
$F$ is continuously differentiable from $\Re^n$ to $\Re^n$.
Instead of the $\ell_2$-norm, arbitrary norms can be used
in the objective function and in the trust region
constraint. The algorithm is shown to be globally
convergent. This research was motivated by the recent work
of \citebb{DuffNoceReid87}. A key point in our analysis is
that the tools from nonsmooth analysis, namely locally
Lipschitz analysis, allow us to establish essentially the
same properties for our algorithm that have been
established for the \citebb{Leve44}--\citebb{Marq63}
algorithm using the tools from smooth optimization. In our
analysis, the sequence generated by the algorithm is the
couple $(x_k, \delta_k)$ where $x_k$ is the iterate and
$\delta_k$ the trust region radius. Since the successor
$(x_{k+1}, \delta_{k+1})$ of $(x_k,\delta_k)$ is not unique
we model our algorithm by a point-to-set map and then apply
Zangwill's theorem of convergence to our case. It is shown
that our algorithm reduces locally to Newton's method.},
summary = {The Levenberg-Morrison-Marquardt algorithm for
approximating zeros of the nonlinear system $F(x)=0$ is
generalized to allow the use of arbitrary norms
n the objective function and in the trust
region constraint. The algorithm, which is motivated by
that of \citebb{DuffNoceReid87}, is globally
convergent. Essentially the same properties apply for the
general and for the Levenberg-Morrison-Marquardt algorithm.
In this analysis, the sequence generated is
the couple $(x_k, \Delta_k)$ where $x_k$ is the iterate and
$\Delta_k$ the trust region radius. Since the successor
$(x_{k+1}, \Delta_{k+1})$ of $(x_k,\Delta_k)$ is not unique
the algorithm is modelled by a point-to-set map and then
Zangwill's convergence theorem is applied. The algorithm
locally reduces to Newton's method.}}
@techreport{ElHa90,
author = {M. El{-}Hallabi},
title = {A Global Convergence Theory for A Class of Trust-Region
Methods for Nonsmooth Optimization.},
institution = CAAM, address = RICE-ADDRESS,
number = {TR90-10}, year = 1990,
abstract = {In this work we define a class of trust-region algorithms
for approximating a minimizer of the function $f=h(F)$
where $F: \Re^n \rightarrow \Re^m$ is continuously
differentiable and $h=\Re^m \rightarrow \Re$ is regular
locally Lipschitz. We show that algorithms from this class
are globally convergent. Our analysis is a generalization
of the recent work of \citebb{ElHaTapi93} and can be
applied to most algorithms in the literature. Our
algorithms are a natural generalization of those for smooth
minimization to nonsmooth optimization.},
summary = {A class of trust-region algorithms is defined for
approximating a minimizer of the function $f=h(F)$ where
$F: \Re^n \rightarrow \Re^m$ is continuously differentiable
and $h=\Re^m \rightarrow \Re$ is regular locally Lipschitz.
Algorithms from this class are globally
convergent. The analysis is a generalization of that given
by \citebb{ElHaTapi93} and can be applied to most
algorithms in the literature. The algorithms are a natural
generalization of those for smooth minimization to
non-smooth optimization.}}
@techreport{ElHa93,
author = {M. El{-}Hallabi},
title = {An Inexact Minimization Trust-Region Algorithm:
globalization of {N}ewton's Method},
institution = CAAM, address = RICE-ADDRESS,
number = {TR93-43}, year = 1993,
abstract = {In this work we define a trust region algorithm for
approximating zeros of the nonlinear system $F(x)=0$, where
$F: \Re^n \rightarrow \Re^n$ is continuously
differentiable. We are concerned with the fact that $n$ may
be large. So we replace the $\ell_2$ norm with arbitrary
norms in the objective function and in the trust region
constraint. In particular, if polyhedral norms are used,
then the algorithm can be viewed as a sequential linear
programming algorithm. At each iteration, the local
trust-region model is only solved within some tolerance.
This research is an extension of \citebb{ElHaTapi93} for
nonlinear equations, where an exact solution of the local
model was required. We demonstrate that the algorithm under
consideration is globally convergent, and that, under mild
assumptions, the iteration sequence generated by the
algorithm converges to a solution of the nonliner system.
We also demonstrate that, under the standard assumptions
for Newton's method theory, the rate of convergence is
$q$-superlinear. Moreover, quadratic convergence can be
obtained by requiring sufficient accuracy in the solution
of the local model.},
summary = {A trust-region algorithm generalizing that of
\citebb{ElHaTapi93} is given, in which the trust-region
subproblem can be solved approximately. The algorithm
considered is globally convergent and its
rate of convergence is $Q$-superlinear. Quadratic
convergence can be obtained by requiring sufficient
accuracy in the solution of the local model.}}
@techreport{ElHa99,
author = {M. El{-}Hallabi},
title = {Globally Convergent Multi-Level Inexact Hybrid Algorithm
for Equality Constrained Optimization},
institution = {D\'{e}partement Informatique et Optimisation, Institut
National des Postes et T\'{e}l\'{e}communications},
address = {Rabat, Morocco},
number = {RT11-98(revised)}, year = 1999,
abstract = {Trust-region globalization strategies have proved to be
powerful tools to design globally convergent algorithms,
but at the cost of allowing the local model to be solved
more than once at each iteration. On the other hand,
beside their poor global convergence properties, linesearch
strategies are quite popular for their low cost of
obtaining an acceptable steplength whenever a search
direction is provided. In this paper we aim to combine
both strategies in a globally convergent multi-level
inexact hybrid algorithm to minimize a contunuouly
differentiable nonlinear function $f: \Re^n \rightarrow \Re$
subject to equality constraint $h_i(x)=0$, $i=1,\ldots,m$
where $h_i:\Re^n \rightarrow \Re$ are continuously
differentiable. First, the trust-region approach is used
to determine a trial step that is shown to ne a descent
direction of the merit function, and second, linesearch
techniques are used to obtain an acceptable steplength
in such a direction. We prove that the hybrid algorithm
is globally convergent in the sense that any accumulation
point of the iteration sequence is a Karush-Kuhn-Tucker
point of the minimization problem. In our algorithm, the
curvature of the local model is taken into account first,
to obtain the trial step direction and second, to accept
or reject the steplength. Both tests are less conservative
than the usual ones. Also, we prove that the penalty
parameter is uniformly bounded away from zero at
nonstationary points, instead of forcing the trust-region
radius to be initialized, at each iteration, as large as
some given positive $\delta_{\min}$, we show that the
internal trust-region increasing strategy, a technique
quite popular for preventing large numbers of gradient and
Hessian evaluations, yields the same important property.
Furthermore, we show that the steplength is bounded away
from zero at nonstaionary points. We do not use the
regularity assumption of linear independent gradients.
On the other hand, we assume that $h(x_k)$ does not belong
to the nullspace of $\nabla h(x_k)$ for non feasible
iterates. Moreover, we assume that if
$\{x_k\mid k \in N\subset\calN\}$ is a subsequence
converging to some accumulation point of the iteration
sequence, say $x_*$, then the normalized constraint vector
is uniformly bounded away from the nullspace of
$\nabla h(x_k)$.},
summary = {The combination of linesearch and trust-region techniques
is investigated in the context of problems with equality
constraints. Beneficial effects of internal doubling in
this context are also discussed, together with an
alternative expression of constraint qualification.}}
@techreport{ElHaTapi93,
author = {M. El{-}Hallabi and R. A. Tapia},
title = {A Global Convergence Theory for Arbitrary Norm Trust-Region
Methods for Nonlinear Equations},
institution = CAAM, address = RICE-ADDRESS,
number = {TR93-41}, year = 1993,
abstract = {In this work, we extend the Levenberg-Marquardt algorithm
for approximating zeros of the nonlinear system $F(x)=0$,
where $F: \Re^n \rightarrow \Re^n$ is continuously
differentiable. Instead of the $\ell_2$ norm, arbitrary
norms can be used in the trust-region constraint. The
algorithm is shown to be globally convergent. This research
is motivated by the recent work of \citebb{DuffNoceReid87}.
A key point in our analysis is that the tools from
nonsmooth analysis and the Zangwill convergence theory
allow us to establish essentially the same properties for
an arbitrary trust-region algorithm that have been
established for the Levenberg-Marquardt algorithm using the
tools from smooth optimization. It is shown that all
members of this class of algorithms locally reduce to
Newton's method and that the iteration sequence actually
converges to a solution.},
summary = {The Levenberg-Morrison-Marquardt algorithm for
approximating zeros of the nonlinear system $F(x)=0$, where
$F: \Re^n \rightarrow \Re^n$ is continuously
differentiable, is extended. Arbitrary norms can be used in
place of the $\ell_2$-norm for the trust-region constraint.
The algorithm is globally convergent. This
algorithm is motivated by the work of
\citebb{DuffNoceReid87}. It locally reduce to Newton's
method and the iteration sequence converges to a
solution.}}
@techreport{ElHaTapi95,
author = {M. El{-}Hallabi and R. A. Tapia},
title = {An Inexact Trust-Region Feasible-Point Algorithm for
Nonlinear Systems of Equalities and Inequalities},
institution = CAAM, address = RICE-ADDRESS,
number = {TR95-09}, year = 1995,
abstract = {In this work we define a trust-region feasible-point
algorithm for approximating solutions of the nonlinear
system of equalities and inequalities $F(x, y)=0, y \ge 0$,
where $F: { \Re^n \times \Re^m } \rightarrow \Re^p$ is
continuously differentiable. This formulation is quite
general; the Karush-Kuhn-Tucker conditions of a general
nonlinear programming problem are an obvious example, and a
set of equalities and inequalities can be transformed,
using slack variables, into such form. We will be concerned
with the possibility that $n$, $m$ and $p$ may be large and
that the Jacobian matrix may be sparse and rank deficient.
Exploiting the convex structure of the local model
trust-region subproblem, we propose a globally convergent
inexact trust-region feasible-point algorithm to minimize
an arbitrary norm of the residual, say $\| F(x, y)\|_a$,
subject to the nonnegativity constraints. This algorithm
uses a trust-region globalization strategy to determine a
descent direction as an inexact solution of the local model
trust-region subproblem and then, it uses linesearch
techniques to obtain an acceptable steplength. We
demonstrate that, under rather weak hypotheses, any
accumulation point of the iteration sequence is a
constrained stationary point for $f=\|F\|_a$, and that the
sequence of constrained residuals converges to zero.},
summary = {A feasible-point trust-region algorithm for approximating
solutions of the nonlinear system of equalities and
inequalities $F(x, y)=0, y \ge 0$, where $F: { \Re^n \times
\Re^m } \rightarrow \Re^p$ is continuously differentiable,
is considered. By exploiting the convex structure of the
local trust-region subproblem, a globally convergent
inexact trust-region feasible-point algorithm is suggested
for minimizing an arbitrary norm of the residual,
$\| F(x, y)\|_a$, subject to non-negativity constraints. This
algorithm uses a descent direction which is an inexact
solution of the trust-region subproblem and then uses
linesearch techniques to obtain an acceptable steplength.
It is shown that, under weak hypotheses, any accumulation
point of the iteration sequence is a constrained stationary
point for $f=\|F\|_a$, and that the sequence of constrained
residuals converges to zero.}}
@article{ElstNeum97,
author = {C. Elster and A. Neumaier},
title = {A method of trust region type for minimizing noisy
functions},
journal = {Computing},
volume = 58, number = 1, pages = {31--46}, year = 1997,
abstract = {The optimization of noisy functions in a few variables only
is a common problem occurring in various applications, for
instance in finding the optimal choice of a few control
parameters in chemical experiments. The traditional tool
for the treatment of such problems is the method of
\citebb{NeldMead65} (NM). In this paper, an alternative
method based on a trust region approach (TR) is offered and
compared to NM. On the standard collection of test
functions for unconstrained optimization by
\citebb{MoreGarbHill81}, TR performs substantially more
robust than NM. If performance is measured by the number of
function evaluations, TR is on the average twice as fast as
NM.},
summary = {The optimization of noisy functions of a few variables is a
commonly occurring problem in application areas such as
finding the optimal choice of a few control parameters in
chemical experiments. The traditional tool for the
treatment of such problems is the method of
\citebb{NeldMead65} (NM). An alternative method based on a
trust-region approach (TR) is proposed and compared to NM.
On a standard collection of test functions for
unconstrained optimization by \citebb{MoreGarbHill81}, TR
is substantially more robust than NM. If performance is
measured by the number of function evaluations, TR is seen
to be, on average, twice as fast as NM.}}
@article{EskoSchn91,
author = {E. Eskow and R. B. Schnabel},
title = {Algorithm 695: software for a new modified {C}holesky
factorization},
journal = TOMS,
volume = 17, number = 3, pages = {306--312}, year = 1991}
@article{Evan68,
author = {D. J. Evans},
title = {The use of pre-conditioning in iterative methods for
solving linear equations with positive definite matrices},
journal = JIMA,
volume = 4, pages = {295--314}, year = 1968}
%%% F %%%
@article{FaccLuci93,
author = {F. Facchinei and S. Lucidi},
title = {Nonmonotone Bundle-Type Scheme for Convex Nonsmooth
Minimization},
journal = JOTA,
volume = 76, number = 2, pages = {241--257}, year = 1993}
@inproceedings{FaccFiscKanz97,
author = {F. Facchinei and A. Fischer and Ch. Kanzow},
title = {A Semismooth {N}ewton Method for Variational Inequalities:
the Case of Box Constraints},
crossref = {FerrPang97}, pages = {76--90}}
@article{FaccKanz97,
author = {F. Facchinei and Ch. Kanzow},
title = {On unconstrained and constrained stationary points of the
implicit {L}agrangian},
journal = JOTA,
volume = 92, number = 1, pages = {99--115}, year = 1997}
@article{FaccSoar97,
author = {F. Facchinei and J. Soares},
title = {A new merit function for nonlinear complementarity problems
and a related algorithm},
journal = SIOPT,
volume = 7, number = 1, pages = {225--247}, year = 1997}
@article{FaccJudiSoar98,
author = {F. Facchinei and J. Judice and J. Soares},
title = {An active set {N}ewton algorithm for large-scale nonlinear
programs with box constraints},
journal = SIOPT,
volume = 8, number = 1, pages = { 158--186}, year = 1998,
abstract = {A new algorithm for large-scale nonlinear programs with box
constraints is introduced. The algorithm is based on an
efficient identification technique of the active set at the
solution and on a nonmonotone stabilization technique. It
possesses global and superlinear convergence properties
under standard assumptions. A new technique for generating
test problems with known characteristics is also
introduced. The implementation of the method is described
along with computational results for large-scale problems.},
summary = {An algorithm for large-scale nonlinear programs with box
constraints is introduced. The algorithm is based on an
efficient identification technique of the active set at the
solution and on a non-monotone stabilization technique. It
possesses global and superlinear convergence properties.
A technique for generating
test problems with known characteristics is also
introduced. The implementation of the method is described
along with computational results for large-scale problems.}}
@article{FanSarkLasd88,
author = {Y. Fan and S. Sarkar and L. S. Lasdon},
title = {Experiments with successive quadratic programming
algorithms},
journal = JOTA,
volume = 56, number = 3, pages = {359--383}, year = 1988,
abstract = {There are many variants of successive quadratic
programming (SQP) algorithms. Important issues
include: the choice of either line search or trust
region strategies and the QP formulation to be used and
how the QP is to be solved. The authors consider the
QPs proposed by Fletcher and Powell and discuss a
specialized reduced-gradient procedure for solving
them. A computer implementation is described, and the
various options are compared on some well-known test
problems. Factors influencing robustness and speed
are identified.},
summary = {Important issues in SQP methods include the choice
of either linesearch or trust-region strategies and the
QP formulation to be used and how the QP is to be solved.
The QPs proposed by Fletcher and Powell are considered
and a specialized reduced-gradient procedure discussed for
solving them. The various options are compared on some
well-known test problems.}}
@article{FeiHuan98,
author = {X. Fei and W. Huanchen},
title = {Integrated algorithm for bilevel nonsmooth
optimization problems},
journal = {Journal of Shanghai Jiaotong University},
volume = 32, number = 12, pages = {115--119}, year = 1998,
note = {(in Chinese)},
abstract = {This paper is concerned with a kind of 1 leader-$N$
followers bilevel nonsmooth optimization problems.
An integrated algorithm is proposed which embeds
adaptively DFP into the inner iteration of the trust
region bundle method and makes the best use of the
global convergence of the bundle method and the local
fast convergence of the DFP. The Lipschitzian
property of functions involved is researched. An
approach of computing a subgradient of the objective
functions of the problems is investigated. The basic
idea and steps of the algorithm are
discussed. Finally, the convergence analysis is
given.},
summary = {A 1 leader-$N$ followers bilevel nonsmooth optimization
problems is considered. A trust-region based bundle
in which appropriate generailized second derivatives are
obtained using a DFP-like formula is given, which
combines the global convergence properties of the
bundle method with the fast local convergence properties
resulting from the use of approximate second derivatives.}}
@inproceedings{Felg97,
author = {U. Felgenhauer},
title = {Algorithmic stability analysis for certain trust region
methods},
booktitle = {Mathematical Programming with Data Perturbations},
editor = {A. V. Fiacco},
publisher = {Marcel Dekker, Inc.}, address = {New York and Basel},
series = {Lecture Notes in Pure and Applied Mathematics},
number = 195, pages = {109--131}, year = 1997,
summary = {Quasi-Newton trust-region methods for unconstrained and
bound-constrained optimization are proven to be robust
with respect to errors in the gradient. Global convergence
and active constraint identifications are proved under the
assumption that this error is bounded by a multiple of the
trust-region radius and that the model's Hessians are
bounded and non-zero.}}
@techreport{Feng98,
author = {G. Feng},
title = {Trust-region method with simplicial decomposition for
linearly constrained problems},
institution = {Department of Applied Mathematics, Tongji University},
address = {Shanghai, China},
number = {December, 17}, year = 1998,
abstract = {For the nonlinear programming problems, in which the
objective function is continuously differentiable,
pseudo-convex and the feasible set is a nonempty polyhedron,
we develop an algorithm of trust region method using
simplicial decomposition. The algorithm solves a linearly
constrained problem in the subprogram and a master program
iteratively. The subprogram is a linear programming similar
to Frank-Wolfe linearization technique, but with a
restricted stepsize, and produces feasible points defining
simplices. the produced simplices are only subsets of the
feasible region of the original programming. The master
program is a trust region method on the produced simplex.
According to the ratio of the actual and predicted reduction
in the master program, we change the stepsize in the
subproblem adaptively per iteration. The resulting
algorithm is proved to be globally convergent. The
advantage of the algorithm over the original trust region
method is that the feasible region under consideration of
the former is only a subset of the latter. But the
algorithm must solve an additional linear programming
problem which is relatively simpler.},
summary = {A trust-region method is presented for the solution of
pseudoconvex optimization problems subject to linear
constraints. The method uses restricted simplicial
decomposition to produce successive simplices that are
included in the feasible domain and a trust-region method
is then employed to minimize the objective function on
those simplices. }}
@techreport{Feng99,
author = {G. Feng},
title = {Combination of trust region method and simplicial
decomposition for linearly constrained problems},
institution = {Department of Applied mathematics, Tongji University},
address = {Shanghai, China},
number = {March}, year = 1999,
abstract = {The algorithm given here incorporates the restricted
decomposition algorithm (RSD) into the trust-region method
(TR). The global convergence is proved. The advantage of
the presented algorithm over RSD is that the former is exact
and finite in every iteration. In comparison with TR the
feasible set of the master problem in the presented
algorithm is only the subset of that in TR and has no
restriction on step. Thus the former is much easier to
solve than the latter.},
summary = {A variant of the method developed in \citebb{Feng98} is
presented, where no restriction on the steplength is
imposed on the master problem.}}
@article{FerrPang97b,
author = {M. C. Ferris and J. S. Pang},
title = {Engineering and Economic Applications of Complementarity
Problems},
journal = SIREV,
volume = 39, number = 4, pages = {669--713}, year = 1997}
@techreport{FerrKanzMuns98,
author = {M. C. Ferris and C. Kanzow and T. S. Munson},
title = {Feasible Descent Algorithms for Mixed Complementarity
Problems},
institution = MADISON, address = MADISON-ADDRESS,
type = {Mathematical Programming Technical Report},
number = {MP-TR-98-04}, year = 1998}
@techreport{FerrZavr96,
author = {M. C. Ferris and S. K. Zavriev},
title = {The linear convergence of a successive linear programming
algorithm},
institution = MADISON, address = MADISON-ADDRESS,
type = {Mathematical Programming Technical Report},
number = {MP-TR-96-12}, year = 1996,
abstract = {We present a successive linear programming algorithm for
solving constrained nonlinear optimization problems. The
algorithm employs an Armijo procedure for updating a trust
region radius. We prove the linear convergence of the
method by relating the solutions of our subproblems to
standard trust region and gradient projection subproblems
and adapting an error bound analysis due to
\citebb{LuoTsen93}. Computational results are provided for
polyhedrally constrained nonlinear programs.},
summary = {A successive linear programming algorithm for solving
constrained nonlinear optimization problems is presented.
that usess an Armijo procedure for updating a
trust region radius. Linear convergence of the method is
proved by relating the solutions of the subproblems to
standard trust-region and gradient projection subproblems
and adapting an error bound analysis of
\citebb{LuoTsen93}. Computational results are provided for
polyhedrally constrained nonlinear programs.}}
@article{Fiac76,
author = {A. V. Fiacco},
title = {Sensitivity analysis for nonlinear programming using
penalty methods},
journal = MP,
volume = 10, number = 3, pages = {287--311}, year = 1976}
@book{Fiac83,
author = {A. V. Fiacco},
title = {Introduction to Sensitivity and Stability Analysis in
Nonlinear Programming},
publisher = AP, address = AP-ADDRESS,
series = {Mathematics in Science and Engineering}, volume = 165,
year = 1983}
@techreport{FiacMcCo63,
author = {A. V. Fiacco and G. P. McCormick},
title = {Programming under Nonlinear Constraints by Unconstrained
Optimization: a Primal-Dual Method},
institution = {Research Analysis Corporation},
address = {McLean, Virginia, USA},
number = {RAC-TP-96}, year = 1963}
@article{FiacMcCo64a,
author = {A. V. Fiacco and G. P. McCormick},
title = {The Sequential Unconstrained Minimization Technique for
Nonlinear Programming: a Primal-Dual Method},
journal = {Management Science},
volume = 10, number = 2, pages = {360--366}, year = 1964}
@article{FiacMcCo64b,
author = {A. V. Fiacco and G. P. McCormick},
title = {Computational Algorithm for the Sequential Unconstrained
Minimization Technique for Nonlinear Programming},
journal = {Management Science},
volume = 10, number = 4, pages = {601--617}, year = 1964}
@book{FiacMcCo68,
author = {A. V. Fiacco and G. P. McCormick},
title = {Nonlinear Programming: Sequential Unconstrained
Minimization Techniques},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1968,
note = {Reprinted as \emph{Classics in Applied Mathematics 4}, SIAM,
Philadelphia, USA, 1990}}
@article{Fisc92,
author = {A. Fischer},
title = {A special {N}ewton-type optimization method},
journal = {Optimization},
volume = 24, number = {3--4}, pages = {269--284}, year = 1992}
@inproceedings{Fisc95,
author = {A. Fischer},
title = {An {NCP}-function and its use for the solution of
complementarity problems},
crossref = {DuQiWome95}, pages = {88--105}}
@article{Flet70,
author = {R. Fletcher},
title = {A New Approach to Variable Metric Algorithms},
journal = COMPJ,
volume = 13, pages = {317--322}, year = 1970}
@incollection{Flet70b,
author = {R. Fletcher},
title = {A class of methods for nonlinear programming with
termination and convergence properties},
booktitle = {Integer and nonlinear programming},
editor = {J. Abadie},
publisher = NH, address = NH-ADDRESS,
pages = {157--175}, year = 1970}
@techreport{Flet70c,
author = {R. Fletcher},
title = {An efficient, globally convergent, algorithm for
unconstrained and linearly constrained optimization
problems},
institution = HARWELL, address = HARWELL-ADDRESS,
number = {TP 431}, year = 1970,
abstract = {An algorithm for minimization of functions of many
variables, subject possibly to linear constraints on the
variables, is described. In it a subproblem is solved in
which a quadratic approximation is made to the object
function and minimized over a region in which the
approximation is valid. A strategy for deciding when this
region should be expanded or contracted is given. The
quadratic approximation involves estimating the hessian
of the object function by a matrix which is updated at
each iteration by a formula recently reported by
\citebb{Powe70a}. This formula enables global convergence
of the algorithm to be proved. Use of such an
approximation, as against using exact second derivatives,
also enables a reduction of about 60\%\ to be made in the
number of operations to solve the subproblem. Numerical
evidence is reported showing that the algorithm is
efficient in the number of function evaluations required
to solve well known test problems.},
summary = {An algorithm is described for minimization of nonlinear
functions, subject possibly to linear constraints on the
variables. At each iteration, a quasi-Newton (PSB)
quadratic approximation of the objective function is
minimized over a region in which the approximation is valid.
A strategy for deciding when this region should be expanded
or contracted is given. Global convergence is proved and
numerical tests show that the algorithm is efficient
in the number of function evaluations.}}
@article{Flet71,
author = {R. Fletcher},
title = {A general quadratic programming algorithm},
journal = JIMA,
volume = 7, pages = {76--91}, year = 1971}
@techreport{Flet71b,
author = {R. Fletcher},
title = {A modified {M}arquardt subroutine for nonlinear
least-squares},
institution = HARWELL, address = HARWELL-ADDRESS,
number = {AERE-R 6799}, year = 1971,
abstract = {A Fortran subroutine is described for minimizing a sum of
squares of functions of many variables. Such problems arise
in nonlinear data fitting, and in the solution of nonlinear
algebraic equations. The subroutine is based on an
algorithm due to \citebb{Marq63}, but with modifications
which improve the performance of the method in certain
circumstances, yet which require negligible extra computer
time and storage.},
summary = {A Fortran subroutine is described for minimizing a sum of
squares of functions of many variables. Such problems arise
in nonlinear data fitting, and in the solution of nonlinear
algebraic equations. The subroutine is based on an
algorithm due to \citebb{Marq63}, but with modifications
which improve the performance of the method, yet which
require negligible extra computer time and storage.}}
@article{Flet73,
author = {R. Fletcher},
title = {An exact penalty function for nonlinear programming with
inequalities},
journal = MP,
volume = 5, number = 2, pages = {129--150}, year = 1973}
@article{Flet76,
author = {R. Fletcher},
title = {Factorizing symmetric indefinite matrices},
journal = LAA,
volume = 14, pages = {257--272}, year = 1976}
@book{Flet80,
author = {R. Fletcher},
title = {Practical Methods of Optimization: Unconstrained
Optimization},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1980}
@book{Flet81,
author = {R. Fletcher},
title = {Practical Methods of Optimization: Constrained Optimization},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1981}
@inproceedings{Flet82,
author = {R. Fletcher},
title = {Second-order corrections for non-differentiable
optimization},
booktitle = {Numerical Analysis, Proceedings Dundee 1981},
editor = {G. A. Watson},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
pages = {85--114}, year = 1982,
note = {Lecture Notes in Mathematics 912}}
@article{Flet82b,
author = {R. Fletcher},
title = {A model algorithm for composite nondifferentiable
optimization problems},
journal = MPS,
volume = 17, pages = {67--76}, year = 1982,
abstract = {Composite functions $\phi(x)=f(x) + h(c(x))$, where $f$ and
$c$ are smooth and $h$ is convex, encompass many
nondifferentiable optimization problems of interest
including exact penalty functions in nonlinear programming,
nonlinear min-max problems, best nonlinear $L_1$, $L_2$ and
$L_\infty$ approximation and finding feasible points of
nonlinear inequalities. The idea is used of making a linear
approximation to $c(x)$ whilst including second order terms
in a quadratic approximation to $f(x)$. This is used to
determine a composite function $\psi$ which approximates
$\phi(x)$ and a basic algorithm is proposed in which $\psi$
is minimized on each iteration. If the technique of step
restriction (or trust region) is incorporated into the
algorithm, then it is shown that global convergence can be
proved. It is also described briefly how the above
approximations ensure that a second order rate of
convergence is achieved by the basic algorithm. },
summary = {Composite functions $\phi(x)=f(x) + h(c(x))$, where $f$ and
$c$ are smooth and $h$ is convex, encompass many
non-differentiable optimization problems of interest. Making
a linear approximation to $c(x)$ whilst including
second-order terms in a quadratic approximation to $f(x)$
is used to determine a composite function $\psi$ which
approximates $\phi(x)$ , and an algorithm is proposed in
which $\psi$ is minimized on each iteration. If the trust
region technique is incorporated into the algorithm, then
global convergence can be proved. It is also described how
the above approximations ensure that a second-order rate of
convergence is achieved.}}
@inproceedings{Flet85,
author = {R. Fletcher},
title = {An $\ell_1$ penalty method for nonlinear constraints},
crossref = {BoggByrdSchn85}, pages = {26--40}}
@book{Flet87,
author = {R. Fletcher},
title = {Practical Methods of Optimization},
publisher = WILEY, address = WILEY-ADDRESS,
edition = {second}, year = 1987}
@inproceedings{Flet87b,
author = {R. Fletcher},
title = {Recent developments in linear and quadratic programming},
crossref = {IserPowe87}, pages = {213--243}}
@article{Flet95,
author = {R. Fletcher},
title = {An Optimal Positive Definite Update for Sparse {H}essian
Matrices},
journal = SIOPT,
volume = 5, number = 1, pages = {192--217}, year = 1995}
@article{FletJack74,
author = {R. Fletcher and M. P. Jackson},
title = {Minimization of a quadratic function of many variables
subject only to lower and upper bounds},
journal = JIMA,
volume = 14, number = 2, pages = {159--174}, year = 1974}
@article{FletSain89,
author = {R. Fletcher and Sainz de la Maza, E.},
title = {Nonlinear programming and nonsmooth optimization by
successive linear programming},
journal = MP,
volume = 43, number = 3, pages = {235--256}, year = 1989,
abstract = {Methods are considered for solving nonlinear programming
problems using an exact $\ell_1$ penalty function. LP-like
subproblems incorporating a trust region constraint are
solved successively both to estimate the active set and to
provide a foundation for proving global convergence. In one
particular method, second-order information is represented
by approximating the reduced Hessian matrix, and
\citebb{ColeConn82b} steps are taken. A criterion for
accepting these steps is given which enables the
superlinear convergence properties of the Coleman-Conn
method to be retained whilst preserving global convergence
and avoiding the \citebb{Mara78} effect. The methods
generalize to solve a wide range of composite nonsmooth
optimization problems and the theory is presented in this
general setting. A range of numerical experiments on small
test problems is described.},
summary = {Methods are considered for solving nonlinear programming
problems using an exact $\ell_1$ penalty function. LP-like
subproblems incorporating a trust-region constraint are
solved successively both to estimate the active set and to
provide a foundation for proving global convergence. In one
particular method, second-order information is represented
by approximating the reduced Hessian matrix, and
\citebb{ColeConn82b} steps are taken. A criterion for
accepting these steps is given which enables the
superlinear convergence properties of the Coleman-Conn
method to be retained whilst preserving global convergence
and avoiding the \citebb{Mara78} effect. The methods
generalize to solve a wide range of composite non-smooth
optimization problems and the theory is presented in this
general setting. A range of numerical experiments on small
test problems is described.}}
@techreport{FletLeyf97,
author = {R. Fletcher and S. Leyffer},
title = {Nonlinear Programming without a penalty function},
institution = DUNDEE, address = DUNDEE-ADDRESS,
type = {Numerical Analysis Report}, number = {NA/171}, year = 1997,
abstract = {In this paper the solution of nonlinear programming
problems by a Sequential Quadratic Programming (SQP)
trust--region algorithm is considered. The aim of the
present work is to promote global convergence without the
need to use a penalty function. Instead, a new concept of a
``filter'' is introduced which allows a step to be accepted
if it reduces either the objective function or the
constraint violation function. Numerical tests on a wide
range of test problems are very encouraging and the new
algorithm compares favourably with LANCELOT and an
implementation of S$l_1$QP.},
summary = {A Sequential Quadratic Programming (SQP) trust-region
algorithm for nonlinear programming is considered, which is
globally convergent without the need to use a penalty
function. Instead, the concept of a ``filter'' is
introduced which allows a step to be accepted if it reduces
either the objective function or the constraint violation
function. Numerical tests on a wide range of test problems
are very encouraging and the new algorithm compares
favourably with {\sf LANCELOT} and an implementation of
S$l_1$QP.}}
@techreport{FletLeyf98,
author = {R. Fletcher and S. Leyffer},
title = {User Manual for filter{SQP}},
institution = DUNDEE, address = DUNDEE-ADDRESS,
type = {Numerical Analysis Report}, number = {NA/181}, year = 1998}
@techreport{FletLeyfToin98,
author = {R. Fletcher and S. Leyffer and Ph. L. Toint},
title = {On the Global Convergence of an {SLP}-Filter Algorithm},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {98/13}, year = 1998,
abstract = {A mechanism for proving global convergence in filter-type
methods for nonlinear programming is described. Such
methods are characterized by their use of the dominance
concept of multiobjective optimization, instead of a
penalty parameter whose adjustment can be problematic. The
main interest is to demonstrate how convergence for NLP can
be induced without forcing sufficient descent in a
penalty-type merit function. The proof technique is
presented in a fairly basic context, but the ideas involved
are likely to be more widely applicable. The technique
allows a wide range of specific algorithmic choices
associated with updating the trust-region radius and with
feasibility restoration.},
summary = {A mechanism for proving global convergence in filter-type
trust-region methods for nonlinear programming is
described. The main interest is to demonstrate how global
convergence can be induced without forcing sufficient
descent in a penalty-type merit function. The technique of
proof allows a wide range of specific algorithmic choices
associated with updating the trust-region radius and with
feasibility restoration.}}
@techreport{FletLeyfToin00,
author = {R. Fletcher and S. Leyffer and Ph. L. Toint},
title = {On the Global Convergence of an {SQP}-Filter Algorithm},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {???}, year = 2000,
abstract = {A mechanism for proving global convergence in filter-type
methods for nonlinear programming is described. Such
methods are characterized by their use of the dominance
concept of multiobjective optimization, instead of a
penalty parameter whose adjustment can be problematic. The
main interest is to demonstrate how convergence for NLP can
be induced without forcing sufficient descent in a
penalty-type merit function. The proof technique is
presented in a fairly basic context, but the ideas involved
are likely to be more widely applicable. The technique
allows a wide range of specific algorithmic choices
associated with updating the trust-region radius and with
feasibility restoration.},
summary = {A mechanism for proving global convergence in filter-type
trust-region methods for nonlinear programming is
described. The main interest is to demonstrate how global
convergence can be induced without forcing sufficient
descent in a penalty-type merit function. The technique of
proof allows a wide range of specific algorithmic choices
associated with updating the trust-region radius and with
feasibility restoration.}}
@techreport{FletGoulLeyfToin99,
author = {R. Fletcher and N. I. M. Gould and
S. Leyffer and Ph. L. Toint},
title = {Global Convergence of Trust-Region {SQP}-Filter Algorithms
for Nonlinear Programming},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {99/03}, year = 1999,
abstract = {Global convergence to first-order critical points is
proved for two trust-region SQP-filter algorithms of
the type introduced by \citebb{FletLeyf97}. The algorithms
allow for an approximate solution of the quadratic
subproblem and incorporate the safeguarding tests
described in \citebb{FletLeyfToin98}. The first algorithm
decomposes the step into its normal and tangential
components, while the second replaces this decomposition
by a stronger condition on the associated model decrease.},
summary = {Global convergence to first-order critical points is
proved for two trust-region SQP-filter algorithms of
the type introduced by \citebb{FletLeyf97}. The algorithms
allow for an approximate solution of the quadratic
subproblem and incorporate the safeguarding tests
described in \citebb{FletLeyfToin98}. The first algorithm
decomposes the step into its normal and tangential
components, while the second replaces this decomposition
by a stronger condition on the associated model decrease.}}
@article{FletWats80,
author = {R. Fletcher and G. A. Watson},
title = {First and second order conditions for a class of
nondifferentiable optimization problems},
journal = MP,
volume = 18, number = 3, pages = {291--307}, year = 1980}
@article{FlipJans96,
author = {O. E. Flippo and B. Jansen},
title = {Duality and Sensitivity in nonconvex quadratic Optimization
over an Ellipsoid},
journal = {European Journal of Operational Research},
volume = 94, number = 1, pages = {167--178}, year = 1996,
abstract = {In this paper, a duality framework is discussed for the
problem of optimizing a nonconvex quadratic function over
an ellipsoid. Additional insight is obtained from the
observation that this nonconvex problem is in a sense
equivalent to a convex problem of the same type, from which
known necessary and sufficient conditions for optimality
readily follow. Based on the duality results, some existing
solution procedures are interpreted as in fact solving the
dual. The duality relations are also shown to provide a
natural framework for sensitivity analysis.},
summary = {A duality framework for the problem of optimizing a
non-convex quadratic function over an ellipsoid is
described. Additional insight is obtained by observing that
this non-convex problem is in a sense equivalent to a convex
problem of the same type, from which known necessary and
sufficient conditions for optimality readily follow. Based
on the duality results, some existing solution procedures
are interpreted as in fact solving the dual. The duality
relations also provide a natural framework for
sensitivity analysis.}}
@article{Font90,
author = {R. Fontecilla},
title = {Inexact Secant Methods for Nonlinear Constrained
Optimization},
journal = SINUM,
volume = 27, number = 1, pages = {154--165}, year = 1990}
@article{FontSteiTapi87,
author = {R. Fontecilla and T. Steihaug and R. A. Tapia},
title = {A convergence theory for a class of quasi-{N}ewton methods
for constrained optimization},
journal = SINUM,
volume = 24, number = 5, pages = {1133--1151}, year = 1987}
@book{FortGlow82,
author = {M. Fortin and R. Glowinski},
title = {M\'ethodes de {L}agrangien Augment\'e},
publisher = {Dunod}, address = {Paris, France},
number = 9,
series = {M\'{e}thodes math\'{e}matiques de l'informatique},
year = 1982}
@phdthesis{Fort00,
author = {Ch. Fortin},
title = {A Survey of the Trust-Region Subproblem within a
Semidefinite Framework},
school = {University of Waterloo},
address = {Waterloo, Ontario, Canada},
year = 2000,
abstract ={Trust region subproblems arise within a class of
unconstrained methods called trust region methods. The
subproblems consist of minimizing a quadratic function
subject to a norm constraint. This thesis is a survey of
different methods developed to find an approximate solution
to the subproblem. We study the well-known method of
\citebb{MoreSore83} and two recent methods for large sparse
problems: the so-called Lanczos method of
\citebb{GoulLuciRomaToin99} and the \citebb{RendWolk97}
algorithm. The common ground to explore these methods will
be semi-definite programming. This approach has been used
by \citebb{RendWolk97} to explain their method and the
Mor\'{e}-Sorensen algorithm; we extend this work to the
Lanczos method. The last chapter of this thesis is
dedicated to some improvements done to the Rendl-Wolkowicz
algorithm and the comparisons between the Lanczos method
and the Rendl and Wolkowicz algorithm. In particular, we
show some weakness of the Lanczos method and show that the
Rendl-Wolkowicz algorithm is more robust.},
summary = {A survey of different methods developed to find an
approximate solution to the trust-region subproblem is
presented. The well-known method of \citebb{MoreSore83}
and two recent methods for large sparse problems, namely
the so-called Lanczos method of \citebb{GoulLuciRomaToin99}
and the \citebb{RendWolk97} algorithm, are studied. The
common ground to explore these methods is semi-definite
programming. This approach has been used by
\citebb{RendWolk97} to explain their method and the
Mor\'{e}-Sorensen algorithm; this work is extended to the
Lanczos method. Some improvements to the Rendl-Wolkowicz
algorithm are also described and the Lanczos method and
the Rendl and Wolkowicz algorithm compared. Some weakness
of the Lanczos method is discussed and the Rendl-Wolkowicz
algorithm is argued to be more robust.}}
@article{FourMehr93,
author = {R. Fourer and S. Mehrotra},
title = {Solving symmetrical indefinite systems for an
interior-point method for linear programming},
journal = MPA,
volume = 62, number = 1, pages = {15--39}, year = 1993}
@article{FoxHallSchr78,
author = {P. A. Fox and A. D. Hall and N. L. Schryer},
title = {The {PORT} mathematical subroutine library},
journal = TOMS,
volume = 4, number = 2, pages = {104--126}, year = 1978}
@techreport{Fral89,
author = {C. Fraley},
title = {Software Performance on Nonlinear Least-Squares Problems},
institution = STANFORD, address = STANFORD-ADDRESS,
number = {CS-TR-89-1244}, year = 1989,
abstract = {This paper presents numerical results for a large and
varied set of problems using software that is widely
available and has undergone extensive testing. The
algorithms implemented in this software include
Newton-based linesearch and trust-region methods for
unconstrained optimization, as well as Gauss-Newton,
Levenberg-Marquardt, and special quasi-Newton methods for
nonlinear least squares. Rather than give a critical
assessment of the software itself, our original purpose was
to use the best available software to compare the
underlying algorithms, to identify classes of problems for
each method on which the performance is either very good or
very poor and to provide benchmarks for future work in
nonlinear least squares and unconstrained optimization. The
variability in the results made it impossible to meet
either of the first two goals; however the results are
significant as a step toward explaining why these aims are
so difficult to accomplish.},
summary = {Numerical results are presented for a large set of problems
using software that is widely available and has undergone
extensive testing. The algorithms implemented include
Newton-based linesearch and trust-region methods for
unconstrained optimization, as well as Gauss-Newton,
Levenberg-Morrison-Marquardt, and special quasi-Newton
methods for nonlinear least-squares. Rather than give a
critical assessment of the software itself, the original
intention was to use the best available software to compare
the underlying algorithms, to identify classes of problems
for each method on which the performance is either very
good or very poor and to provide benchmarks for future work
in nonlinear least-squares and unconstrained optimization.
The variability in the results makes it impossible to meet
either of the first two goals; however the results are
significant as a step toward explaining why these aims are
so difficult to accomplish.}}
@article{FranWolf56,
author = {M. Frank and P. Wolfe},
title = {An algorithm for quadratic programming},
journal = {Naval Research Logistics Quarterly},
volume = 3, pages = {95--110}, year = 1956}
@article{Freu91,
author = {R. M. Freund},
title = {Theoretical efficiency of a shifted-barrier-function
algorithm for linear programming},
journal = LAA,
volume = 152, pages = {19--41}, year = 1991}
@article{FrieMart94,
author = {A. Friedlander and J. M. Mart\'{\i}nez},
title = {On the Maximization of a Concave Quadratic Function with
Box Constraints},
journal = SIOPT,
volume = 4, number = 1, pages = {177--192}, year = 1994}
@article{FrieMartRayd95,
author = {A. Friedlander and J. M. Mart\'{\i}nez and M. Raydan},
title = {A new method for large-scale box constrained convex
quadratic minimization problems},
journal = OMS,
volume = 5, number = 1, pages = {57--74}, year = 1995}
@article{FrieMartSant94,
author = {A. Friedlander and J. M. Mart\'{\i}nez and S. A. Santos},
title = {A new Trust Region Algorithm for Bound Constrained
Minimization},
journal = {Applied Mathematics and Optimization},
volume = 30, number = 3, pages = {235--266}, year = 1994,
abstract = {A new method for maximizing a concave quadratic function
with bounds on the variables is described. The new
algorithm combines conjugate gradients with gradient
projection techniques, as the algorithm of
\citebb{MoreTora91} and other well-known method do. A new
strategy for the decision of leaving the current face is
introduced that makes it possible to obtain finite
convergence even for a singular Hessian and in the presence
of dual degeneracy. Numerical experiments are presented.},
summary = {A method for maximizing a concave quadratic function with
bounds on the variables is described, which combines
conjugate gradients with gradient projection techniques, as
in \citebb{MoreTora91}. A strategy for the decision of
leaving the current face is introduced that ensures finite
convergence even for a singular Hessian and in the presence
of dual degeneracy. Numerical experiments are presented.}}
@article{FrieMartSant94b,
author = {A. Friedlander and J. M. Mart\'{\i}nez and S. A. Santos},
title = {On the resolution of linearly constrained convex
minimization problems},
journal = SIOPT,
volume = 4, number = 2, pages = {331--339}, year = 1994,
abstract = {The problem of minimizing a twice continuously
differentiable convex function $f$ is considered, subject
to $Ax=b$, $x \geq 0$, where $A \in \Re^{m \times n}$, $m$,
$n$ are large and the feasible region is bounded. It is
proven that this problem is equivalent to a ``primal-dual''
box-constrained problem with $2n+m$ variables. This problem
is solved using an algorithm for bound constrained
minimization that can deal with many variables. Numerical
experiments are presented.},
summary = {The problem of minimizing a twice continuously
differentiable convex function $f$ is considered, subject
to $Ax=b$, $x \geq 0$, where $A \in \Re^{m \times n}$, $m$,
$n$ are large and the feasible region is bounded. It is
proven that this problem is equivalent to a ``primal-dual''
box-constrained problem with $2n+m$ variables. This problem
is solved using an algorithm for bound constrained
minimization that can deal with many variables. Numerical
experiments are presented.}}
@techreport{Fris54,
author = {K. R. Frisch},
title = {Principles of Linear Programming---With Particular
Reference to the Double Gradient Form of the Logarithmic
Potential Method},
institution = {University Institute for Economics}, address = {Oslo},
type = {Memorandum of October 18}, year = 1954}
@techreport{Fris55,
author = {K. R. Frisch},
title = {The Logarithmic Potential Function for Convex Programming},
institution = {University Institute for Economics}, address = {Oslo},
type = {Memorandum of May 13}, year = 1955}
@mastersthesis{Fugg96,
author = {P. Fugger},
title = {Trust region subproblems and comparison with quasi-{N}ewton
methods},
school = {Technical University of Graz}, address = {Graz, Austria},
year = 1996,
abstract = {In the field of unconstrained nonlinear optimization
Quasi-Newton methods (e.g. the DFP and BFGS formulae) are
well-known and extensively studied in the literature.
Another approach to the minimization of nonlinear problems
is done by the so-called restricted step or trust region
methods. One possibility for solving the Trust Region
subproblem (TRS) is to apply a parametric eigenvalue
problem (Rendl's method). The choice of the parameter of
Rendl's method is decisive for the number of iterations
needed to obtain a minimizer. The computational amount of
the method is compared with the costs of the DFP algorithm},
summary = {One possibility for solving the trust-region subproblem is
to apply a parametric eigenvalue problem (Rendl's method).
The choice of the parameter of Rendl's method is decisive
for the number of iterations needed to obtain a minimizer.
The computational efficiency of this method is compared
with that of the DFP algorithm}}
@article{FujiKojiNaka97,
author = {K. Fujisawa and M. Kojima and K. Nakata},
title = {Exploiting sparsity in primal-dual interior-point methods
for semidefinite programming},
journal = MPB,
volume = 79, number = {1--3}, pages = {235--253}, year = 1997}
@article{Fuku86b,
author = {M. Fukushima},
title = {A successive quadratic-programming algorithm with global
and superlinear convergence properties},
journal = MP,
volume = 35, number = 3, pages = {253--264}, year = 1986}
@article{Fuku92,
author = {M. Fukushima},
title = {Equivalent differentiable optimization problems and descent
methods for asymmetric variational inequality problems},
journal = MPA,
volume = 53, number = 1, pages = {99--110}, year = 1992}
@inproceedings{Fuku96,
author = {M. Fukushima},
title = {Merit functions for variational inequality and
complementarity problems},
crossref = {DiPiGian96}, pages = {155--170}}
@article{FukuHaddvanStroSugiYama96,
author = {M. Fukushima and M. Haddou and Nguyen, H. v. and
J. J. Strodiot and T. Sugimoto and E. Yamakawa},
title = {A parallel descent algorithm for convex programming},
journal = COAP,
volume = 5, number = 1, pages = {5--37}, year = 1996,
abstract = {In this paper, we propose a parallel decomposition
algorithm for solving a class of convex optimization
problems, which is broad enough to contain ordinary
convex programming problems with a strongly convex
objective function. The algorithm is a variant of the
trust region method applied to the Fenchel dual of
the given problem. We prove global convergence of the
algorithm and report some computational experience
with the proposed algorithm on the Connection Machine
Model CM-5.},
summary = {A parallel decomposition algorithm for solving a class
of convex optimization problem is proposed that contains
convex programming problems with a strongly convex
objective function. The algorithm is a variant of the
trust-region method applied to the Fenchel dual of the
problem. We prove global convergence and report
computational experience on the Connection Machine Model
CM-5.}}
@article{FukuYama86,
author = {M. Fukushima and Y. Yamamoto},
title = {A second-order algorithm for continuous-time nonlinear
optimal control problems},
journal = {IEEE Transactions on Automatic Control},
volume = {AC-31}, number = 7, pages = {673--676}, year = 1986,
abstract = {A second-order algorithm is presented for the
solution of continuous-time nonlinear optimal control
problems. The algorithm is an adaptation of the trust
region modifications of Newton's method and solves at
each iteration a linear-quadratic control problem
with an additional constraint. Under some
assumptions, the proposed algorithm is shown to
possess a global convergence property. A numerical
example illustrates the method.},
summary = {A second-order algorithm is presented for the
solution of continuous-time nonlinear optimal control
problems. The algorithm is a trust-region variant of
Newton's method and solves at each iteration a
linear-quadratic control problem with an additional
constraint. The algorithm is globally convergent. A
numerical example illustrates the method.}}
@techreport{FuLuoYe96,
author = {M. Fu and Z. Q. Luo and Y. Ye},
title = {Approximation algorithms for quadratic programming},
institution = {Department of Management Science, University of Iowa},
year = 1996,
abstract = {We consider the problem of approximating the global minimum
of a general quadratic program (QP) with $n$ variables
subject to $m$ ellipsoidal constraints. For $m=1$, we
rigorously show that and $\epsilon$-minimizer, where error
$\epsilon \in (0,1)$, can be obtained in polynomial time,
meaning that the number of arithmetic operations is a
polynomial in $n$, $m$ and $\log(1/\epsilon)$. For $m \geq
2$, we present a polynomial-time $(1-
\frac{1}{m^2})$-approximation algorithm as well as a
semidefinite programming relaxation for this problem. In
addition, we present approximation algorithms for solving
QP under the box constraints and the assignment polytope
constraints.},
summary = {The problem of approximating the global minimum of a
general quadratic program (QP) with $n$ variables subject
to $m$ ellipsoidal constraints is considered. For $m=1$, it
is shown that and $\epsilon$-minimizer, where error
$\epsilon \in (0,1)$, can be obtained in polynomial time,
meaning that the number of arithmetic operations is a
polynomial in $n$, $m$ and $\log(1/\epsilon)$. For $m \geq
2$, a polynomial-time $(1- \frac{1}{m^2})$-approximation
algorithm is presented, as well as a semidefinite
programming relaxation for this problem. In addition,
approximation algorithms for solving QP under the box
constraints and the assignment polytope constraints are
given.}}
@article{Fure93,
author = {B. P. Furey},
title = {A sequential quadratic programming-based algorithm for
optimization of gas networks},
journal = {Automatica},
volume = 29, number = 6, pages = {1439-1450}, year = 1993,
abstract = {British Gas uses a complex, heavily looped network of
pipes and controllable units (compressors and
regulators) to transmit gas from coastal supply
terminals to regional demand points. Computer
algorithms are required for efficient management of
the system. This paper describes an algorithm for
optimal control over periods of up to a day. The
problem is large scale and highly nonlinear in both
objective function and constraints. The method is
based on sequential quadratic programming and takes
account of the structure of the pipeflow equations by
means of a reduced gradient technique which
eliminates most of the variables from the quadratic
subproblems. The latter involve only simple bound
constraints, which are handled efficiently by a
conjugate gradient-active set algorithm. Trust region
techniques permit use of the exact Hessian,
preserving sparsity. More general constraints are
handled at an outer level by a truncated augmented
Lagrangian method. Results are included for some
realistic problems. The algorithm is generally
applicable to problems with a control structure.},
summary = {An algorithm for optimal control of the British Gas
network of pipes and controllable units over periods
of up to a day is described. The
problem is large-scale and highly nonlinear in both
objective function and constraints. The method is
based on sequential quadratic programming and takes
account of the structure of the pipeflow equations by
means of a reduced gradient technique. which
eliminates most of the variables from the quadratic
subproblems. The latter involve only simple bound
constraints, which are handled efficiently by a
conjugate gradient-active set algorithm. Trust-region
techniques permit use of the exact Hessian,
preserving sparsity. More general constraints are
handled at an outer level by a truncated augmented
Lagrangian method. Results are included for some
realistic problems.}}
%%% G %%%
@techreport{GabrMore95,
author = {S. A. Gabriel and J. J. Mor\'{e}},
title = {Smoothing of mixed complementarity problems},
institution = ANL, address = ANL-ADDRESS,
number = {MCS-P541-00995}, year = 1995}
@inproceedings{GabrPang94,
author = {S. A. Gabriel and J. S. Pang},
title = {A trust region method for constrained nonsmooth equations},
crossref = {HageHearPard94}, pages = {155--181},
abstract = {In this paper, we develop and analyze the convergence of a
fairly general trust region method for solving a system of
nonsmooth equations subject to some linear constraints. The
method is based on the existence of an iteration function
for the nonsmooth equations and involves the solution of a
sequence of subproblems defined by this function. A
particular realization of the method leads to an
arbitrary-norm trust region method. Applications of the
latter method to the nonlinear complementarity and related
problems are discussed. Sequential convergence of the
method and its rate of convergence are established under
certain regularity conditions similar to those used in the
NE/SQP method and its generalization. Some computational
results are reported.},
summary = {The convergence of a trust-region method for solving a
system of non-smooth equations subject to linear
constraints is considered. The method is based on the
existence of an iteration function for the non-smooth
equations and involves the solution of a sequence of
subproblems defined by this function. A particular
realization of the method leads to an arbitrary-norm
trust-region method. Applications of the latter method to
the nonlinear complementarity and related problems are
discussed. Global convergence of the method and its rate of
convergence are established under certain regularity
conditions similar to those used in the NE/SQP method and
its generalization. Computational results are reported.}}
@phdthesis{Gand78,
author = {W. Gander},
title = {On the linear least squares problem with a quadratic
constraint},
school = {Computer Science Department, Stanford University},
address = {Stanford, California, USA},
number = {STAN-CS-78-697}, year = 1978}
@article{Gand81,
author = {W. Gander},
title = {Least squares with a quadratic constraint},
journal = NUMMATH,
volume = 36, pages = {291--307}, year = 1981,
abstract = {We present the theory of the linear least squares problem
with a quadratic constraint. New theorems characterizing
properties of the solutions are given. A numerical
application is discussed.},
summary = {Properties of the solutions of the linear least-squares
problem with a quadratic constraint are given and a
numerical applcation discussed. The paper summarizes
\citebb{Gand78}}}
@inproceedings{Gao98,
author = {L. Gao},
title = {Using {H}uber Method to solve $L_1$-norm problem},
crossref = {Yuan98}, pages = {263--272},
abstract = {The nondifferentiable $L_1$ function is approximated by the
Huber function, such that the original $L_1$ estimation
problem is transformed to a sequence of unconstrained
minimization problems. An algorithm is consider for the
Huber problem. Numerical experiments are reported and
comparisons within different methods are made},
summary = {The non-differentiable $\ell_1$ estimation problem is
replaced by a sequence of smooth minimization problems
using the Huber function to approximate the absolute value.
An algorithm is proposed that uses a trust-region method
to solve each of the subproblems. Numerical comparisons are
made with competing approaches.}}
@book{GareJohn79,
author = {M. R. Garey and D. S. Johnson},
title = {Computers and Intractibility},
publisher = FREEMAN, address = FREEMAN-ADDRESS,
year = 1979}
@article{Garf90,
author = {E. Garfield},
title = {The most cited papers of all time, {SCI} 1945--1988.
{P}art 3.
{A}nother 100 from the Citation Classics Hall of Fame},
journal = {Current Contents},
volume = 34, pages = {3--13}, year = 1990}
@article{Gay81,
author = {D. M. Gay},
title = {Computing optimal locally constrained steps},
journal = SISSC,
volume = 2, pages = {186--197}, year = 1981,
abstract = {In seeking to solve an unconstrained minimization problem,
one often computes steps based on a quadratic approximation
$q$ to the objective function. A reasonable way to choose
such steps is by minimizing $q$ constrained to a
neighbourhood of the current iterate. This paper considers
ellipsoidal neighbourhoods and presents a new way to handle
certain computational details when the Hessian of $q$ is
indefinite, paying particular attention to a special case
which may then arise. The proposed step computing algorithm
provides an attractive way to deal with negative curvature.
Implementations of this algorithm have proved very
satisfactory in the nonlinear least-squares solver NL2SOL.},
summary = {The solution of the trust-region subproblem with
ellipsoidal norms is considered and a way to handle
certain computational details when the Hessian of $q$ is
indefinite is presented, paying particular attention to the
hard case. The proposed step computing algorithm provides
an attractive way to deal with negative curvature.
Implementations of the algorithm have proved very
satisfactory in the nonlinear least-squares solver NL2SOL.}}
@techreport{Gay82,
author = {D. M. Gay},
title = {On the convergence in Model/Trust-Region Algorithms for
Unconstrained Optimization},
institution = BELLLABS, address = BELLLABS-ADDRESS,
type = {Computing Science Technical Report}, number = 104,
year = 1982,
abstract = {This paper discusses convergence tests in the context of
model/trust-region algorithms for solving unconstrained
optimization problems. It presents a general theorem that
supports a diagnostic test for possible convergence to a
minimizer at which the Hessian of the objective function is
singular. (This is an event with which one must be prepared
to deal in problems that arise from fitting a mathematical
model to data.) Given just continuity of the objective
function's gradient, the theorem assures that either the
``singular convergence'' test is satisfied infinitely often
for any positive convergence tolerance, or else the lengths
of the steps taken tend to zero; moreover, if the model
Hessians are locally bounded, then any limit point of the
iterates is a critical point. One can use this information
in a suite of convergence tests that in my opinion involve
easily understood tolerances and provide helpful
diagnostics.},
summary = {Convergence tests in the context of model/trust-region
algorithms for solving unconstrained optimization problems
are discussed. A general theorem that supports a diagnostic
test for possible convergence to a minimizer at which the
Hessian of the objective function is singular is given. If
the gradient of the objective function is continuous, the
theorem assures that either the ``singular convergence''
test is satisfied infinitely often for any positive
convergence tolerance, or else the lengths of the steps
taken tend to zero; moreover, if the model Hessians are
locally bounded, then any limit point of the iterates is a
critical point. This information can be used in a suite of
convergence tests that involve easily understood tolerances
and which provide helpful diagnostics.}}
@article{Gay83b,
author = {D. M. Gay},
title = {Algorithm 611: subroutines for unconstrained minimization
using a model/trust-region approach},
journal = TOMS,
volume = 9, number = 4, pages = {503--524}, year = 1983,
summary = {A set of subroutines for the minimization of a smooth
function are provided. These codes work with exact or
finite-difference gradients, and exact or secant
approximations to Hessians, using the reverse-communication
paradigm. An approximation to the trust-region subproblem
is computed using the double-dogleg technique of
\citebb{DennMei79}.}}
@inproceedings{Gay84,
author = {D. M. Gay},
title = {A trust region approach to linearly constrained
optimization},
booktitle = {Numerical Analysis: Proceedings Dundee 1983},
editor = {D. F. Griffiths},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
pages = {72--105}, year = 1984,
note = {Lecture Notes in Mathematics 1066},
abstract = {This paper suggests a class of trust-region algorithms for
solving linearly constrained optimization problems. The
algorithms use a ``local'' active-set strategy to select
the steps they try. This strategy is such that degeneracy
and zero Lagrange multipliers do not slow convergence (to a
first-order stationary point) and that no anti-zigzagging
precautions are necessary. (Unfortunately, when there are
zero Lagrange multipliers, convergence to a point failing
to satisfy second-order necessary conditions remains
possible.) We discuss specialization of the algorithms to
the case of simple bounds on the variables and report
preliminary computational experience.},
summary = {A class of trust-region algorithms for solving linearly
constrained optimization problems is suggested. The
algorithms use a ``local'' active-set strategy to select
their steps. This strategy is such that degeneracy and zero
Lagrange multipliers do not slow convergence (to a
first-order stationary point) and that no anti-zigzagging
precautions are necessary---when there are zero Lagrange
multipliers, convergence to a point failing to satisfy
second-order necessary conditions remains possible.
Specialization of the algorithms to the case of simple
bounds on the variables are discussed, and preliminary
computational experience reported.}}
@inproceedings{GayOverWrig98,
author = {D. M. Gay and M. L. Overton and M. H. Wright},
title = {A Primal-Dual Interior Method for Nonconvex Nonlinear
Programming},
crossref = {Yuan98}, pages = {31--56}}
@article{GermToin99,
author = {M. Germain and Ph. L. Toint},
title = {An Iterative Process for International Negotiations
on Acid Rain in Northern Europe Using a General Convex
Formulations},
journal = {Environmental and Resource Economics},
note = {To appear.}, year = 1999,
abstract = {This paper proposes a game theoretical approach of
international negotiations on transboundary pollution.
This approach is distinguished by a discrete time
formulation and by a suitable formulation of the local
information assumption on cost and damage functions: at
each stage of the negotiation, the parties assign the best
possible cooperative state, given the available information,
as an objective for the next stage. It is shown that the
resulting sequences of states converges from a
non-cooperative situation to an international optimum in a
finite number of stages. Furthermore, a financial transfer
structure is also presented that makes the desired sequence
of states individually rational and strategically stable.
The concepts are applied in a numerical simulation of the
$SO_2$ transboundary pollution problem related to
acid rain in Northern Europe.},
summary = {A game theoretical approach of international negotiations
on transboundary pollution is proposed, that uses
a discrete time formulation. The resulting sequences of
states is shown to converge from a non-cooperative
situation to an international optimum in a finite number
of stages. A financial transfer structure is also presented
that makes the desired sequence of states individually
rational and strategically stable. The concepts are applied
in a numerical simulation of the $SO_2$ transboundary
pollution problem related to acid rain in Northern Europe,
using a trust-region method to calculate the optimum at
each negotiation stage.}}
@article{GeorLiu79,
author = {A. George and J. W. H. Liu},
title = {The design of a user interface for a sparse matrix package},
journal = TOMS,
volume = 5, number = 2, pages = {139--162}, year = 1979}
@book{GeorLiu81,
author = {A. George and J. W. H. Liu},
title = {Computer Solution of Large Sparse Positive Definite Systems},
publisher = PH, address = PH-ADDRESS,
year = 1981}
@phdthesis{Gert99,
author = {E. M. Gertz},
title = {Combination Trust-Region Line-Search Methods for
Unconstrained Optimization},
school = {Department of Mathematics, University of California},
address = {San Diego, California, USA},
year = 1999,
abstract = {Many important problems may be expressed in terms of
nonlinear multivariate unconstrained optimization. The
basic unconstrained optimization problem is to minimize a
real-valued function $f(x)$ over all vectors $x \in \Re^n$.
Many techniques for solving these types of problems are
available if $f$ is twice continuously differentiable.
Two broad classes of algorithms for the unconstrained
minimization problem are trst-region algorithms and
line-search algorithms. These two classes may be combined
by performing a line search in the direction proposed by the
solution to the trust-region subproblem. We develop three
combination methods which require that a sufficient
decrease condition is met at each step. The first of the
new algorithms uses a backtracking line search based on the
Armijo condition. In all these algorithms the line search
is used to control the trust-region radius. We present
strong first and second order convergence theorems for
these new methods, an analysis of their asymptotic
convergence properties and the results of numerical
experiments using the new algorithms. It is possible to
use the Wolfe-condition based algorithms to define
quasi-Newton methods which use the BFGS update. The
quasi-Newton methods are robust and efficient.},
summary = {Three unconstrained minimization methods are presented that
combine trust-region and linesearch techniques, in the
sense that a linesearch is performed in the direction
obtained from the solution of the trust-region subproblem.
The linesearch is also used to control the trust-region
radius. Strong global convergence to first- and second-order
points is proved and the asymptotic convergence properties
of the algorithms analyzed. Numerical results are presented,
that include a quasi-Newton BFGS variant of the algorithms.}}
@article{Gilb91,
author = {J. Ch. Gilbert},
title = {Maintaining the positive definiteness of the matrices in
reduced secant methods for equality constrained
optimization},
journal = MP,
volume = 50, number = 1, pages = {1--28}, year = 1991}
@article{GillGoulMurrSaunWrig84,
author = {P. E. Gill and N. I. M. Gould and W. Murray
and M. A. Saunders and M. H. Wright},
title = {A weighted {G}ram-{S}chmidt method for convex
quadratic programming},
journal = MP,
volume = 30, number = 2, pages = {176--195}, year = 1984}
@article{GillMurr74,
author = {P. E. Gill and W. Murray},
title = {{N}ewton-type methods for unconstrained and linearly
constrained optimization},
journal = MP,
volume = 7, number = 3, pages = {311--350}, year = 1974}
@techreport{GillMurr76,
author = {P. E. Gill and W. Murray},
title = {Minimization subject to bounds on the variables},
institution = NPL, address = NPL-ADDRESS,
type = {NPL Report}, number = {NAC 72}, year = 1976}
@article{GillMurr78,
author = {P. E. Gill and W. Murray},
title = {Numerically stable methods for quadratic programming},
journal = MP,
volume = 14, number = 3, pages = {349--372}, year = 1978}
@article{GillMurrPoncSaun92,
author = {P. E. Gill and W. Murray and D. B. Poncele\'{o}n and M. A.
Saunders},
title = {Preconditioners for indefinite systems arising in
optimization},
journal = SIMAA,
volume = 13, number = 1, pages = {292--311}, year = 1992}
@article{GillMurrSaunWrig83,
author = {P. E. Gill and W. Murray and M. A. Saunders
and M. H. Wright},
title = {Computing forward-difference intervals for numerical
optimization},
journal = SISSC,
volume = 4, number = 2, pages = {310-321}, year = 1983}
@article{GillMurrSaunWrig85,
author = {P. E. Gill and W. Murray and M. A. Saunders
and M. H. Wright},
title = {Some issues in implementing a sequential quadratic
programming algorithm},
journal = {SIGNUM Newsletter},
volume = 20, number = 2, pages = {13--19}, year = 1985}
@article{GillMurrSaunStewWrig85,
author = {P. E. Gill and W. Murray and M. A. Saunders and G. W.
Stewart and M. H. Wright},
title = {Properties of a representation of a basis for the null
space},
journal = MP,
volume = 33, number = 2, pages = {172--186}, year = 1985}
@article{GillMurrSaunTomlWrig86,
author = {P. E. Gill and W. Murray and M. A. Saunders and J. A.
Tomlin and M. H. Wright},
title = {On projected {N}ewton barrier methods for linear
programming and an equivalence to {K}armarkar's projective
method},
journal = MP,
volume = 36, number = 2, pages = {183--209}, year = 1986}
@techreport{GillMurrSaunWrig88,
author = {P. E. Gill and W. Murray and M. A. Saunders and M. H.
Wright},
title = {Shifted barrier methods for linear programming},
institution = STANFORD, address = STANFORD-ADDRESS,
type = {Technical Report}, number = {SOL88-9}, year = 1988}
@inproceedings{GillMurrSaunWrig90,
author = {P. E. Gill and W. Murray and M. A. Saunders
and M. H. Wright},
title = {A {S}chur-complement method for sparse
quadratic programming},
crossref = {CoxHamm90}, pages = {113--138}}
@article{GillMurrSaunWrig91,
author = {P. E. Gill and W. Murray and M. A. Saunders
and M. H. Wright},
title = {Inertia-controlling methods for general quadratic
programming},
journal = SIREV,
volume = 33, number = 1, pages = {1--36}, year = 1991}
@inproceedings{GillMurrSaunWrig92,
author = {P. E. Gill and W. Murray and M. A. Saunders and M. H.
Wright},
title = {Some theoretical properties of an augmented {L}agrangian
merit function},
booktitle = {Advances in Optimizations and Parallel Computing},
editor = {P. M. Pardalos},
publisher = {Elsevier}, address = {Amsterdam},
pages = {127--143}, year = 1992}
@book{GillMurrWrig81,
author = {P. E. Gill and W. Murray and M. H. Wright},
title = {Practical Optimization},
publisher = AP, address = AP-ADDRESS,
year = 1981}
@article{GladPola79,
author = {T. Glad and E. Polak},
title = {A multiplier method with automatic limitation of penalty
growth},
journal = MP,
volume = 17, number = 2, pages = {140--155}, year = 1979}
@article{Goem97,
author = {M. X. Goemans},
title = {Semidefinite programming in combinatorial optimization},
journal = MPB,
volume = 79, number = {1--3}, pages = {143--161}, year = 1997}
@article{Gold70,
author = {D. Goldfarb},
title = {A Family of Variable Metric Methods Derived by Variational
Means},
journal = MC,
volume = 24, pages = {23--26}, year = 1970}
@techreport{Gold80,
author = {D. Goldfarb},
title = {The Use of Negative Curvature in Minimization Algorithms},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR80-412}, year = 1980,
abstract = {In this paper we examine existing algorithms for
minimizing a nonlinear function of many variables which
make use of negative curvature. These algorithms can all be
viewed as modified versions of Newton's method and their
merits and drawbacks are discussed to help identify new and
more promising methods. The algorithms considered include
ones which compute and search along nonascent directions of
negative curvature and ones which search along curvi-linear
paths generated by these directions and descent directions.
Versions of the \citebb{GoldQuanTrot66}, or equivalently,
methods based upon a trust region strategy, and gradient
path methods are also considered. When combined with the
numerically stable \citebb{BuncParl71} factorization of a
symmetric indefinite matrix the latter two approaches give
rise to new, and what appears to be, efficient and robust
minimization methods which can take advantage of negative
curvature when it is encountered. Several suggestions are
made for further research in this area.},
summary = {Algorithms for minimizing a nonlinear function of many
variables which make use of negative curvature are
examined. These algorithms can all be viewed as modified
versions of Newton's method and their merits and drawbacks
are discussed to help identify new and more promising
methods. The algorithms considered include ones which
compute and search along non-ascent directions of negative
curvature and ones which search along curvilinear paths
generated by these directions and descent directions.
Versions of the \citebb{GoldQuanTrot66}, or equivalently,
methods based upon a trust-region strategy, and gradient
path methods are also considered. When combined with the
numerically stable \citebb{BuncParl71} factorization of a
symmetric indefinite matrix the latter two approaches give
rise to efficient and robust minimization methods
which can take advantage of negative curvature.}}
@article{GoldLiuWang91,
author = {D. Goldfarb and S. C. Liu and S. Wang},
title = {A Logarithmic Barrier Function Algorithm for Quadratically
Constrained Convex Quadratic Programs},
journal = SIOPT,
volume = 1, number = 2, pages = {252--267}, year = 1991}
@article{GoldWang93,
author = {D. Goldfarb and S. Wang},
title = {Partial-update {N}ewton methods for unary, factorable
and partially separable optimization},
journal = SIOPT,
volume = 3, number = 2, pages = {383--397}, year = 1993}
@article{GoldIdna83,
author = {D. Goldfarb and A. Idnani},
title = {A numerically stable dual method for solving strictly convex
quadratic programs},
journal = MP,
volume = 27, number = 1, pages = {1--33}, year = 1983}
@article{GoldLiu93,
author = {D. Goldfarb and S. C. Liu},
title = {An $O(n^3 {L})$ primal dual potential reduction algorithm for
solving convex quadratic programs},
journal = MP,
volume = 61, number = 2, pages = {161--170}, year = 1993}
@article{GoldToin84,
author = {D. Goldfarb and Ph. L. Toint},
title = {Optimal Estimation of {J}acobian and {H}essian Matrices
That Arise in Finite Difference Calculations},
journal = MC,
volume = 43, number = 167, pages = {69--88}, year = 1984}
@article{GoldQuanTrot66,
author = {S. M. Goldfeldt and R. E. Quandt and H. F. Trotter},
title = {Maximization by quadratic hill-climbing},
journal = {Econometrica},
volume = 34, pages = {541--551}, year = 1966,
abstract = {The purpose of this paper is to describe a new gradient
method for maximizing general functions. After a brief
discussion of various known gradient methods the
mathematical foundation is laid for the new algorithm which
rests on maximizing a quadratic approximation to the
function on a suitably chosen spherical region. The method
requires no assumptions about the concavity of the function
to be maximized and automatically modifies the step size in
the light of the success of the quadratic approximation to
the function. The paper further discusses some practical
problems of implementing the algorithm and presents recent
computational experience with it.},
summary = {A gradient method for maximizing general functions is
discussed. After a brief discussion of various known
gradient methods the mathematical foundation is laid for
the algorithm which rests on maximizing a quadratic
approximation to the function on a suitably chosen
spherical region. The method requires no assumptions about
the concavity of the function to be maximized and
automatically modifies the step size in the light of the
success of the quadratic approximation to the function.
Practical problems of implementing the algorithm are
discussed, and computational experience presented.}}
@article{Gold64,
author = {A. A. Goldstein},
title = {Convex programming in {H}ilbert space},
journal = {Bull. Amer. Math. Soc.},
volume = 70, pages = {709--710}, year = 1964}
@article{GoluOLea89,
author = {G. H. Golub and D. P. O'Leary},
title = {Some history of the conjugate gradient and {L}anczos
methods},
journal = SIREV,
volume = 31, number = 1, pages = {50--102}, year = 1989}
@book{GoluvanL89,
author = {G. H. Golub and Van Loan, C. F.},
title = {Matrix Computations},
publisher = {Johns Hopkins University Press}, address = {Baltimore},
edition = {second}, year = 1989}
@article{GoluvonM91,
author = {G. H. Golub and U. von Matt},
title = {Quadratically constrained least squares and quadratic
problems},
journal = NUMMATH,
volume = 59, pages = {561--580}, year = 1991}
@article{GomeMaciMart99,
author = {F. A. M. Gomes and M. C. Maciel and J. M. Mart\'{\i}nez},
title = {Nonlinear Programming algorithms using trust regions and
augmented {L}agrangians with nonmonotone penalty parameters},
journal = MP,
volume = 84, number = 1, pages = {161--200}, year = 1999,
abstract = {A model algorithm based on the successive quadratic
programming method for solving the general nonlinear
programming problem is presented. The objective function
and the constraints of the problem are only required to be
differentiable and their gradients to satisfy a Lipschitz
condition. The strategy for obtaining global convergence is
based on the trust region approach. The merit function is a
type of augmented Lagrangian. A new updating scheme is
introduced for the penalty parameter, by means of which
monotone increase is not necessary. Global convergence
results are proved and numerical experiments are presented.},
summary = {An algorithm based on the SQP method for solving the
general nonlinear programming problem is presented. The
objective function and the constraints of the problem are
only required to be differentiable and their gradients to
satisfy a Lipschitz condition. The strategy for obtaining
global convergence is based on a trust-region.
The merit function is a type of augmented Lagrangian. An
updating scheme is introduced for the penalty parameter, by
means of which monotone increase is not necessary. Global
convergence results are proved and numerical experiments
are presented.}}
@article{Gonz91,
author = {C. C. Gonzaga},
title = {An interior trust region method for linearly constrained
optimization},
journal = {COAL Newsletter},
volume = 19, pages = {55--66}, year = 1991,
summary = {The link between the ellipsoids associated with
interior-point methods scaling and trust regions is
exposed.}}
@article{GonzTapiPotr98,
author = {M. D. Gonzalez{-}Lima and R. A. Tapia and F. A. Potra},
title = {On Effectively Computing the Analytic Center of the
Solution Set by Primal-Dual Interior-Point Methods},
journal = SIOPT,
volume = 8, number = 1, pages = {1--25}, year = 1998}
@article{GopaBieg97,
author = {V. Gopal and L. T. Biegler},
title = {Nonsmooth dynamic simulation with linear programming based
methods},
journal = {Computers and Chemical Engineering},
volume = 21, number = 7, pages = {675--689}, year = 1997,
abstract = {Process simulation has emerged as a valuable tool for
process design, analysis and operation. In this work, we
extend the capabilities of iterated linear programming (LP)
for dealing with problems encountered in dynamic nonsmooth
process simulation. A previously developed LP method is
refined with the addition of a new descent strategy which
combines line search with a trust region approach. This
adds more stability and efficiency to the method. The LP
method has the advantage of naturally dealing with profile
bounds as well. This is demonstrated to avoid the
computational difficulties which arise from the iterates
going into physically unrealistic regions. A new method for
the treatment of discontinuities occurring in dynamic
simulation problems is also presented in this paper. The
method ensures that any event which has occurred within the
time interval in consideration is detected and if more than
one event occurs, the detected one is indeed the earliest
one. A specific class of implicitly discontinuous process
simulation problems, phase equilibrium calculations, is
also examined. A new formulation is introduced to solve
multiphase problems.},
summary = {A previously developed LP method is refined with the
addition of a descent strategy which combines line
search with a trust-region approach. The LP method has
the advantage of naturally dealing with additional profile
bounds. A method for the treatment of discontinuities
occurring in dynamic simulation problems is also presented.
The method ensures that any event which has occurred within
the time interval in consideration is detected and if more
than one event occurs, the detected one is indeed the
earliest one. A specific class of implicitly discontinuous
process simulation problems, phase equilibrium
calculations, is also examined. A formulation is
introduced to solve multiphase problems.}}
@article{GopaBieg98,
author = {V. Gopal and L. T. Biegler},
title = {A successive linear programming approach for initialization
and reinitialization after discontinuities of
differential-algebraic equations },
journal = SISC,
volume = 20, number = 2, pages = {447--467}, year = 1998,
abstract = {Determination of consistent initial conditions is an
important aspect of the solution of differential-algebraic
equations (DAEs). Specification of inconsistent initial
conditions, even if they are only slightly
inconsistent, often leads to a failure in the
initialization problem. We present a successive
linear programming (SLP) approach for the solution of
the DAE derivative array equations for the
initialization problem. The SLP formulation handles
roundoff errors and inconsistent user specifications,
among other things, and allows for reliable
convergence strategies that incorporate variable
bounds and trust region concepts. A new consistent
set of initial conditions is obtained by minimizing
the deviation of the variable values from the
specified ones. For problems with discontinuities
caused by a step change in the input functions, a new
criterion is presented for identifying the subset of
variables which are continuous across the
discontinuity. The SLP formulation is then applied to
determine a consistent set of initial conditions for
further solution of the problem in the domain after
the discontinuity. Numerous example problems are
solved to illustrate these concepts.},
summary = {Determination of consistent initial conditions is an
important aspect of the solution of differential-algebraic
equations (DAEs). Specification of inconsistent initial
conditions, even if they are only slightly
inconsistent, often leads to a failure in the
initialization problem. A successive linear
programming (SLP) approach for the solution of the
DAE derivative array equations for the initialization
problem is proposed. The SLP formulation handles
roundoff errors and inconsistent user specifications,
among other things, and allows for reliable
convergence strategies that incorporate variable
bounds and trust region concepts. A new consistent
set of initial conditions is obtained by minimizing
the deviation of the variable values from the
specified ones. For problems with discontinuities
caused by a step change in the input functions, a new
criterion is presented for identifying the subset of
variables which are continuous across the
discontinuity. The SLP formulation is then applied to
determine a consistent set of initial conditions for
further solution of the problem in the domain after
the discontinuity. Numerous example problems are
solved to illustrate these concepts.}}
@article{Goul85,
author = {N. I. M. Gould},
title = {On practical conditions for the existence and uniqueness of
solutions to the general equality quadratic-programming
problem},
journal = MP,
volume = 32, number = 1, pages = {90--99}, year = 1985}
@article{Goul86,
author = {N. I. M. Gould},
title = {On the accurate determination of search directions for
simple differentiable penalty functions},
journal = IMAJNA,
volume = 6, pages = {357--372}, year = 1986}
@article{Goul89,
author = {N. I. M. Gould},
title = {On the convergence of a sequential penalty function method
for constrained minimization},
journal = SINUM,
volume = 26, number = 1, pages = {107--128}, year = 1989}
@article{Goul91,
author = {N. I. M. Gould},
title = {An algorithm for large-scale quadratic programming},
journal = IMAJNA,
volume = 11, number = 3, pages = {299--324}, year = 1991}
@article{Goul99,
author = {N. I. M. Gould},
title = {On modified factorizations for large-scale
linearly-constrained optimization},
journal = SIOPT,
volume = 9, number = 4, pages = {1041--1063}, year = 1999}
@inproceedings{Goul99b,
author = {N. I. M. Gould},
title = {Iterative methods for ill-conditioned linear systems from
optimization},
crossref = {DiPiGian99}, pages = {123--142}}
@inproceedings{GoulLuciRomaToin98,
author = {N. I. M. Gould and S. Lucidi and M. Roma and Ph. L. Toint},
title = {A linesearch algorithm with memory for unconstrained
optimization},
crossref = {DeLeMurlPardTora98}, pages = {207--223}}
@article{GoulLuciRomaToin99,
author = {N. I. M. Gould and S. Lucidi and M. Roma and Ph. L. Toint},
title = {Solving the trust-region subproblem using the {L}anczos
method},
journal = SIOPT,
volume = 9, number = 2, pages = {504--525}, year = 1999,
abstract = {The approximate minimization of a quadratic function within
an ellipsoidal trust region is an important subproblem for
many nonlinear programming methods. When the number of
variables is large, the most widely-used strategy is to
trace the path of conjugate gradient iterates either to
convergence or until it reaches the trust-region boundary.
In this paper, we investigate ways of continuing the
process once the boundary has been encountered. The key is
to observe that the trust-region problem within the
currently generated Krylov subspace has very special
structure which enables it to be solved very efficiently.
We compare the new strategy with existing methods. The
resulting software package is available as {\tt HSL\_VF05}
within the \citebb{HSL00}.},
summary = {When the number of variables is large, the most widely-used
strategy for solving the trust-region subproblem is to
trace the path of conjugate gradient iterates either to
convergence or until it reaches the trust-region boundary.
Means of continuing the process once the boundary has been
encountered are investigated. One observes that the
trust-region problem within the currently generated Krylov
subspace has very special structure which enables it to be
solved efficiently. The proposed strategy is compared with
existing methods.}}
@inproceedings{GoulNoce98,
author = {N. I. M. Gould and J. Nocedal},
title = {The modified absolute-value factorization norm for
trust-region minimization},
crossref = {DeLeMurlPardTora98}, pages = {225--241},
abstract = {A trust-region method for unconstrained minimization, using
a trust-region norm based upon a modified absolute-value
factorization of the model Hessian, is proposed. It is
shown that the resulting trust-region subproblem may be
solved using a single factorization. In the convex case,
the method reduces to a backtracking Newton linesearch
procedure. The resulting software package is available as
{\tt HSL\_VF06} within the \citebb{HSL00}. Numerical
evidence shows that the approach is effective in the
nonconvex case.},
summary = {A trust-region method for unconstrained minimization,
using a trust-region norm based upon a modified
absolute-value factorization of the model Hessian, is
proposed. It is shown that the resulting trust-region
subproblem may be solved using a single
factorization. In the convex case, the method reduces
to a backtracking Newton linesearch procedure. Numerical
experience suggests that the approach is effective in the
non-convex case.}}
@techreport{GoulOrbaSartToin99,
author = {N. I. M. Gould and D. Orban and A. Sartenaer
and Ph. L. Toint},
title = {On the Practical Dependency of a Trust-Region Algorithm
on its Parameters},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {(in preparation)}, year = 1999,
abstract = {In this paper, it is shown through numerical tests that
commonly accepted values for the parameters of a
trust-region algorithm might not be the best ones. Better
ranges of values for these parameters are exhibited on a
statistical basis. It is also shown what improvements can
be hoped for when using a band preconditioner.},
summary = {An extensive numerical study of the statistically best
ranges for the trust-region algorithm parameters is
described, whose conclusions differ from folklore
knowledge. The impact of preconditioning on parameter
choice and performance is also discussed.}}
@techreport{GoulOrbaSartToin99b,
author = {N. I. M. Gould and D. Orban and A. Sartenaer
and Ph. L. Toint},
title = {On the Local Convergenve of a Primal-Dual Trust-Region
Interior-Point Algorithm for Constrained Nonlinear
Programming},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {(in preparation)}, year = 1999,
abstract = {The local convergence properties of the trust-region
interior-point method described in
\citebb{ConnGoulOrbaToin99} are analyzed. It is shown that
the method asymptotically require a single inner iteration
per outer iteration and converges 2-step supercubically.},
summary = {The local convergence properties of the trust-region
interior-point method described in
\citebb{ConnGoulOrbaToin99} are analyzed. It is shown that
the method asymptotically require a single inner iteration
per outer iteration and converges 2-step supercubically.}}
@article{GoulToin99,
author = {N. I. M. Gould and Ph. L. Toint},
title = {A Note on the Second-Order Convergence of Optimization
Algorithms Using Barrier Functions},
journal = MP,
volume = 85, number = 2, pages = {433--438}, year = 1999}
@inproceedings{GoulToin99b,
author = {N. I. M. Gould and Ph. L. Toint},
title = {{SQP} methods for large-scale nonlinear programming},
booktitle = {System Modelling and Optimization,
Methods, Theory and Applications},
editor = {M. J. D. Powell and S. Scholtes},
publisher = KLUWER, address = KLUWER-ADDRESS,
pages = {149--178}, year = 2000,
abstract = {We compare and contrast a number of recent sequential
quadratic programming (SQP) methods that have been proposed
for the solution of large-scale nonlinear programming
problems. Both line-search and trust-region approaches
are considered, as are the implications of interior-point
and quadratic programming methods.},
summary = {A comparison is proposed of a number of recent SQP methods
for the solution of large-scale nonlinear programming
problems. Both linesearch and trust-region approaches
are considered, as are the implications of interior-point
and quadratic programming methods.}}
@techreport{GoulHribNoce98,
author = {N. I. M. Gould and M. E. Hribar and J. Nocedal},
title = {On the solution of equality constrained quadratic problems
arising in optimization},
institution = RAL, address = RAL-ADDRESS,
number = {RAL-TR-98-069}, year = 1998}
@article{GoulToll72,
author = {F. J. Gould and J. W. Tolle},
title = {Geometry of optimality conditions and constraint
qualifications},
journal = MP,
volume = 2, number = 1, pages = {1--18}, year = 1972}
@article{GowGuoLiuLuci97,
author = {A. S. Gow and X. Z. Guo and D. L. Liu and A. Lucia},
title = {Simulation of refrigerant phase equilibria},
journal = {Industrial and Engineering Chemistry Research},
volume = 36, number = 7, pages = {2841--2848}, year = 1997,
abstract = {Vapor-liquid equilibria for refrigerant mixtures modeled by
an equation of state are studied. Phase behavior calculated
by the Soave-Redlich-Kwong (SRK) equation with a single
adjustable binary interaction parameter is compared with
experimental data for binary refrigerant mixtures, two with
a supercritical component and one that exhibits azeotropic
behavior. It is shown that the SRK equation gives an
adequate description of the-phase envelope for binary
refrigerant systems. The complex domain trust region
methods of Lucia and co- workers (\bciteb{LuciGuoWang93},
\bciteb{LuciXu94}) are applied to fixed vapor, isothermal
flash model equations, with particular attention to root
finding and root assignment at the equation of state (EOS)
level of the calculations, and convergence in the
retrograde and azeotropic regions of the phase diagram.
Rules far assigning roots to the vapor and liquid phases in
the case where all roots to the EOS are complex- valued are
proposed and shown to yield correct results, even in
retrograde regions. Convergence of the flash model
equations is also studied. It is shown that the complex
domain trust region algorithms outperform Newton's method
in singular regions of the phase diagram (i.e., at near
azeotropic conditions and in the retrograde loop),
primarily due to the eigenvalue-eigenvector decomposition
strategy given in Sridhar and Lucia (1995). A variety of
geometric figures are used to illustrate salient points.},
summary = {Vapor-liquid equilibria for refrigerant mixtures modeled by
an equation of state are studied. Phase behaviour
calculated by the Soave-Redlich-Kwong (SRK) equation with a
single adjustable binary interaction parameter is compared
with experimental data for binary refrigerant mixtures, two
with a supercritical component and one that exhibits
azeotropic behaviour. It is shown that the SRK equation
gives an adequate description of the-phase envelope for
binary refrigerant systems. The complex domain trust-region
methods of \citebb{LuciGuoWang93} and \citebb{LuciXu94} are
applied to fixed vapor, isothermal flash model equations,
with particular attention to root finding and root
assignment at the equation of state (EOS) level of the
calculations, and convergence in the retrograde and
azeotropic regions of the phase diagram. Rules for
assigning roots to the vapor and liquid phases in the case
where all roots to the EOS are complex- valued
yield correct results, even in retrograde
regions. Convergence of the flash model equations is also
studied. It is shown that the complex domain trust-region
algorithms outperform Newton's method in singular regions
of the phase diagram. A variety of geometric figures are
used to illustrate salient points.}}
@article{GreeStra92,
author = {A. Greenbaum and Z. Strako\v{s}},
title = {Predicting the behaviour of finite precision {L}anczos and
conjugate gradient computations},
journal = SIMAA,
volume = 13, number = 1, pages = {121--137}, year = 1992}
@article{Gree67,
author = {J. Greenstadt},
title = {On the relative efficiencies of gradient methods},
journal = MC,
volume = 21, pages = {360--367}, year = 1967}
@book{Gree97,
author = {A. Greenbaum},
title = {Iterative Methods for Solving Linear Systems},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1997}
@inproceedings{Grie89,
author = {A. Griewank},
title = {On automatic differentiation},
crossref = {IriTana89}, pages = {83--108}}
@inproceedings{Grie94,
author = {A. Griewank},
title = {Computational Differentiation and Optimization},
booktitle = {Mathematical Programming: State of the Art 1994},
editor = {J. R. Birge and K. G. Murty},
publisher = {The University of Michigan}, address = {Ann Arbor, USA},
pages = {102--131}, year = 1994}
@book{GrieCorl91,
author = {A. Griewank and G. Corliss},
title = {Automatic Differentiation of Algorithms: Theory,
Implementation and Application},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1991}
@inproceedings{GrieToin82a,
author = {A. Griewank and Ph. L. Toint},
title = {On the unconstrained optimization of partially separable
functions},
crossref = {Powe82}, pages = {301--312}}
@article{GrieToin82b,
author = {A. Griewank and Ph. L. Toint},
title = {Partitioned variable metric updates for large structured
optimization problems},
journal = NUMMATH,
volume = 39, pages = {119--137}, year = 1982}
@article{GrifStew61,
author = {R. E. Griffith and R. A. Stewart},
title = {A nonlinear programming technique for the optimization of
continuous processing systems},
journal = {Management Science},
volume = 7, pages = {379--392}, year = 1961,
abstract = {A description is given of a method for solving some
nonlinear programming problems. The mathematics of this
method are quite simple and are easy to apply to electronic
computation. A numerical example, a model construction
example, and a description of a particular existing
computer system are included in order to clarify the mode
of operation of the method.},
summary = {A simple method for solving nonlinear programming problems
is given. A numerical example, a model construction
example, and a description of a particular existing
computer system are included in order to clarify the mode
of operation of the method.}}
@article{GripLampLuci86,
author = {L. Grippo and F. Lampariello and S. Lucidi},
title = {A nonmonotone line search technique for {N}ewton's method},
journal = SINUM,
volume = 23, number = 4, pages = {707--716}, year = 1986}
@article{GripLampLuci89,
author = {L. Grippo and F. Lampariello and S. Lucidi},
title = {A Truncated {N}ewton Method with Nonmonotone Line Search
for Unconstrained Optimization},
journal = JOTA,
volume = 60, number = 3, pages = {401--419}, year = 1989}
@article{GripLampLuci91,
author = {L. Grippo and F. Lampariello and S. Lucidi},
title = {A class of nonmonotone stabilization methods in
unconstrained optimization},
journal = NUMMATH,
volume = 59, pages = {779--805}, year = 1991}
@techreport{GrotMcKi98,
author = {A. Grothey and K. McKinnon},
title = {A Superlinearly Convergent Trust Region Bundle Method},
institution = {Department of Mathematics and Statistics},
address = {Univeristy of Edinburgh, Scotland},
number = {MS 98-015}, year = 1998,
abstract = {Bundle methods for the minimization of non-smooth
functions have been around for almost 20 years.
Numerous variations have been proposed. But until very
recently they all suffered from the drawback of only
linear convergence. The aim of this paper is to show
how exploiting an analogy with SQP gives rise to a
superlinearly convergent bundle method. Our algorithm
features a trust region philosophy and is expected to
converge superlinearly even for non-convex problems.},
summary = {Current bundle methods for the minimization of non-smooth
functions converge at a linear rate. A superlinearly
convergent bundle method, using a trust region, is
proposed for nonconvex problems. Numerical experience
on a power-generation problem is reported.}}
@article{Gurw94,
author = {C. Gurwitz},
title = {Local Convergence of a Two-Piece Update of a Projected
{H}essian Matrix},
journal = SIOPT,
volume = 4, number = 3, pages = {461--485}, year = 1994}
@book{GruvSach80,
author = {W. A. Gruver and E. W. Sachs},
title = {Algorithmic Methods in Optimal Control},
publisher = {Pitman}, address = {Boston, USA},
year = 1980}
%%% H %%%
@book{Hack94,
author = {W. Hackbusch},
title = {Iterative Solution of Large Sparse Systems of Equations},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
series = {Springer Series in Applied Mathematical Sciences},
year = 1994}
@article{Hage87,
author = {W. W. Hager},
title = {Dual techniques for constrained optimization},
journal = JOTA,
volume = 55, pages = {37--71}, year = 1987}
@article{Hage99,
author = {W. W. Hager},
title = {Stabilized sequential quadratic programming},
journal = COAP,
volume = 12, number = {1--2}, pages = {253--273}, year = 1999}
@techreport{Hage99b,
author = {W. W. Hager},
title = {Minimizing a Quadratic Over a Sphere},
institution = {Mathematics Department, University of Florida},
address = {Gainesville, Florida, USA},
month = {May}, year = 1999,
abstract = {A new method, the sequential subspace method (SSM),
is developed for minimizing a quadratic over a
sphere. In each iteration of the scheme, the
quadratic is minimized over a subspace that contains
the prior iterate, a sequential quadratic programming
iterate, and a projected gradient. A low dimensional
subspace with these properties is obtained using the
transformed minimal residual algorithm (TMRES). We
prove that the SSM is locally quadratically
convergent. Numerical experiments indicate that the
SSM requires far fewer matrix-vector operations than
other recently developed algorithms.},
summary = {A sequential subspace method (SSM), is developed for
minimizing a quadratic over a sphere. In each iteration,
the quadratic is minimized over a subspace that contains
the prior iterate, a sequential quadratic programming
iterate, and a projected gradient. A low dimensional
subspace with these properties is obtained using the
transformed minimal residual algorithm (TMRES).
The SSM is proved to be locally quadratically
convergent. Numerical experiments indicate that the
SSM requires few matrix-vector operations.}}
@article{Han77,
author = {S. P. Han},
title = {A Globally Convergent Method for Nonlinear Programming},
journal = JOTA,
volume = 22, number = 3, pages = {297--309}, year = 1977}
@article{HanMang79,
author = {S. P. Han and O. L. Mangasarian},
title = {Exact penalty functions in nonlinear programming},
journal = MP,
volume = 17, number = 3, pages = {251--269}, year = 1979}
@article{HanMang83,
author = {S. P. Han and O. L. Mangasarian},
title = {A dual differentiable exact penalty-function},
journal = MP,
volume = 25, number = 3, pages = {293--306}, year = 1983}
@article{HanPardYe92,
author = {C. Han and P. Pardalos and Y. Ye},
title = {On the solution of indefinite quadratic problems using an
interior point method},
journal = {Informatica},
volume = 3, pages = {474--496}, year = 1992}
@techreport{HanHan99,
author = {Q. Han and J. Han},
title = {Modified Quasi-{N}ewton Method with Collinear Scaling for
Unconstrained Optimization},
institution = {Institute of Computational Mathematics and
Scientific/Engineering Computing,
Chinese Academy of Sciences},
address = {Beijing},
number = {February}, year = 1999,
abstract = {It is well known that among the current methods for
unconstrained optimization problems, Newton or quasi-Newton
method with global strategy may be the most efficient
method, which have local quadratic or superlinear
convergence. However, when the iterate point is far away
from the a solution of the problem, Newton or quasi-Newton
method may proceed slowly for the general nonlinear
objective function. In the paper, we present a modified
quasi-Newton method with trust region using the collinear
scaling for unconstrained optimization. Not only the
gradient information but the values of the objective
function are used to construct the local model at the
current iteration point. Moreover, the information about
the super steepest descent direction is embedded into the
local model. the amount of computation in each iteration
of the modified quasi-Newton method algorithm with trust
region is the same as that of the standard quasi-Newton
method with trust region. And some numerical results show
that the modified method needs very fewer iterations to
reach the solution of the optimization problem. Global
and local convergence of the method is also analyzed.},
summary = {A trust-region method is proposed for unconstrained
minimization, where the model is obtained by a conic
quasi-Newton update. The trust region is defined not in
the original space but in the space of collinearly scaled
variables. Limited numerical experience illustrates the
practical potential of the method.}}
@article{Hank97,
author = {M. Hanke},
title = {A regularizing {L}evenberg-{M}arquardt scheme, with
applications to inverse groundwater filtration problems},
journal = {Inverse Problems},
volume = 13, number = 1, pages = {79--95}, year = 1997,
abstract = {The first part of this paper studies a Levenberg-Marquardt
scheme for nonlinear inverse problems where the
corresponding Lagrange (or regularization) parameter is
chosen from an inexact Newton strategy. While the
convergence analysis of standard implementations based on
trust region strategies always requires the invertibility
of the Frechet derivative of the nonlinear operator at the
exact solution, the new Levenberg-Marquardt scheme is
suitable for ill-posed problems as long as the Taylor
remainder is of second order in the interpolating metric
between the range and domain topologies. Estimates of this
type are established in the second part of the paper for
ill-posed parameter identification problems arising in
inverse groundwater hydrology. Both transient and
steady-state data are investigated. Finally, the numerical
performance of the new Levenberg-Marquardt scheme is
studied and compared to a usual implementation on a
realistic but synthetic two-dimensional model problem from
the engineering literature.},
summary = {A Levenberg-Morrison-Marquardt scheme for nonlinear
inverse problems is considered, where the corresponding
Lagrange parameter is chosen from an inexact Newton
strategy. This scheme is suitable for ill-posed problems as
long as the Taylor remainder is of second order in the
interpolating metric between the range and domain
topologies. Estimates of this type are established for
ill-posed parameter identification problems arising in
inverse groundwater hydrology. Both transient and
steady-state data are investigated. The performance of the
scheme is compared to a usual implementation on a
two-dimensional engineering model problem.}}
@article{HansKrog92,
author = {R. J. Hanson and F. T. Krogh},
title = {A Quadratic-Tensor Model Algorithm for Nonlinear
Least-Squares Problems with Linear Constraints},
journal = TOMS,
volume = 18, number = 2, pages = {115--133}, year = 1992,
abstract = {A new algorithm is presented for solving nonlinear
least-squares and nonlinear equation problems. The
algorithm is based on approximating the nonlinear
functions using the quadratic-tensor model proposed
by Schnabel and Frank (1984). The problem statement
may include simple bounds or more general linear
constraints on the unknowns. The algorithm uses a
trust-region defined by a box containing the current
values of the unknowns. The objective function
(Euclidean length of the functions) is allowed to
increase at intermediate steps. These increases are
allowed as long as the predictor indicates that a new
set of best values exists in the trust-region. There
is logic provided to retreat to the current best
values, should that be required. The computations for
the model-problem require a constrained nonlinear
least-squares solver. This is done using a simpler
version of the algorithm. In its present form the
algorithm is effective for problems with linear
constraints and dense Jacobian matrices. Results on
standard test problems are presented in the
Appendix. The new algorithm appears to be efficient
in terms of function and Jacobian evaluations.},
summary = {A new algorithm is presented for solving linearly
constrained nonlinear least-squares and nonlinear equation
problems, based on approximating the nonlinear
functions using the quadratic-tensor model.
The algorithm uses a box-shaped
trust-region. The objective function
is allowed to increase at intermediate steps,
as long as the predictor indicates that a new
set of best values exists in the trust-region. There
is logic provided to retreat to the current best
values, if necessary. The algorithm is effective for
problems with linear constraints and dense Jacobian
matrices and appears to be efficient
in terms of function and Jacobian evaluations.}}
@article{HarkPang90,
author = {P. T. Harker and J. S. Pang},
title = {Finite-dimensional variational inequality and nonlinear
complementarity problems: a survey of theory, algorithms
and applications},
journal = MPB,
volume = 48, number = 2, pages = {161--220}, year = 1990}
@article{HarkXiao90,
author = {P. T. Harker and B. Xiao},
title = {{N}ewton's method for the nonlinear complementarity
problem: a {B}-differentiable equation approach},
journal = MPB,
volume = 48, number = 3, pages = {339--358}, year = 1990}
@article{HeDiaoGao97,
author = {G. He and B. Diao and Z. Gao},
title = {An {SQP} algorithm with nonmonotone line search for general
nonlinear constrained optimization problem},
journal = JCM,
volume = 15, number = 2, pages = {179--192}, year = 1997}
@techreport{Hebd73,
author = {M. D. Hebden},
title = {An Algorithm for Minimization Using Exact Second
Derivatives},
institution = HARWELL, address = HARWELL-ADDRESS,
number = {T.P. 515}, year = 1973,
abstract = {A review of the methods currently available for the
minimization of a function whose first and second
derivatives can be calculated shows either that the method
requires the eigensolution of the Hessian, or with one
exception that a simple example can be found which causes
the method to fail. In this paper one of the successful
methods that requires the eigensolution is modified so that
at each iteration the solution of a number (approximately
two) of systems of linear equations is required, instead of
the eigenvalue calculation.},
summary = {A method for the solution of the $\ell_2$ trust-region
subproblem is proposed so that at each iteration the
solution of a number (roughly two) of systems of linear
equations is required, instead of the eigenvalue
calculation.}}
@article{Hein93,
author = {M. Heinkenschloss},
title = {Mesh Independence for Nonlinear Least Squares Problems with
Norm Constraints},
journal = SIOPT,
volume = 3, number = 1, pages = {81--117}, year = 1993}
@article{Hein94,
author = {M. Heinkenschloss},
title = {On the solution of a two ball trust region subproblem},
journal = MP,
volume = 64, number = 3, pages = {249--276}, year = 1994,
abstract = {In this paper we investigate the structure of a two ball
trust region subproblem arising in nonlinear parameter
identification problems and propose a method for its
solution. The method decomposes the subproblem and allows
the application of efficient, well studied methods for the
solution of the trust region subproblems arising in
unconstrained optimization. In the discussion of the
structure we focus on the case where both constraints are
active and on the treatment of the unconstrained problem.},
summary = {The structure of a two-ball trust-region subproblem arising
in nonlinear parameter identification problems is
investigated, and a method for its solution is proposed.
The method decomposes the subproblem, and allows the
application of efficient methods for the
solution of the trust-region subproblems. The discussion
of the structure
focuses on the case where both constraints are active and
on the treatment of the unconstrained problem.}}
@article{Hein98,
author = {M. Heinkenschloss},
title = {A Trust-Region Method for Norm Constrained Problems},
journal = SINUM,
volume = 35, number = 4, pages = {1594--1620}, year = 1998,
abstract = {In this paper a trust region method for the solution of
nonlinear optimization problems with norm constraints is
presented and analyzed. Such problems often arise in
parameter identification or nonlinear eigenvalue problems.
The algorithms studied here allow for inexact gradient
information and the use of subspace methods for the
approximate solution of subproblems. Characterizations and
the descent properties of trust region steps are given,
criteria for the existence of successful iterations under
inexact gradient information and under the use of subspace
methods are established, and global convergence of the
method is proven.},
summary = {A trust-region method for the solution of nonlinear
optimization problems with norm constraints is presented
and analyzed. Such problems often arise in parameter
identification or nonlinear eigenvalue problems. The
algorithms studied allow for inexact gradient information
and the use of subspace methods for the approximate
solution of subproblems. Characterizations and the descent
properties of trust-region steps are given, criteria for
the existence of successful iterations under inexact
gradient information and under the use of subspace methods
are established, and global convergence of the method is
proven.}}
@techreport{HeinUlbrUlbr97,
author = {M. Heinkenschloss and M. Ulbrich and S. Ulbrich},
title = {Superlinear and quadratic convergence of affine-scaling
interior-point {N}ewton methods for problems with simple
bounds without strict complementarity},
institution = CAAM, address = RICE-ADDRESS,
number = {TR97-30}, year = 1997,
abstract = {A class of affine-scaling interior-point methods for bound
constrained optimization problems is introduced which are
locally q-superlinearly or q-quadratically convergent. It
is assumed that the strong second order sufficient
optimality conditions at the solution are satisfied, but
strict complementarity is not required. The methods are
modifications of the affine-scaling interior-point Newton
methods introduced by \citebb{ColeLi94}. There are two
modifications. One is a modification of the scaling matrix,
the other one is the use of a projection of the step to
maintain strict feasibility rather than a simple scaling of
this step. A comprehensive local convergence analysis is
given. A few simple examples are presented to illustrate
the pitfalls of the original approach of
\citeauthor{ColeLi94} in the degenerate case and to
demonstrate the performance of the fast converging
modifications developed in this paper.},
summary = {A class of affine-scaling interior-point methods for
bound-constrained optimization problems is introduced which
are locally Q-superlinearly or Q-quadratically convergent,
even without assuming strict complementarity. The methods
are derived from \citebb{ColeLi94} but use a different
scaling matrix and a projection of the step to maintain
strict feasibility. Simple examples are presented to
illustrate the pitfalls of \citeauthor{ColeLi94}'s approach
in the degenerate case and to demonstrate the performance
of the fast converging modifications.}}
@techreport{HeinVice95,
author = {M. Heinkenschloss and L. N. Vicente},
title = {Analysis of Inexact Trust Region Interior-Point {SQP}
Algorithms},
institution = CRPC, address = RICE-ADDRESS,
type = {Technical Report}, number = {CRPC-TR95546}, year = 1995,
abstract = {In this paper we analyze inexact trust-region
interior-point (TRIP) sequential quadratic programming
(SQP) algorithms for the solution of optimization problems
with nonlinear equality constraints and simple bound
constraints on some of the variables. Such problems arise
in many engineering applications, in particular in optimal
control problems with bounds on the control. The nonlinear
constraints often come from the discretization of partial
differential equations. In such cases the calculation of
derivative information and the solution of the linearized
equations is expensive. Often, the solution of linear
systems and directional derivatives are computed inexactly
yielding nonzero residuals. This paper analyzes the effect
of the inexactness onto the convergence of TRIP SQP and
gives practical rules to control the size of the residuals
of these inexact calculations. It is shown that if the size
of the residuals is of the order of both the size of the
constraints and the trust region radius, then the TRIP SQP
algorithms are globally first-order convergent. Numerical
experiments with two optimal control problems governed by
nonlinear partial differential equations are reported.},
summary = {Inexact trust-region interior-point (TRIP) sequential
quadratic programming (SQP) algorithms for the solution of
optimization problems with nonlinear equality constraints
and simple bound constraints are analysed. The effect of
the inexactness in the computation of derivative
information on the convergence of TRIP SQP is analysed, and
practical rules to control the size of the associated
residuals are given. It is shown that if the size of the
residuals is of the order of both the size of the
constraints and the trust region radius, then the TRIP SQP
algorithms are globally first-order convergent. Numerical
experiments with two optimal control problems governed by
nonlinear partial differential equations are reported.}}
%also
%institution = CAAM, address = RICE-ADDRESS,
%number = {TR95-18}, year = 1995,
@techreport{HeinVice99,
author = {M. Heinkenschloss and L. N. Vicente},
title = {Analysis of Inexact Trust Region {SQP} Algorithms},
institution = COIMBRA, address = COIMBRA-ADDRESS,
number = {99-15}, year = 1999,
abstract = {In this paper we study the global convergence behavior of
a class of composite-step trust-region SQP methods that
allow inexact problem information. The inexact problem
information can result from iterative linear systems
solves within the trust-region SQP method or from
approximations of first-order derivatives. Accuracy
requirements in our trust-region SQP methods are adjusted
based on feasibility and optimality of the iterates. In
the absence of inexactness, our global convregence theory
is equal to that of \citebb{DennElAlMaci97}. If all
iterates are feasible, i.e.\ if all iterates satisfy the
equality constraints, then our results are related to the
known convergence analyses for trust-region methods with
inexact gradient information for unconstrained
optimization.},
summary = {The global convergence behaviour of a class of
composite-step trust-region SQP methods that allow inexact
problem information is studied. This inexact
information can result from iterative linear systems
solves within the trust-region SQP method or from
approximations of first-order derivatives. Accuracy
requirements are based on feasibility and optimality of
the iterates. In the absence of inexactness, the analysis
reduces to that of \citebb{DennElAlMaci97}. If all iterates
satisfy the equality constraints, then the results are
related to the known convergence properties for trust-region
methods with inexact gradient information in unconstrained
optimization.}}
@article{HeinSpel94,
author = {J. Heinz and P. Spellucci},
title = {A successful implementation of the {P}antoja-{M}ayne {SQP}
method},
journal = OMS,
volume = 4, number = 1, pages = {1--28}, year = 1994}
@article{HelfZwic95,
author = {H.-P. Helfrich and D. Zwick},
title = {Trust region algorithms for the nonlinear least distance
problem},
journal = {Numerical Algorithms},
volume = 9, number = {1-2}, pages = {171--179}, year = 1995,
abstract = {The nonlinear least distance problem is a special
case of equality constrained optimization. Let a
curve or surface be given in implicit form via the
equation $f(x)=0$, $x \in \Re^d$, and let $z
\in $R^d$ be a fixed data point. We discuss
two algorithms for solving the following problem:
Find a point $x^*$ such that $f(x^*)=0$ and
$\|z-x^*\|_2$ is minimal among all such
$x$. The algorithms presented use the trust region
approach in which, at each iteration, an
approximation to the objective function or merit
function is minimized in a given neighborhood (the
trust region) of the current iterate. Among other
things, this allows one to prove global convergence
of the algorithm.},
summary = {Trust-region algorithms are presented for the nonlinearly
constrained least-distance problem. Global convergence is
proved.}}
@article{HelfZwic96,
author = {H. P. Helfrich and D. Zwick},
title = {A trust region algorithm for parametric curve and surface
fitting},
journal = JCAM,
volume = 73, number = {1--2}, pages = {119--134}, year = 1996,
abstract = {Let a family of curves or surfaces be given in parametric
form via the model equation $x=f(s,\beta)$ where $x \in
\Re^n$, $\beta \in \Re^m$, and $s \in S \subset \Re^d$,
$d<n$. We present an algorithm for solving the problem:
\emph{Find a shape vector $\beta_*$ such that the manifold
$M^*$ is a best fit to scattered data $\{z_i\}_{i=1}^N
\subset \Re^n$ in the sense that, for some
$\{s_i^*\}_{i=1}^N$, the sum of the squared least distances
` $\sum_{i=1}^N \| z_i - f(s_i^*,\beta^*) \||_2^2$ from the
data points to the manifold $M^*$ is minimal among all such
manifolds.} For robustness, our algorithm uses a globally
convergent trust region approach in which, at each
iteration, an approximation to the objective function is
minimized in a given neighborhood of the current iterate.
The set $S$ may be all of $\Re^d$ or a closed, convex
subset. In particular, it may be chosen so that our theory
is applicable to splines.},
summary = {An algorithm is presented for solving the problem of
finding a parameter (shape) vector such that the associated
manifold is a best least-squares fit to scattered data.
For robustness, the algorithm uses a globally convergent
trust-region approach. The support set for the manifold may
be all of $\Re^d$ or a closed, convex subset. In
particular, it may be chosen so that our theory is
applicable to splines.}}
@article{Helg91,
author = {T. Helgaker},
title = {Transition-State Optimizations by Trust-Region Image
Minimization},
journal = {Chemical Physics Letters},
volume = 182, number = 5, pages = {503--510}, year = 1991,
abstract = {A new method for optimizing transition states is presented.
The method combines Smith's image function with
trust-region minimization. Calculations on HCN and
C$_2$H$_6$ illustrate the usefulness of the method for ab
initio potential energy surfaces. It is found that
second-order image optimizations of transition states are
as fast as conventional minimizations.},
summary = {A method for optimizing transition states is presented.
The method combines Smith's image function with
trust-region minimization. Calculations on HCN and C2H6
illustrate the usefulness of the method for ab-initio
potential energy surfaces. It is found that second-order
image optimizations of transition states are as fast as
conventional minimizations.}}
@article{HelgAlml88,
author = {T. Helgaker and J. Almlof},
title = {Molecular wave functions and properties calculated using
floating Gaussian orbitals},
journal = {Journal of Chemical Physics},
volume = 89, number = 8, pages = {4889-4902}, year = {1988},
abstract = {The calculation of molecular electronic wave
functions and properties using floating Gaussian
orbitals (i.e., orbitals whose positions are
optimized in space) is described. The wave function
is optimized using a second-order convergent scheme
(the trust-region method), and molecular properties
up to second order are calculated analytically. The
method is applied to a series of small molecules (HF,
H/sub 2/O, NH/sub 3/, CH/sub 4/, CO, H/sub 2/CO, and
C/sub 2/H/sub 4/) at the Hartree-Fock level using
four different floating basis sets (double zeta,
double zeta plus polarization, double zeta plus
diffuse, and double zeta plus polarization and
diffuse). Geometries are fully optimized, and dipole
moments, static polarizabilities, harmonic
frequencies, and double-harmonic infrared intensities
are calculated at the optimized geometries. The
results are compared with those obtained using the
corresponding fixed basis sets, and also with the
results from a large basis of near-Hartree-Fock
quality (6-311++G(3df,3pd)). Floating produces only
minor changes in the electronic energy, but other
properties are often significantly improved. In
particular, properties involving external field
variations (dipole moments, polarizabilities, and
intensities) converge considerably faster to the
Hartree-Fock limit when floating is
allowed. Properties calculated using the floating
double-zeta basis set augmented with one set of
polarization functions and one set of diffuse
orbitals are close to the Hartree-Fock limit.},
summary = {The calculation of molecular electronic wave
functions and properties using floating Gaussian
orbitals (i.e., orbitals whose positions are
optimized in space) is described. The wave function
is optimized using a second-order convergent trust-region
method and molecular properties
up to second order are calculated analytically. The
method is applied to a series of small molecules
at the Hartree-Fock level using four different floating
basis sets. Properties calculated using the floating
double-zeta basis set augmented with one set of
polarization functions and one set of diffuse
orbitals are close to the Hartree-Fock limit.}}
@article{Hest69,
author = {M. R. Hestenes},
title = {Multiplier and gradient methods},
journal = JOTA,
volume = 4, pages = {303--320}, year = 1969}
@book{Hest80,
author = {M. R. Hestenes},
title = {Conjugate Direction Methods in Optimization},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
year = 1980}
@article{HestStie52,
author = {M. R. Hestenes and E. Stiefel},
title = {Methods of conjugate gradients for solving linear systems},
journal = {J. Res. N.B.S.},
volume = 49, pages = {409--436}, year = 1952}
@article{High87,
author = {N. J. Higham},
title = {A survey of condition number estimation for triangular
matrices},
journal = SIREV,
volume = 29, number = 4, pages = {575--596}, year = 1987}
@book{High93,
author = {N. J. Higham},
title = {Writing for the Mathematical Sciences},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1993}
@article{High95a,
author = {N. J. Higham},
title = {Stability of the diagonal pivoting method with partial
pivoting},
journal = SIMAA,
volume = 18, number = 1, pages = {52--65}, year = 1997}
@book{High96,
author = {N. J. Higham},
title = {Accuracy and Stability of Numerical Algorithms},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1996}
@techreport{High98,
author = {D. J. Higham},
title = {Trust Region Algorithms and Timestep Selection},
institution = {Department of Mathematics},
address = {University of Strathclyde, Glasgow, Scotland},
type = {Research Report}, number = 3, year = 1998,
abstract = {Unconstrained optimization problems are closely related to
systems of ordinary differential equations (ODEs) with
gradient structure. In this work, we prove results that
apply to both areas. We analyze the convergence properties
of a trust region, or Levenberg-Marquadt, algorithm for
optimization. The algorithm may also be regarded as a
linearized implicit Euler method with adaptive timestep for
gradient ODEs. We establish global convergence, and show
that the rate of convergence is superlinear, but not
quadratic. The precise form of superlinear convergence is
exhibited---the ratio of successive displacements from
the limit point is bounded above and below by geometrically
decreasing sequences. In the gradient ODE context this
result contributes to the theory of \emph{gradient
stability}. The result also introduces the notion of
adapting the timestep in order to control the rate at which
equilibrium is approached. A related timestepping algorithm
is developed for general ODEs that guarantees fast
superlinear local convergence to a stable equilibrium. This
algorithm has many applications to ODEs and
semi-discretized partial differential equations where a
steady state solution is required.},
summary = {The close relation between unconstrained optimization and
systems of ordinary differential equations (ODEs) with
gradient structure is exploited. The convergence properties
of a trust region, or Levenberg-Morrison-Marquardt,
algorithm are examined. The algorithm may also be regarded
as a linearized implicit Euler method with adaptive
timestep for gradient ODEs. Global convergence is
established and the rate of convergence is superlinear, but
not quadratic. The precise form of superlinear convergence
is exhibited. In the gradient ODE context this
result contributes to the theory of ``gradient stability''.
It also introduces the notion of adapting the
timestep in order to control the rate at which equilibrium
is approached. A related timestepping algorithm
for general ODEs guarantees fast superlinear
local convergence to a stable equilibrium. This algorithm
has many applications to ODEs and semi-discretized partial
differential equations where a steady state solution is
required.}}
@article{HighChen98,
author = {N. J. Higham and S. Cheng},
title = {Modifying the Inertia of Matrices Arising in Optimization},
journal = LAA,
volume = {275--276}, pages = {261--279}, year = 1998}
@book{HiriLema93,
author = {J.-B. Hiriart-Urruty and C. Lemar\'{e}chal},
title = {Convex Analysis and Minimization Algorithms. {P}art 1:
{F}undamentals},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
year = 1993}
@book{HiriLema93b,
author = {J.-B. Hiriart-Urruty and C. Lemar\'{e}chal},
title = {Convex Analysis and Minimization Algorithms. {P}art 2:
{A}dvanced Theory and Bundle Methods},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
year = 1993}
@misc{LeTh97,
author = {Le Thi, H. A.},
title = {Contribution \`{a} \l'optimisation non convexe et
l'optimisation globale},
howpublished = {Habilitation thesis, University of Rouen, Rouen, France},
year = 1997}
@article{LeTh99,
author = {Le Thi, H. A.},
title = {An efficient algorithm for globally minimizing a quadratic
function under convex quadratic constraints},
journal = MP,
note = {To appear.}, year = 1999,
abstract = {In this paper, we investigate two approaches to minimizing
a quadratic form subject to the intersection of finitely
many ellipsoids. The first approach is the
d.c. (difference of convex functions) optimization
algorithm (DCA) whose main tools are the proximal
point algorithm and/or the projection subgradient method
in convex optimization. The second is a branch-and-bound
scheme using Lagrangian duality for bounding and
ellipsoidal bisection in branching. The DCA was
first introduced by Pham Dinh Tao in 1986 for general
d.c. program and later developed by our various work is
a local method but, from a good starting point, it often
provides a global solution. This motivates us to combine
the DCA and our branch and bound algorithm in order to
obtain a good initial point for the DCA and to prove the
globality of the DCA. In both two approaches we
attempt to use the ellipsoidal constrained quadratic
programs as the main subproblems. The idea is based upon
the fact that these programs can be efficiently solved by
the procedure recently proposed by Pham Dinh Tao and Le
Thi H. An, which has been shown to be robust and fast to
large-scale problems. Several numerical experiments with
dimension up to 200 are given which show the effectiveness
and the robustness of the DCA and the combined
DCA-branch-and-bound algorithm.},
summary = {A combination of the DCA (algorithm for the difference of
convex functions) and branch-and-bound is developed to
globally solve the problem of minimizing a quadratic
function subject to a finite set of convex quadratic
constraints. The trust-region subproblem is used as the
main subproblem of the new algorithm. Numerical experiments
illustrate the proposal.}}
@article{HuanNg94,
author = {L. R. Huang and K. F. Ng},
title = {2nd-order necessary and sufficient conditions in nonsmooth
optimization},
journal = MP,
volume = 66, number = 3, pages = {379--402}, year = 1994}
@manual{HSL00,
author = {{H}arwell {S}ubroutine {L}ibrary},
title = {A catalogue of subroutines (release 2000)},
organization = HARWELL, address = HARWELL-ADDRESS,
year = 2000}
%%% I %%%
@manual{IMSL99,
author = {{IMSL}},
title = {Fortran Numerical Library},
organization = {Visual Numerics, Inc.}, address = {Houston, Texas, USA},
year = 1999}
@article{Iron70,
author = {B. M. Irons},
title = {A frontal solution program for finite-element analysis},
journal = IJNME,
volume = 2, pages = {5--32}, year = 1970}
@article{Iuse91,
author = {A. N. Iusem},
title = {On the dual convergence and the rate of primal convergence
of {B}regman's convex programming method},
journal = SIOPT,
volume = 1, number = 3, pages = {401--423}, year = 1991}
%%% J %%%
@inproceedings{Jage95,
author = {M. Jagersand},
title = {Visual servoing using trust region methods and estimation
of the full coupled visual-motor {J}acobian},
booktitle = {Proceedings of the IASTED International Conference.
Applications of Control and Robotics},
publisher = {IASTED-ACTA Press}, address = {Anaheim, California, USA},
pages = {105-108}, year = 1995,
abstract = {We present an algorithm for visual servoing, capable of
learning the robot kinematics and camera calibration.
The approach differs from previous work in that a
full coupled Jacobian is estimated online without
prior models, and that a trust region method is used,
improving stability and convergence of the
controller. We present experimental results on the
positioning accuracy and convergence of this
controller showing an up to 5 fold improvement in
repeatability on PUMA 761 and 762 robots and
successful estimation of the Jacobian for 3, 6 and 12
controlled DOF with highly non-linear transfer
functions.},
summary = {An algorithm for visual servoing, capable of
learning the robot kinematics and camera calibration,
is presented. The approach differs from previous
work in that a full coupled Jacobian is estimated
online without prior models, and that a trust region
method is used, improving stability and convergence
of the controller. Experimental results on the
positioning accuracy and convergence of this
controller, showing an up to 5 fold improvement in
repeatability on PUMA 761 and 762 robots and
successful estimation of the Jacobian for 3, 6 and 12
controlled DOF with highly non-linear transfer
functions, are prsented.}}
@inproceedings{JageFuenNels96,
author = {M. Jagersand and O. Fuentes and R. Nelson},
title = {Acquiring visual-motor models for precision manipulation
with robot hands},
editor = {B. Buxton and R. Cipolla},
booktitle = {Computer Vision -- ECCV `96. 4th Eurpean Conference on
Computer Proceedings},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
volume = 2, pages = {603--612}, year = 1996,
abstract = {Dextrous high degree of freedom (DOF) robotic hands
provide versatile motions for fine manipulation of
potentially very different objects. However, fine
manipulation of an object grasped by a multifinger
hand is much more complex than if the object is
rigidly attached to a robot arm. Creating an accurate
model is difficult if not impossible. We instead
propose a combination of two techniques: the use of
an approximate estimated motor model, based on the
grasp tetrahedron acquired when grasping an object,
and the use of visual feedback to achieve accurate
fine manipulation. We present a novel active vision
based algorithm for visual serving, capable of
learning the manipulator kinematics and camera
calibration online while executing a manipulation
task. The approach differs from previous work in that
a full, coupled image Jacobian is estimated online
without prior models, and that a trust region control
method is used, improving stability and
convergence. We present an extensive experimental
evaluation of visual model acquisition and visual
serving in 3, 4 and 6 DOF.},
summary = {Dextrous high degree of freedom (DOF) robotic hands
provide versatile motions for fine manipulation of
potentially very different objects. However, fine
manipulation of an object grasped by a multifinger
hand is much more complex than if the object is
rigidly attached to a robot arm. Creating an accurate
model is difficult if not impossible. We instead
propose a combination of two techniques: the use of
an approximate estimated motor model, based on the
grasp tetrahedron acquired when grasping an object,
and the use of visual feedback to achieve accurate
fine manipulation. A novel active vision
based algorithm for visual serving is presented, capable of
learning the manipulator kinematics and camera
calibration online while executing a manipulation
task. The approach differs from previous work in that
a full, coupled image Jacobian is estimated online
without prior models, and that a trust-region control
method is used, improving stability and
convergence. Extensive experimental
evaluation of visual model acquisition and visual
serving in 3, 4 and 6 DOF. is presented}}
@inproceedings{JageFuenNels97,
author = {M. Jagersand and O. Fuentes and R. Nelson},
title = {Experimental evaluation of uncalibrated visual servoing
for precision manipulation},
booktitle = {Proceedings 1997 IEEE International Conference on Robotics
and Automation},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 4, pages = {2874--2880}, year = 1997,
abstract = {We present an experimental evaluation of adaptive and
non-adaptive visual servoing in 3, 6 and 12 degrees
of freedom (DOF), comparing it to traditional joint
feedback control. While the purpose of experiments in
most other work has been to show that the particular
algorithm presented indeed also works in practice, we
do not focus on the algorithm but rather on
properties important to visual servoing in
general. Our main results are: positioning of a 6
axis PUMA 762 arm is up to 5 times more precise under
visual control than under joint control and positioning
of a Utah/MIT dextrous hand is better under visual
control than under joint control by a factor of 2 and
and a trust-region-based adaptive visual feedback
controller is very robust. For m tracked visual
features the algorithm can successfully estimate
online the m*3 (m>or=3) image Jacobian (J) without
any prior information, while carrying out a 3 DOF
manipulation task. For 6 and higher DOF manipulation,
a rough initial estimate of J is beneficial. We also
verified that redundant visual information is
valuable. Errors due to imprecise tracking and goal
specification were reduced as the number of visual
features, m, was increased. Furthermore highly
redundant systems allow us to detect outliers in the
feature vector and deal with partial occlusion.},
summary = {An experimental evaluation of adaptive and
non-adaptive visual servoing in 3, 6 and 12 degrees
of freedom (DOF) is compared to traditional joint
feedback control. The main results are: positioning of a 6
axis PUMA 762 arm is up to 5 times more precise under
visual control than under joint control and positioning
of a Utah/MIT dextrous hand is better under visual
control than under joint control by a factor of 2 and
and a trust-region-based adaptive visual feedback
controller is very robust. For $m$ tracked visual
features, the algorithm can successfully estimate
online the $3m$ ($m \geq 3$) image Jacobian ($J$) without
any prior information, while carrying out a 3 DOF
manipulation task. For 6 and higher DOF manipulation,
a rough initial estimate of $J$ is beneficial.
Redundant visual information is also shown to be
valuable. Errors due to imprecise tracking and goal
specification were reduced as the number of visual
features, $m$, was increased. Furthermore highly
redundant systems allow the detection of outliers in the
feature vector and dealing with partial occlusion.}}
@inproceedings{JainMcClSark86,
author = {V. K. Jain and T. E. McClellan and T. K. Sarkar},
title = {Half-{F}ourier transform and application to radar signals},
booktitle = {ICASSP 86 Proceedings. IEEE-IECEJ-ASJ International
Conference on Acoustics, Speech and Signal Processing},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 1, pages = {241--244}, year = {1986},
abstract = {The authors discuss the half Fourier transform (HFT)
and explore its application to radar-return signals
with specular components. It is shown that this
transform enables the desired part to be separated
from the specular impulsive components. The
effectiveness of this technique and the computer
program developed is demonstrated by simulation
examples. The program uses an optimization package
which minimizes a nonlinear sum-of-squares functional
with a model trust region strategy.},
summary = {The half Fourier transform is discussed
and its application to radar-return signals
with specular components examined. This
transform enables the desired part to be separated
from the specular impulsive components. The
problem is numerically solved by applying a trust-region
algorithm to a nonlinear least-squares formulation.}}
@article{JaraMack87,
author = {H. Jarausch and W. Mackens},
title = {Solving large nonlinear systems of equations by an adaptive
condensation process},
journal = NUMMATH,
volume = 50, number = 6, pages = {633--653}, year = 1987,
abstract = {The authors present an algorithm which efficiently
solves large nonlinear systems of the form $Au=F(u)$, $u \in
\Re^n$ whenever an (iterative) solver '$A^{-1}$'
for the symmetric positive definite matrix $A$ is
available and $F'(u)$ is symmetric. Such problems arise
from the discretization of nonlinear elliptic partial
differential equations. By means of an adaptive
decomposition process the authors split the original
system into a low dimensional system-to be treated by
any sophisticated solver-and a remaining
high-dimensional system, which can easily be solved
by fixed point iteration. Specifically the authors
choose a Newton-type trust region algorithm for the
treatment of the small system. They show global
convergence under natural assumptions on the
nonlinearity. The convergence results typical for
trust-region algorithms carry over to the full
iteration process. The only large systems to be
solved are linear ones with the fixed matrix $A$. Thus
existing software for positive definite sparse linear
systems can be used.},
summary = {The authors present an algorithm which efficiently
solves large nonlinear systems of the form $Au=F(u)$, $u \in
\Re^n$ whenever an (iterative) solver '$A^{-1}$'
for the symmetric positive definite matrix $A$ is
available and $F'(u)$ is symmetric. Such problems arise
from the discretization of nonlinear elliptic partial
differential equations. By means of an adaptive
decomposition process the original
system is split into a low-dimensional system-to and a
remaining high-dimensional system, which can easily
be solved by fixed point iteration.
A Newton-type trust-region algorithm is chosen for the
treatment of the small system. Convergence results typical for
trust-region algorithms carry over to the full
iteration process.}}
@article{Jarr91,
author = {F. Jarre},
title = {On the convergence of the method of analytic centers when
applied to convex quadratic programs},
journal = MP,
volume = 49, number = 3, pages = {341--358}, year = 1991}
@misc{Jarr98,
author = {F. Jarre},
title = {An {QQP}-Minimization Method for Semidefinite and Smooth
nonconvex programs},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {Recent international research has focussed on semidefinite
programs where the entries of an unknown matrix $X$ need to
be chosen in a certain optimal way under the constraint
that $X$ be symmetric and positive semi-definite. The
present frameworks for solving semidefinite programs allow
further convex constraints on the matrix entries. In many
real world applications, however, also smooth nonconvex
constraints occur. For such problems the standard approach
still often used to date is to impose an equality
constraint $X=LL^T$ where $L$ is lower tringular. It is not
surprising that standard nonlinear optimization software
fails for such a formulation; the Cholesky factor $L$ is
not unique when $X$ is singular; it lies in a nonlinear
nonconvex manifold, and methods based on linearizing the
equation $=LL^T$ perform very poorly in this situation. We
present an interior-point approach for solving such
problems, discuss local convergence properties, and present
some preliminary numerical results.},
summary = {An interior-point approach is presented for problems where
the entries of a positive semidefinite matrix $X$ have to
be optimally determined in the presence of non-convex
constraints on the entries of $X$. The method combines
ideas of a predictor-corrector interior-point method, of
the SQP method and of trust-region methods. Some
convergence results are given and very preliminary
numerical experiments discussed.}}
@article{JarrSaun95,
author = {F. Jarre and M. A. Saunders},
title = {A practical interior-point method for convex programming},
journal = SIOPT,
volume = 5, number = 1, pages = {149--171}, year = 1995}
@article{JensAgre86,
author = {J. J. A. Jensen and H. Agren},
title = {A direct, restricted-step, second-order {MC} {SCF} program
for large scale \emph{ab initio} calculations},
journal = {Chemical Physics},
volume = 104, number = 2, pages = {229--250}, year = 1986,
abstract = {A general purpose MC SCF program with a direct, fully
second-order and step-restricted algorithm is
presented. The direct character refers to the
solution of an MC SCF eigenvalue equation by means of
successive linear transformations where the
norm-extended hessian matrix is multiplicated onto a
trial vector without explicitly constructing the
hessian. This allows for applications to large
wavefunctions. In the iterative solution of the
eigenvalue equation a norm-extended optimization
algorithm is utilized in which the number of negative
eigenvalues of the Hessian is monitored. The step
control is based on the trust region concept and is
accomplished by means of a simple modification of the
Davidson-Liu simultaneous expansion method (1978) for
iterative calculation of an eigenvector. Convergence
to the lowest state of a symmetry is thereby
guaranteed, and test calculations also show reliable
convergence for excited states. The authors outline
the theory and describe in detail an efficient
implementation, illustrated with sample
calculations.},
summary = {A general purpose MC SCF program with a direct, fully
second-order and step-restricted algorithm is
presented. The step control is based on the trust-region
concept. Convergence to the lowest state of a symmetry is
guaranteed, and test calculations also show reliable
convergence for excited states. }}
@article{JensPoly94,
author = {D. Jensen and R. Polyak},
title = {On the convergence of a modified barrier method for convex
programming},
journal = {IBM J. Res. Develop.},
volume = 38, number = 3, pages = {307--320}, year = 1994}
@techreport{JensPolySchn92,
author = {D. Jensen and R. Polyak and R. Schneur},
title = {Numerical experience with modified barrier functions method
for linear programming},
institution = IBMWATSON, address = IBMWATSON-ADDRESS,
type = {Research Report}, number = {RC 18415}, year = 1992}
@article{JiKritAbouTont99,
author = {X. S. Ji and W. Kritpiphat and A. Aboudheir and
P. Tontiwachwuthikul},
title = {Mass transfer parameter estimation using optimization
technique: Case study in {CO}2 absorption with chemical
reaction},
journal = {Canadian Journal of Chemical Engineering},
volume = 77, number = 1, pages = {69--73}, year = 1999,
abstract = {This paper proposes a new approach of applying an
optimization technique to simultaneously determine a
physical liquid-film mass transfer coefficient
($k(L)(o)$) and effective interfacial area ($a(v)$) from
a pilot plant data. The mass transfer mechanism of
the CO2-NaOH system was modeled using the two-film
theory to represent the behaviors of packed
absorbers. The model presents an overall absorption
rate (R-v) as a function of $k(L)(o)$ and $a(v)$. The
optimization algorithm used in this study follows a
modified Levenberg-Marquardt method with a trust
region approach. The R-v predictions from the model
are in good agreement with the experimental data,
with an average error of 6.5\%.},
summary = {A new approach of applying an optimization technique
to simultaneously determine a physical liquid-film
mass transfer coefficient ($k(L)(o)$) and effective
interfacial area ($a(v)$) from a pilot plant data is
considered. The mass transfer mechanism of the
CO2-NaOH system is modeled using the two-film theory
to represent the behaviors of packed absorbers. The
model presents an overall absorption rate (R-v) as a
function of $k(L)(o)$ and $a(v)$. The optimization
algorithm used in this study follows a modified
Levenberg-Morrison-Marquardt method with a trust
region approach. The R-v predictions from the model
are in good agreement with the experimental data,
with an average error of 6.5\%.}}
@article{JianQi96,
author = {H. Jiang and L. Qi},
title = {Globally and superlinearly convergent trust-region
algorithms for convex {SC$^1$}-minimization problems and
its application to stochastic programs},
journal = JOTA,
volume = 90, number = 3, pages = {649--669}, year = 1996,
abstract = {A function mapping from R(n) to R is called an SC1-function
if it is differentiable and its derivative is semismooth. A
convex SC1-minimization problem is a convex minimization
problem with an SC objective function and linear
constraints. Applications of such minimization problems
include stochastic quadratic programming and minimax
problems. In this paper, we present a globally and
superlinearly convergent trust-region algorithm for solving
such a problem. Numerical examples are given on the
application of this algorithm to stochastic quadratic
programs.},
summary = {A globally and superlinearly convergent trust-region
algorithm for solving SC1 problems is presented.
Numerical examples are given on the application of this
algorithm to stochastic quadratic programs.}}
@article{JianQi97,
author = {H. Jiang and L. Qi},
title = {A new nonsmooth equations approach to nonlinear
complementarity problems},
journal = SICON,
volume = 35, number = 1, pages = {178--193}, year = 1997}
@article{JianFukuQiSun98,
author = {H. Jiang and M. Fukushima and L. Qi and D. Sun},
title = {A trust region method for solving generalized
complementarity problems},
journal = SIOPT,
volume = 8, number = 1, pages = {140--158}, year = 1998,
abstract = {Based on a semismooth equation reformulation using
Fischer's function, a trust region algorithm is proposed
for solving the generalized complementarity problem (GCP).
The algorithm uses a generalized Jacobian of the function
involved in the semismooth equation and adopts the squared
natural residual of the semismooth equation as a merit
function. The proposed algorithm is applicable to the
nonlinear complementarity problem, because the latter
problem is a special case of the GCP. Global convergence
and, under a nonsingularity assumption, local Q-superlinear
(or quadratic) convergence of the algorithm are
established. Moreover, calculation of a generalized
Jacobian is discussed and numerical results are presented.},
summary = {Based on a semi-smooth equation reformulation using
Fischer's function, a trust-region algorithm is proposed
for solving the generalized complementarity problem (GCP).
It uses a generalized Jacobian of the function
involved in the semi-smooth equation and adopts the squared
natural residual of the semi-smooth equation as a merit
function. Global convergence and, under a nonsingularity
assumption, a local Q-superlinear (or quadratic) rate of
convergence are established. Calculation of a generalized
Jacobian is discussed and numerical results presented.}}
@article{JittOsbo80,
author = {K. Jittorntrum and M. R. Osborne},
title = {A modified barrier function method with improved rate of
convergence for degenerate problems},
journal = {Journal of the Australian Mathematical Society (Series B)},
volume = 21, pages = {305--329}, year = 1980}
@article{JonsLars90,
author = {O. Jonsson and T. Larsson},
title = {A note on step-size restrictions in approximation procedures
for structural optimization},
journal = {Computers and Structures},
volume = 37, number = 3, pages = {259--263}, year = 1990,
abstract = {Different possibilities are discussed for the
restriction of the step size in the well known
iterative approximation concept for solving
structural optimization problems. Such restrictions
might for some problem instances be necessary to
stabilize the behaviour of the solution procedure and
to ensure convergence. There are two basic means to
achieve such a restriction: either by using trust
region constraints or by inducting a penalty on
getting remote from the approximation point. In
structural optimization solution procedures, the
first of these possibilities is commonly used. It is
demonstrated how a penalty term can be used instead,
still making it possible to use the efficient dual
concept. Relations to other mathematical programming
methods are outlined and a small numerical example is
presented.},
summary = {Different possibilities are discussed for the
restriction of the step size in
iterative approximation methods for
structural optimization problems: by using either
trust-region constraints or a penalty on
the distance from the approximation point. The
first approach is commonly used. It is
demonstrated how a penalty term can be used instead.}}
@article{JudiPire89,
author = {J. J. J\'{u}dice and F. M. Pires},
title = {Direct methods for convex quadratic programs subject to box
constraints},
journal = {Investigaci\'{o}n Operacional},
volume = 9, pages = {23--56}, year = 1989}
%%% K %%%
@article{Kani66,
author = {S. Kaniel},
title = {Estimates for some computational techniques in linear
algebra},
journal = MC,
volume = 20, number = 95, pages = {369--378}, year = 1966}
@article{Kant48,
author = {L. Kantorovich},
title = {Functional analysis and applied mathematics},
journal = {Uspehi Matematicheskih Nauk},
volume = 3, pages = {89--185}, year = 1948}
@article{KanzYamaFuku97,
author = {Ch. Kanzow and N. Yamashita and M. Fukushima},
title = {New {NCP}-Functions and Their Properties},
journal = JOTA,
volume = 94, number = 1, pages = {115--135}, year = 1997}
@inproceedings{KanzZupk98,
author = {Ch. Kanzow and M. Zupke},
title = {Inexact Trust-Region Methods for Nonlinear Complementarity
Problems},
crossref = {FukuQi98}, pages = {211--235},
abstract = {In order to solve the nonlinear complementarity problem, we
first reformulate it as a nonsmooth system of equations by
using a recently introduced NCP-function. We then apply a
trust-region-type method to this system of equations. Our
trust-region method allows an inexact solution of the
trust-region-subproblem. We show that the algorithm is
well-defined for a general nonlinear complementarity
problem and that it has some nice global and local
convergence properties. Numerical results indicate that the
new method is quite promising.},
summary = {The nonlinear complementarity problem is reformulated as a
non-smooth system of equations by using a recently
introduced NCP-function. A trust-region-type method is then
applied to the resulting system of equations, that allows
an inexact solution of the trust-region subproblem.
The algorithm is well-defined for a general
nonlinear complementarity problem and has
global and local convergence properties. Numerical results
are discussed.}}
@inproceedings{KariRendWolk94,
author = {S. E. Karisch and F. Rendl and H. Wolkowicz},
title = {Trust regions and relaxations for the quadratic assignment
problem},
editor = {P. M. Pardalos and H. Wolkowicz},
booktitle = {Quadratic Assignment and Related Problems. DIMACS Workshop.},
publisher = AMS, address = AMS-ADDRESS,
pages = {199--219}, year = 1994,
abstract = {General quadratic matrix minimization problems, with
orthogonal constraints, arise in continuous
relaxations for the (discrete) quadratic assignment
problem (QAP). Currently, bounds for QAP are obtained
by treating the quadratic and linear parts of the
objective function, of the relaxations,
separately. This paper handles general objectives as
one function. The objectives can be both
nonhomogeneous and nonconvex. The constraints are
orthogonal or Loewner partial order (positive
semidefinite) constraints. Comparisons are made to
standard trust region subproblems. Numerical results
are obtained using a parametric eigenvalue
technique.},
summary = {General quadratic matrix minimization problems, with
orthogonal constraints, arise in continuous
relaxations for the (discrete) quadratic assignment
problem (QAP). Currently, bounds for QAP are obtained
by treating the quadratic and linear parts of the
objective function, of the relaxations,
separately. It is shown how to handle general objectives as
one function. The objectives can be both
non-homogeneous and non-convex. The constraints are
orthogonal or Loewner partial order (positive
semidefinite) constraints. Comparisons are made with
standard trust-region subproblems. Numerical results
are obtained using a parametric eigenvalue
technique.}}
@article{Karm84,
author = {N. Karmarkar},
title = {A new polynomial-time algorithm for linear programming},
journal = {Combinatorica},
volume = 4, pages = {373--395}, year = 1984}
@mastersthesis{Karu39,
author = {W. Karush},
title = {Minima of functions of several variables with inequalities
as side conditions},
school = {Department of Mathematics},
address = {University of Chicago, Illinois, USA},
year = 1939}
@article{Kauf00,
author = {L. Kaufman},
title = {A reduced storage, quasi-{N}ewton trust region approach to
function optimization},
journal = SIOPT,
volume = 10, number = 1, pages = {56--69}, year = 2000,
abstract = {In this paper we consider several algorithms for reducing
the storage when using a quasi-Newton method in a dogleg
trust region setting for minimizing functions of many
variables. Secant methods require 0 (n 2 ) locations to
store an approximate Hessian and 0 (n 2 ) operations per
iteration when minimizing a function of n variables. This
storage requirement becomes worrisome when n becomes
large. Our algorithms use a BFGS update and require
kn storage and 4kn + 0 (k 2 ) operations per iteration,
but may require more iterations than the standard
trust region techniques. Typically k is between 10 and
100. Our dogleg trust region strategies involve
expressions with matrix products with both the inverse
of this Hessian and with the Hessian itself. Our techniques
for updating expressions for the Hessian and its
inverse can be used to improve the performance of
line search, limited memory algorithms.},
summary = {A limited-memory quasi-Newton BFGS algorithm for
unconstrained optimization is described that uses a dogleg
trust-region scheme. This technique uses products both with
the approximate Hessian and its inverse.}}
@article{KeHan95a,
author = {X. Ke and J. Han},
title = {A class of nonmonotone trust region algorithms for
constrained optimizations},
journal = CSB,
volume = 40, number = 16, pages = {1321--1324}, year = 1995,
abstract = {In this note, we consider the following constrained
optimization problem (CQP), where $f: R''. R$ is
continuously differentiable function on a closed convex set
$\Omega$. For the constrained optimization problem (CQP), a
class of nonmonotone trust region algorithms is proposed in
sec. 1. In sec.2, the global convergence of this class of
algorithms is proved. In sec. 3, some results about the
Cauchy point are provided. The nonmonotone technique in
this algorithm differs from those in existing nonmonotone
algorithms, i.e. nonmonotone algorithms with linear search
for unconstrained and constrained optimizations , and the
nonmonotone trust region algorithm for unconstrained
optimization.},
summary = {The constrained optimization problem of minimizing a
continuously differentiable function over a closed convex
set is considered. A class of globally convergent
non-monotone trust-region algorithms is proposed for this
problem.}}
@article{KeHan95b,
author = {X. Ke and J. Han},
title = {A nonmonotone trust region algorithm for equality
constrained optimization},
journal = {Science in China Series A-Mathematics Physics Astronomy and
Technological Sciences},
volume = 38, number = 6, pages = {683--695}, year = 1995,
abstract = {A trust region algorithm for equality constrained
optimization is proposed, which is a nonmonotone one in a
certain sense. The augmented Lagrangian function is used as
a merit function. Under certain conditions, the global
convergenece theorems of the algorithm are proved.},
summary = {A non-monotone trust region algorithm is proposed for the
minimization of smooth functions subject to nonlinear
equality constraints. It handles feasibility as
\citebb{CeliDennTapi85} and allows non-monotonicity in the
augmented Lagrangian which is used as a merit function.}}
@article{KeHan96,
author = {X. Ke and J. Han},
title = {A nonmonotone trust region algorithm for unconstrained
nonsmooth optimization},
journal = CSB,
volume = 41, number = 3, pages = {197--201}, year = 1996,
summary = {A non-monotone trust-region method is presented for the
solution of non-smooth unconstrained problems. This
algorithm uses the concept of ``iteration functions'' of
\icitebb{QiSun94}. Global convergence to a Dini stationary
point is proved.}}
@inproceedings{KehtWinMull87,
author = {N. Kehtarnavaz and M. Z. Win and N. Mullani},
title = {Estimation of diastole to systole changes from cardiac
{PET} images},
booktitle = {Proceedings of the Ninth Annual Conference of the IEEE
Engineering in Medicine and Biology Society},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 2, pages = {850--851}, year = {1987},
abstract = {The changes in the myocardium thickness, left
ventricle diameter and tracer activity between
diastole and systole are estimated from cardiac
positron-emission-tomography (PET) images. A
comparative study has been carried out between the
widely used international mathematical subroutine
library (IMSL) and the model-trust-region (MTR)
parameter estimation algorithm. It has been shown
that the MTR algorithm converges regardless of the
initial parameter values (IPV) chosen. To reduce the
number of iterations, a preprocessor has been
developed to provide close IPV to the true parameter
values (TPV). Myocardial, left ventricular and tracer
activity changes are plotted as functions of bipolar
angle. The graphs can be used as a diagnostic tool
for abnormal heart conditions.},
summary = {The changes in the myocardium thickness, left
ventricle diameter and tracer activity between
diastole and systole are estimated from cardiac
positron-emission-tomography (PET) images. A
comparative study is carried out between the IMSL
mathematical subroutine library and the model-trust-region
(MTR) parameter estimation algorithm. It is shown
that the MTR algorithm converges regardless of the
initial parameter values chosen. To reduce the
number of iterations, a preprocessor has been
developed to provide better starting values.
The method is used as a diagnostic tool
for abnormal heart conditions.}}
@article{Kell73,
author = {E. L. Keller},
title = {The general quadratic programming problem},
journal = MP,
volume = 5, number = 3, pages = {311--337}, year = 1973}
@article{KellSach87,
author = {C. T. Kelley and E. W. Sachs},
title = {Quasi-{N}ewton methods and unconstrained optimal control
problems},
journal = SICON,
volume = 25, number = 6, pages = {1503--1517}, year = 1987}
@article{KellSach99,
author = {C. T. Kelley and E. W. Sachs},
title = {A trust region method for parabolic boundary control
problems},
journal = SIOPT,
volume = 9, number = 4, pages = {1064--1081}, year = 1999,
abstract = {In this paper, we develop a trust region algorithm for
constrained parabolic boundary control problems. The method
is a projected form of the Steihaug trust-region CG method
with a smoothing step added at each iteration to improve
performance in the global phase and provide
mesh-independent sup-norm convergence in the terminal
phase.},
summary = {A trust-region algorithm for constrained parabolic boundary
control problems is developed. The method is a projected
form of the Steihaug-Toint method with a
smoothing step added at each iteration to improve
performance in the global phase and provide
mesh-independent sup-norm convergence in the terminal
phase.}}
@article{KellKeye98,
author = {C. T. Kelley and D. E. Keyes},
title = {Convergence analysis of pseudo-transient continuation},
journal = SINUM,
volume = 35, number = 2, pages = {508--523}, year = 1998,
abstract = {Pseudo-transient continuation is a well-known and
physically motivated technique for computation of
steady-state solutions of time-dependent partial
differential equations. Standard globalization strategies
such as line search or trust-region methods often stagnate
at local minima. Pseudo-transient continuation succeeds in
many of these cases by taking advantage of the underlying
PDE structure of the problem. Though widely employed, the
convergence of this scheme is rarely discussed. In this
paper we prove convergence for a generic form of
pseudo-transient continuation and illustrate it with two
practical strategies.},
summary = {Pseudo-transient continuation is a well-known and
physically motivated technique for computation of
steady-state solutions of time-dependent partial
differential equations. Standard globalization strategies
such as linesearch or trust-region methods often stagnate
at local minima. Pseudo-transient continuation succeeds in
many of these cases by taking advantage of the underlying
PDE structure of the problem. Convergence
for a generic form of pseudo-transient continuation is
proved, and illustrated with two practical strategies.}}
@article{KeLiuXu96,
author = {X. Ke and G. Liu and D. Xu},
title = {A nonmonotone trust region algorithm for unconstrained
nonsmooth optimization},
journal = CSB,
volume = 41, number = 3, pages = {197--201}, year = 1996,
summary = {A globally convergent trust-region algorithm is proposed
for unconstrained minimization of locally Lipschitzian
functions, that generalizes the approach of
\citebb{QiSun94} by allowing a non-monotone
sequence of objective function values.}}
@techreport{KhalByrdSchn99,
author = {H. F. Khalfan and R. H. Byrd and R. B. Schnabel},
title = {Retaining Convergence Properties of Trust Region Methods
Without Extra Gradient Evaluations},
institution = {Department of Mathematics and Computer Science,
University of Colorado},
address = {Boulder, Colorado, USA},
year = 1999,
abstract = {Several recent computational studies have shown that
trust-region quasi-Newton methods using the SR1, PSB,
and BFGS updates are effective methods for solving
unconstrained optimization problems. In addition, the
analyses in \citebb{Powe75} and \citebb{ByrdKhalSchn96}
demonstrate strong convergence properties for some
trust-region quasi-Newton methods. A computational
disadvantage of the methods analyzed in these papers, for
which the strongest convergence properties among
trust-region quasi-Newton methods have been shown, is that
the update at rejected points requires a gradient
evaluation that would not otherwise be made. In this
paper, we propose a modification of the PSB method that
uses only the function value at rejected points to make the
update at those points. We then show how to modify
Powell's analysis of the PSB method to prove the same
$q$-superlinear convergence result for the new method.
Finally, we discuss the issues and difficulties involved in
extending this approach to trust region methods using
updates in the Broyden class, such as the BFGS, SR1 and
DFP.},
summary = {A modification of a trust-region method based on the PSB
quasi-Newton update is proposed method that uses only the
function value at rejected points to make the update at
those points. The same $q$-superlinear convergence result
as for the original algorithm holds for the new method.}}
@inproceedings{Kiwi89,
author = {K. C. Kiwiel},
title = {A survey of bundle methods for non-differentiable
optimization},
crossref = {IriTana89}, pages = {263-282}}
@article{Kiwi89b,
author = {K. C. Kiwiel},
title = {An ellipsoid trust region bundle method for nonsmooth
convex minimization},
journal = SICON,
volume = 27, number = 4, pages = {737--757}, year = 1989,
abstract = {This paper presents a bundle method of descent for
minimizing a convex (possibly nonsmooth) function f
of several variables. At each iteration the algorithm
finds a trial point by minimizing a polyhedral model
of f subject to an ellipsoid trust region
constraint. The quadratic matrix of the constraint,
which is updated as in the ellipsoid method, is
intended to serve as a generalized 'Hessian' to
account for 'second-order' effects, thus enabling
faster convergence. The interpretation of generalized
Hessians is largely heuristic, since so far this
notion has been made precise by J.L. Goffin only in
the solution of linear inequalities. Global
convergence of the method is established and
numerical results are given.},
summary = {A bundle method of descent for
minimizing a convex (possibly non-smooth) function
of several variables. At each iteration the algorithm
finds a trial point by minimizing a polyhedral model
subject to an ellipsoid trust-region
constraint. The quadratic matrix of the constraint,
which is updated as in the ellipsoid method, is
interpreted as a generalized 'Hessian' to
account for 'second-order' effects, thus enabling
faster convergence. Global
convergence of the method is established and
numerical results are given.}}
@article{Kiwi96,
author = {K. C. Kiwiel},
title = {Restricted step and {L}evenberg-{M}arquardt techniques in
proximal Bundle Methods for nonconvex nondifferentiable
optimization},
journal = SIOPT,
volume = 6, number = 1, pages = {227--249}, year = 1996,
abstract = {Two methods are given for minimizing locally Lipschitzian
upper semidifferentiable functions. They employ extensions
of restricted step (trust region) and Levenberg-Marquardt
techniques that are widely used in other contexts.
Extensions to linearly constrained optimization are
discussed. Preliminary numerical experience is reported.},
summary = {Two methods are given for minimizing locally Lipschitzian
upper semidifferentiable functions. They employ extensions
of restricted step (trust region) and
Levenberg-Morrison-Marquardt techniques. Extensions to
linearly constrained optimization are discussed. Preliminary
numerical experience is reported.}}
@phdthesis{Knot83,
author = {O. Knoth},
title = {{M}arquardt-\"{a}hnliche {V}erfahren zur {M}inimierung
nichtlinearer {F}unktionen},
school = {Martin-Luther University},
address = {Halle-Wittenberg, Germany},
year = 1983,
note = {(in German)},
summary = {Accumulation points of the
trust-region Newton method for unconstrained optimization
are proved to satisfy second order optimality
conditions. The use of negative curvature in a
``projected'' Marquardt algorithm is also introduced.
Finally, the characterization of local minima for the
$\ell_2$ trust-region subproblem is considered together
with the use of an exact penalty function for this
subproblem.}}
@book{Knut73,
author = {D. E. Knuth},
title = {The Art of Computer Programming, Volume 3,
Sorting and Searching},
publisher = ADW, address = ADW-ADDRESS,
year = 1973}
@article{Knut76,
author = {D. E. Knuth},
title = {Big {O}micron and {B}ig {O}mega and {B}ig {T}heta},
journal = {ACM SIGACT News},
volume = 8, number = 2, pages = {18--24}, year = 1976}
@article{Koji93,
author = {F. Kojima},
title = {Back-propagation learning using the trust region algorithm
and application to nondestructive testing in applied
electromagnetics},
journal = {International Journal of Applied Electromagnetics in
Materials},
volume = 4, number = 7, pages = {27--33}, year = 1993,
abstract = {An artificial neural network is applied by nondestructive
inspections in aerospace materials. The use of an
artificial neural network is presented for classifying
testing data as corresponding to sample materials with
defect and without defect. The back-propagation learning
for a multi-layer feed-forward neural network is applied to
this classification. The trust region method is adopted to
the back-propagation learning problem. Results of numerical
tests are summarized.},
summary = {An artificial neural network is applied by non-destructive
inspections in aerospace materials. The back-propagation
learning for a multi-layer feed-forward neural network is
applied to the resulting classification problem. The
trust-region method is adapted to the back-propagation
learning problem. Numerical tests are discussed.}}
@inproceedings{KojiKawa93,
author = {F. Kojima and H. Kawaguchi},
title = {Backpropagation learning algorithm for nondestructive
testing by thermal imager (aerospace materials)},
booktitle = {IJCNN '93-Nagoya. Proceedings of 1993 International Joint
Conference on Neural Networks},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 1, pages = {955--958}, year = 1993,
abstract = {An artificial neural network is applied to
nondestructive inspections using thermal imager. The
use of an artificial neural network is presented for
classifying test data as corresponding to bonded and
disbonded regions in sample materials. The
backpropagation learning for a multi-layer
feedforward neural network is applied to this
classification. The trust region method is adopted to
the backpropagation learning problem. Results of
numerical tests are summarized.},
summary = {An artificial neural network is applied to
non-destructive inspections using thermal imager. The
use of an artificial neural network is presented for
classifying test data as corresponding to bonded and
disbonded regions in sample materials. The
backpropagation learning for a multi-layer
feedforward neural network is applied to this
classification. The trust-region method is adopted to
the backpropagation learning problem. Numerical results
are summarized.}}
@inproceedings{KrukWolk98,
author = {S. Kruk and H. Wolkowicz},
title = {{SQ}${}^2${P}, Sequential Quadratic Constrained Quadratic
Programming},
crossref = {Yuan98}, pages = {177--204},
abstract = {We follow the popular approach for unconstrained
minimization, i.e. we develop a local quadratic model at a
current approximate minimizer in conjunction with a trust
region. We then minimize this local model in order to find
the next approximate minimizer. Asymptotically, finding the
local minimizer of the quadratic model is equivalent to
applying Newton's method to the stationarity condition. For
constrained problems, the local quadratic model corresponds
to minimizing a quadratic expression of the objective
subject to quadratic approximations of the constraints
(Q${}^2$P), with an additional trust region. This quadratic
model is intractable in general and is usually handled by
using linear approximations of the constraints and
modifying the Hessian of the objective function using the
Hessian of the Lagrangian, i.e. a SQP approach. Instead, we
solve the Lagrangian relaxation of Q${}^2$P using
semi-definite programming. We develop this framework and
present an example which illustrates the advantages over
the standard SQP approach.},
summary = {An algorithm is proposed for constrained nonlinear
programming in which a quadratic model of the
objective function is minimized, at each iteration, subject
to quadratic approximations of the constraints (Q${}^2$P)
and an additional trust region. As this subproblem is in
general intractable, the Lagrangian relaxation of Q${}^2$P
is instead solved using semi-definite programming. An
example illustrates the advantages over the standard SQP
approach.}}
@inproceedings{KuhnTuck51,
author = {H. W. Kuhn and A. W. Tucker},
title = {Nonlinear Programming},
booktitle = {Proceedings of the second Berkeley symposium on
mathematical statistics and probability},
editor = {J. Neyman},
publisher = {University of Berkeley Press}, address = {California, USA},
year = 1951}
@article{KwokKamaWats85,
author = {H. H. Kwok and M. P. Kamat and L. T. Watson},
title = {Location of stable and unstable equilibrium-configurations
using a model trust region quasi-{N}ewton method and
tunnelling},
journal = {Computers and Structures},
volume = 21, number = 5, pages = {909--916}, year = 1985,
abstract = {A hybrid method for locating multipole equilibrium
configurations has been proposed recently. The hybrid
method combined the efficiency of a quasi-Newton method
capable of locating stable and unstable equilibrium
solutions with a robust homotopy method capable of tracking
equilibrium paths, with turning points and exploiting
sparsity of the Jacobian matrix at the same time. A
quasi-Newton method in conjunction with a deflation
technique is proposed here as an alternative to the hybrid
method. The proposed method not only exploits sparsity and
symmetry, but also represents an improvement in efficiency},
summary = {The conjunction of a quasi-Newton method with a deflation
technique is proposed as an alternative to the hybrid
method for locating multipole equilibrium configurations.
The proposed method not only exploits sparsity and
symmetry, but also represents an improvement in efficiency.
It uses a double dogleg globalization strategy.}}
%%% L %%%
@article{LaleNocePlan98,
author = {M. Lalee and J. Nocedal and T. D. Plantenga},
title = {On the implementation of an algorithm for large-scale
equality constrained optimization},
journal = SIOPT,
volume = 8, number = 3, pages = {682--706}, year = 1998,
abstract = {This paper describes a software implementation of Byrd and
\citebb{Omoj89}'s trust-region algorithm for solving
nonlinear equality constrained optimization problems. The
code is designed for the efficient solution of large
problems and provides the user with a variety of linear
algebra techniques for solving the subproblems occurring in
the algorithm. Second derivative information can be used,
but when it is not available, limited memory quasi-Newton
approximations are made. The performance of the code is
studied using a set of difficult test problems from the
{\sf CUTE} collection.},
summary = {A software implementation of Byrd and \citebb{Omoj89}'s
trust-region algorithm for nonlinear equality constrained
optimization is described. The
code is designed for the efficient solution of large
problems and provides the user with a variety of linear
algebra techniques for solving the subproblems occurring in
the algorithm. Second derivative information can be used,
as well as limited memory quasi-Newton approximations. The
performance of the code is studied using a set of difficult
test problems from the {\sf CUTE} collection.}}
@book{LancTism85,
author = {P. Lancaster and M. Tismenetsky},
title = {The Theory of Matrices},
publisher = AP, address = AP-ADDRESS,
edition = {second}, year = 1985}
@article{Lanc50,
author = {C. Lanczos},
title = {An iteration method for the solution of the eigenvalue
problem of linear differential and integral operators},
journal = {Journal of research of the National Bureau of Standards B},
volume = 45, pages = {225--280}, year = 1950}
@article{Lann97,
author = {A. Lannes},
title = {Phase-closure imaging in algebraic graph theory: a new
class of phase-calibration algorithms},
journal = {Journal of the Optical Society of America A-Optics Image
Science and Vision},
volume = 15, number = 2, pages = {419--429}, year = 1997,
abstract = {A new class of phase-calibration algorithms is presented.
The originality of these algorithms, as well as their
efficiency, results from certain particular structures, the
analysis of which calls on algebraic graph theory. The
corresponding optimization process, which is based on the
principle of the trust-region methods, proves to be well
suited to these structures. The main message that emerges
from the study is very clear: the traditional notions of
phase closure imaging can be understood and refined in a
wider framework. The implications of this research
therefore concern all the fields in which the notion of
phase closure plays a key role: weak-phase imaging in
optical interferometry, radio imaging, remote sensing by
aperture synthesis, etc.},
summary = {A class of phase-calibration algorithms is presented.
Their originality, as well as their efficiency, results
from certain particular structures, the analysis of which
calls on algebraic graph theory. The corresponding
optimization process, which is based on the principle of
the trust-region methods, proves to be well suited to these
structures. The main result is that the traditional notions
of phase closure imaging can be understood and refined in a
wider framework. This has implications in all the fields
where the notion of phase closure plays a key role, such as
weak-phase imaging in optical interferometry, radio imaging
and remote sensing by aperture synthesis.}}
@article{Lann98,
author = {A. Lannes},
title = {Weak-phase imaging in optical interferometry},
journal = {Journal of the Optical Society of America A-Optics Image
Science and Vision},
volume = 15, number = 4, pages = {811--824}, year = 1998,
abstract = {The first imaging devices of optical interferometry are
likely to be of weak phase, typically: a set of
three-element arrays independently observing the same
object. The study of their imaging capabilities refers to
appropriate optimization methods, which essentially address
the self-calibration process and its stability. A general
survey of these techniques is given, and it is shown, in
particular, how the related algorithms can be used for
examining the imaging capabilities of weak-phase
interferometric devices. The phase-calibration algorithm
involved in the self-calibration cycles is based on the
principle underlying the trust-region methods. It benefits
from certain remarkable properties, the analysis of which
appeals to algebraic graph theory. The Fourier synthesis
operation, which is also involved in these cycles, is
performed by means of WIPE, a methodology recently
introduced in radio imaging and optical interferometry.
(WIPE is reminiscent Of CLEAN, a widely used technique in
astronomy). In the related theoretical framework the
stability of the image-reconstruction process is controlled
by considering certain elements of the singular-value
decomposition of the derivative of the self- calibration
operator. For example, the largest singular value of this
derivative, which depends on the interferometric
configuration and on the object thus imaged, provides a key
indication of the observational limits of these
experimental devices.},
summary = {The first imaging devices of optical interferometry are
likely to be of weak phase, typically: a set of
three-element arrays independently observing the same
object. The study of their imaging capabilities refers to
appropriate optimization methods, which essentially address
the self-calibration process and its stability. A general
survey of these techniques is given, and it is shown, in
particular, how the related algorithms can be used for
examining the imaging capabilities of weak-phase
interferometric devices. The phase-calibration algorithm
involved in the self-calibration cycles is based on the
principle underlying the trust-region methods. Its
remarkable properties are shown using
the algebraic graph theory. The Fourier synthesis
operation, which is also involved in these cycles, is
performed by means of WIPE, a methodology
introduced in radio imaging and optical interferometry.
In the related theoretical framework the
stability of the image-reconstruction process is controlled
by considering certain elements of the singular-value
decomposition of the derivative of the self- calibration
operator. For example, the largest singular value of this
derivative, which depends on the interferometric
configuration and on the object thus imaged, provides a key
indication of the observational limits of these
experimental devices.}}
@article{LasdPlumYu95,
author = {L. S. Lasdon and J. Plummer and G. Yu},
title = {Primal-dual and primal interior point algorithms for
general nonlinear programs},
journal = ORSAC,
volume = 7, number = 3, pages = {321--332}, year = 1995,
abstract = {An interior point algorithm for general nonlinear
programs is presented. Inequality constraints are
converted to equalities with slack variables. All
bounds are handled with a barrier term in the
objective. The Kuhn-Tucker system of the resulting
equality constrained barrier problem is solved
directly by Newton's method. Primal-dual, primal, and
primal-dual with trust region variants are developed
and evaluated. An implementation which utilizes the
true Lagrangian Hessian and exploits Jacobian and
Hessian sparsity is described. Computational results
are presented and discussed.},
summary = {An interior point algorithm for general nonlinear
programs is presented. Inequality constraints are
converted to equalities with slack variables. All
bounds are handled with a barrier term in the
objective. The Kuhn-Tucker system of the resulting
equality constrained barrier problem is solved
directly by Newton's method. Primal-dual, primal, and
primal-dual with trust-region variants are developed
and evaluated. An implementation which utilizes the
true Lagrangian Hessian and exploits Jacobian and
Hessian sparsity is described. Computational results
are presented and discussed.}}
@book{LawsHans74,
author = {C. L. Lawson and R. J. Hanson},
title = {Solving Least Squares Problems},
publisher = PH, address = PH-ADDRESS,
year = 1974,
note = {Reprinted as \emph{Classics in Applied Mathematics 15}, SIAM,
Philadelphia, USA, 1995}}
@misc{LeibSach99,
author = {F. Leibfritz and E. W. Sachs},
title = {Optimal Static Output Feedback Design using a Trust Region
Interior Point Method},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999,
abstract = {We consider the problem of designing feedback control laws
when a complete set of state variables is not available.
The resulting nonlinear and nonconvex matrix optimization
problem including SDP-constraints for determining the
optimal feedback gain will be solved by a trust region
interior point approach. The algorithm will be discussed
in some details. Finally, using test examples from optimal
output feedback design we demonstrate the usefulness of
this approach numerically.},
summary = {The problem of designing feedback control laws is considered
when a complete set of state variables is not available.
The resulting nonlinear and non-convex matrix optimization
problem including semi-definiteness constraints for
determining the optimal feedback gain is solved by a
trust-region interior-point approach. Test examples from
optimal output feedback design numerically demonstrate the
usefulness of the approach.}}
@inproceedings{LemaZowe94,
author = {C. Lemar\'{e}chal and J. Zowe},
title = {A condensed introduction to bundle methods in nonsmooth
optimizations},
crossref = {Sped94}, pages = {357--382}}
@article{Lesc91,
author = {M. Lescrenier},
title = {Convergence of trust region algorithms for optimization
with bounds when strict complementarity does not hold},
journal = SINUM,
volume = 28, number = 2, pages = {476--495}, year = 1991,
abstract = {\citebb{ConnGoulToin88a} have proposed a class of trust
region algorithms for minimizing nonlinear functions whose
variables are subjected to simple bound constraints. In
their convergence analysis, they show that if the strict
complementarity condition holds, the considered algorithms
reduce to an unconstrained calculation after finitely many
iterations, allowing fast asymptotic rates of convergence.
This paper analyzes the behaviour of these iterative
processes in the case where the strict complementarity
condition is violated. It is proved that inexact Newton
methods lead to superlinear or quadratic rates of
convergence, even if the set of active bounds at the
solution is not entirely detected. Practical criteria for
stopping the inner iterations of the algorithms are
deduced, ensuring these rates of convergence.},
summary = {The behaviour of the trust-region algorithms of
\citebb{ConnGoulToin88a} for optimization with simple
bounds is analyzed in the case where the strict
complementarity condition is violated. It is proved that
inexact Newton methods lead to superlinear or quadratic
rates of convergence, even if the set of active bounds at
the solution is not entirely detected. Practical criteria
for stopping the inner iterations of the algorithms are
deduced.}}
@book{LeTaGlow89,
author = {Le Tallec, P. and Glowinski, R.},
title = {Augmented Lagrangian and Operator-Splitting Methods in
Nonlinear Mechanics},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1989}
@article{Leve44,
author = {K. Levenberg},
title = {A Method For The Solution Of Certain Problems In Least
Squares},
journal = {Quarterly Journal on Applied Mathematics},
volume = 2, pages = {164--168}, year = 1944,
summary = {The standard method for solving least-squares problems
which leads to nonlinear normal equations depends on a
reduction of the residuals to linear form by first-order
Taylor approximations. taken about a trial solution for the
parameters. If the usual least-squares procedure with these
linear approximations yields new values for the parameters
which are not sufficiently close to the trial values, the
neglect of second and higher order terms may invalidate the
process. This difficulty may be alleviated by limiting the
absolute values of the parameters and to simultaneously
minimize the sum of squares of the approximating residuals
under these ``damped'' conditions.}}
@article{LeviPoly66,
author = {E. S. Levitin and B. T. Polyak},
title = {Constrained minimization problems},
journal = {U.S.S.R. Comput. Math. Math. Phys.},
volume = 6, pages = {1--50}, year = 1966}
@misc{Lewi96,
author = {R. M. Lewis},
title = {A trust region framework for managing approximation models
in engineering optimization},
howpublished = {AIAA paper 96-4101, presented at the Sixth AIAA/NASA/ISSMO
Symposium on Multidisplinary Analysis and Design, Bellevue,
Washington},
year = 1996,
summary = {Non-quadratic models are proposed for the trust-region
minimization of expensive functions from engineering
applications.}}
@techreport{Li93,
author = {Y. Li},
title = {Centering, trust region, reflective techniques for
nonlinear minimization subject to bounds},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR93-1385}, year = 1993,
abstract = {Bound-constrained nonlinear minimization problems occur
frequently in practice. Most existing methods belong to an
active set type which can be slow for large scale problems.
Recently, we proposed a new approach (\citebb{ColeLi94},
\citebb{ColeLi96} \citebb{ColeLi96b}) which generates
iterates within the strictly feasible region. The method in
\citebb{ColeLi96b} is a trust region type and, unlike the
existing trust region method for bound-constrained
problems, the conditions for its strong convergence
properties are consistent with algorithm implementation. A
reflective technique can be included in the method. In this
paper, we motivate techniques which are important for our
new approach. Numerical experience on some medium size
problems is included.},
summary = {Motivation is provided for techniques which are important
for the method of \citebb{ColeLi96b}. Numerical experience
on some medium size problems is included.}}
@techreport{Li94a,
author = {Y. Li},
title = {A Trust-Region and Affine Scaling Method for Nonlinearly
Constrained Minimization},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR94-1463}, year = 1994,
abstract = {A nonlinearly constrained minimization problem can be
solved by the exact penalty approach involving
nondifferentiable functions $\sum_i|c_i(x)|$ and $\sum_i
\max(0,c_i(x))$. In this paper, a trust region approach
based on a 2-norm subproblem is proposed for solving a
nonlinear $\ell_1$ problem. The (quadratic) approximation
and the trust region subproblem are defined using affine
scaling techniques. Explicit sufficient decrease conditions
based on the approximations are suggested for obtaining a
limit point satisfying complementarity, Kuhn-Tucker
conditions, and the second order necessary conditions. The
global convergence of the method is presented in
\citebb{Li94b}.},
summary = {A trust-region approach based on a $\ell_2$ norm subproblem
is proposed for solving a nonlinear $\ell_1$ problem. The
(quadratic) approximation and the trust-region subproblem
are defined using affine scaling techniques. Explicit
sufficient decrease conditions based on the approximations
are suggested for obtaining a limit point satisfying
complementarity, Kuhn-Tucker conditions, and the second
order necessary conditions.}}
@techreport{Li94b,
author = {Y. Li},
title = {On Global Convergence of a Trust-Region and Affine Scaling
Method for Nonlinearly Constrained Minimization},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR94-1462}, year = 1994,
abstract = {A nonlinearly constrained minimization problem can be
solved by the exact penalty approach involving
non-differentiable functions $\sum_i|c_i(x)|$ and $\sum_i
\max(0,c_i(x))$. In \citebb{Li94a}, a trust region affine
scaling approach based on a 2-norm subproblem is proposed
for solving a nonlinear $\ell_1$ problem. The (quadratic)
approximation and the trust region subproblem are defined
using affine scaling techniques. Explicit sufficient
decrease conditions based on the approximations are
proposed to obtain a limit point satisfying
complementarity, Kuhn-Tucker conditions, and the second
order necessary conditions. In this paper, we present the
global convergence properties of this new approach.},
summary = {Global convergence properties of the method by
\citebb{Li94a} are presented.}}
@article{Li96,
author = {W. Li},
title = {Differentiable Piecewise Quadratic Exact Penalty Functions
for Quadratic Programs With Simple Bound Constraints},
journal = SIOPT,
volume = 6, number = 2, pages = {299--315}, year = 1996}
@inproceedings{Li97,
author = {W. Li},
title = {A Merit Function and a {N}ewton-Type Method for Symmetric
Linear Complementarity Problems},
crossref = {FerrPang97}, pages = {181--203}}
@article{LianXu97,
author = {X. Liang and C. Xu},
title = {A trust region algorithm for bound constrained minimization},
journal = {Optimization},
volume = 41, number = 3, pages = {279--289}, year = 1997,
abstract = {A trust region algorithm is proposed for box constrained
nonlinear optimization. At each step of the algorithm a
quadratic model problem in box is minimized. Global
convergence and quadratic convergence rate to a strong
local minimizer are given. Computational results are
presented to show the efficiency of the algorithm.},
summary = {A trust region algorithm is proposed for box constrained
nonlinear optimization. At each step of the algorithm a
quadratic model problem is minimized in a box. Global
convergence and quadratic convergence rate to a strong
local minimizer are given. Computational results are
presented to show the efficiency of the algorithm.}}
@techreport{Liao95,
author = {A. Liao},
title = {Solving unconstrained discrete-time
optimal-control-problems using trust method},
institution = ACRI-CORNELL, address = CORNELL-ADDRESS,
number = {CTC95TR230}, year = 1995,
abstract = {trust-region method for a class of large-scale minimization
problems, the unconstrained discrete-time optimal control
(DTOC) problems, is considered. We show that the
trust-region subproblem can be solved within an acceptable
accuracy without forming the Hessian explicitly. The new
approach is based on the inverse power method for
eigenvalue problem and possesses the ability to handle the
hard case. Our proposed approach leads to more efficient
algorithms for DTOC problems.},
summary = {A trust-region method is considered for solving
unconstrained discrete-time optimal control (DTOC)
problems, in which the trust-region subproblem can be
solved within an acceptable accuracy without forming the
Hessian explicitly. The approach is based on the inverse
power method for eigenvalue problem and can handle the hard
case. It leads to more efficient algorithms for DTOC
problems.}}
@article{Liao97,
author = {A. Liao},
title = {Some efficient algorithms for unconstrained discrete-time
optimal control problems},
journal = {Applied Mathematics and Computation},
volume = 87, number = {2-3}, pages = {175--198}, year = 1997,
abstract = {The differential dynamic programming algorithm (DDP) and
the stagewise Newton procedure are two typical examples of
efficient local procedures for discrete-time optimal
control (DTOC) problems. It is desirable to generalize
these local procedures to globally convergent methods. One
successful globalization was recently proposed by
\citebb{ColeLiao95} which combines the trust region idea
with \citebb{Pant88}'s stagewise Newton procedure. In this
paper we propose several algorithms for DTOC problems which
combine a modified ''dogleg'' algorithm with DDP or
Pantoja's Newton procedure. These algorithms possess
advantages of both the dogleg algorithm and the DDP or the
stagewise procedure, i.e., they have strong global and
local convergence properties yet remain economical.
Numerical results are presented to compare these algorithms
and the Coleman-Liao algorithm.},
summary = {Several algorithms for discrete time optimal control
problems are proposed, which combine a modified dogleg
algorithm with the differential dynamic programming method
or \citebb{Pant88}'s Newton procedure. These algorithms
possess advantages of both the dogleg algorithm and the DDP
or the stagewise procedure, i.e., they have strong global
and local convergence properties yet remain economical.
Numerical results are presented to compare these algorithms
and the \citebb{ColeLiao95} algorithm.}}
@article{LinMore99,
author = {C. Lin and J. J. Mor\'{e}},
title = {{N}ewton's Method for Large Bound-Constrained Optimization
Problems},
journal = SIOPT,
volume = 9, number = 4, pages = {1100--1127}, year = 1999,
abstract = {We analyze a trust region version of Newton's method for
bound-constrained problems. Our approach relies on the
geometry of the feasible set, not on the particular
representation in terms of constraints. The convergence
theory holds for linearly-constrained problems, and yields
global and superlinear convergence theory without assuming
neither strict complementarity nor linear independence of
the active constraints. We also show that the convergence
theory leads to an efficient implementation for large
bound-constrained problems.},
summary = {A trust region version of Newton's method for
bound-constrained problems is proposed, that relies on the
geometry of the feasible set, not on the particular
representation in terms of constraints. The convergence
theory holds for linearly-constrained problems, and yields
global and superlinear convergence theory without assuming
neither strict complementarity nor linear independence of
the active constraints. The theory also leads to an
efficient implementation for large bound-constrained
problems.}}
%institution = ANL, address = ANL-ADDRESS,
%number = {MCS-P724-0898}, year = 1998,
@article{LiuHanWang98,
author = {G. Liu and J. Han and S. Wang},
title = {A trust region algorithm for bilevel programming problems},
journal = {Chinese Science Bulletin},
volume = 43, number = 10, pages = {820--824}, year = 1998,
abstract = {A trust region algorithm is proposed for solving bilevel
programming problems where the lower level programming
problem is a strongly convex programming problem with
linear constraints. This algorithm is based on a trust
region algorithm for nonsmooth unconstrained optimization
problems, and its global convergence is also proved.},
summary = {A trust region algorithm is proposed for solving bilevel
programming problems where the lower level programming
problem is a strongly convex programming problem with
linear constraints. This algorithm is based on a trust
region algorithm for non-smooth unconstrained optimization
problems, and its global convergence is also proved.}}
@article{LiuNoce89,
author = {D. Liu and J. Nocedal},
title = {On the limited memory {BFGS} method for large scale
optimization},
journal = MPB,
volume = 45, number = 3, pages = {503--528}, year = 1989}
@misc{LiuYuan98,
author = {X. Liu and Y. Yuan},
title = {A robust trust-region algorithm for solving general
nonlinear programming problems},
howpublished = {Presentation at the International Conference on Nonlinear
Programming and Variational Inequalities, Hong Kong},
year = 1998,
abstract = {The trust-region approach has been extended to solving
nonlinear constrained optimization. Most of these
extensions consider only equality constraints and require
strong global regularity assumptions. In this report, a
trust-region algorithm for solving general nonlinear
programming is presented, which solves a trust-region
subproblem and a quadratic programming trust-region
subproblem at each iteration. The algorithm is similar
to the methods presented by \citebb{Burk92} and
\citebb{Yuan95}. For the equality constrained case, our
algorithm is similar to the methods of
\citebb{DennElAlMaci97} and \citebb{DennVice97}. A new
penalty parameter updating procedure is introduced. Under
very milds conditions, the global convergence results are
proved. Some local convergence results are also proved.
The preliminary numerical results show that our algorithm
is comparable to {\tt VF02AD}.},
summary = {A trust-region methods for general constrained optimization
is discussed, that uses a composite step technique. The
normal step is obtained by solving an ordinary trust-region
subproblem, while the tangential step results from a
trust-region constrained quadratic program. The method has
similarities with those of \citebb{Burk92}, \citebb{Yuan95},
\citebb{DennElAlMaci97} and \citebb{DennVice97}, but
features a new updating rule for the penalty parameter.
It is globally convergent. Numerical results indicate that
its efficiency is comparable to that of {\tt VF02AD}.}}
@article{LiSwet93,
author = {W. Li and J. Swetits},
title = {A {N}ewton Method for Convex Regression, Data Smoothing,
and Quadratic Programming with Bounded Constraints},
journal = SIOPT,
volume = 3, number = 3, pages = {466--488}, year = 1993}
@article{Loot69,
author = {F. A. Lootsma},
title = {Hessian matrices of penalty functions for solving
constrained optimization problems},
journal = {Philips Research Reports},
volume = 24, pages = {322--331}, year = 1969}
@article{Lots84,
author = {P. L\"{o}tstedt},
title = {Solving the minimal least squares problem subject to bounds
on the variables},
journal = BIT,
volume = 24, pages = {206--224}, year = 1984}
@article{LuciGuoWang93,
author = {A. Lucia and X. Z. Guo and X. Wang},
title = {Process Simulation in the complex-domain},
journal = {Aiche Journal},
volume = 39, number = 3, pages = {461--470}, year = 1993,
abstract = {The asymptotic behavior of fixed-point methods in the
complex domain is studied. Both direct substitution and
Newton's method exhibit stable periodic and aperiodic
behavior from real- or complex-valued starting points.
Moreover, multiple stable periodic orbits can exist for
direct substitution. Traditional trust region (or dogleg)
methods, on the other hand, often terminate at singular
points, which correspond to nonzero-valued saddlepoints in
the least-squares function that can be arbitrarily far from
a solution. Furthermore, the basins of attraction of these
singular points are usually dispersed throughout the basin
boundaries in the complex domain, clearly illustrating that
singular points (via the dogleg strategy) also attract
either real- or complex-valued starting points. In light of
this, an extension of the dogleg strategy to the complex
domain, based on a simple norm-reducing, singular point
perturbation, is proposed. This extended trust region
method removes all forms of nonconvergent behavior and
always terminates at a fixed point, even from critical
point (worst-case) initial values. Many numerical results
and geometric illustrations using chemical process
simulation examples are presented.},
summary = {The asymptotic behaviour of fixed-point methods in the
complex domain is studied. Both direct substitution and
Newton's method exhibit stable periodic and aperiodic
behaviour from real- or complex-valued starting points.
Moreover, multiple stable periodic orbits can exist for
direct substitution. Traditional trust-region (or dogleg)
methods, on the other hand, often terminate at singular
points, which correspond to non-zero-valued saddlepoints in
the least-squares function that can be arbitrarily far from
a solution. Furthermore, the basins of attraction of these
singular points are usually dispersed throughout the basin
boundaries in the complex domain, clearly illustrating that
singular points (via the dogleg strategy) also attract
either real- or complex-valued starting points. In this
light, an extension of the dogleg strategy to the complex
domain, based on a simple norm-reducing, singular point
perturbation, is proposed. This extended trust-region
method always terminates at a fixed point, even from
critical point (worst-case) initial values. Numerical
results and geometric illustrations using chemical process
simulation examples are presented.}}
@article{LuciXu90,
author = {A. Lucia and J. Xu},
title = {Chemical process optimization using {N}ewton-like methods},
journal = {Computers and Chemical Engineering},
volume = 14, number = 2, pages = {119--138}, year = 1990,
abstract = {Various interrelated issues that effect the
reliability and efficiency of Newton-like methods for
chemical process optimization are studied. An
algorithm for solving large, sparse quadratic
programming (QP) problems that is based on an active
set strategy and a symmetric, indefinite
factorization is presented. The QP algorithm is fast
and reliable. A simple asymmetric trust region method
is proposed for improving the reliability of
successive QP methods. Ill-defined QP subproblems are
avoided by adjusting the size of the trust region in
an automatic way. Finally, it is shown that reliable
initial values of the unknown variables and
multipliers can be generated automatically using
generic problem information, short-cut techniques and
simulation tools. Many relevant numerical results and
illustrations are presented.},
summary = {An algorithm for solving large, sparse quadratic
programming (QP) problems that is based on an active
set strategy and a symmetric, indefinite
factorization is presented. A simple asymmetric
trust-region method is proposed for improving the
reliability of successive QP methods. Ill-defined QP
subproblems are avoided by adjusting the size of the
trust region in an automatic way. Finally, it is shown
that reliable initial values of the unknown variables
and multipliers can be generated automatically using
generic problem information, short-cut techniques and
simulation tools. Relevant numerical results and
illustrations are presented.}}
@article{LuciXu94,
author = {A. Lucia and J. Xu},
title = {Methods of successive quadratic programming},
journal = {Computers and Chemical Engineering},
volume = 18, pages = {S211-215}, year = 1994,
abstract = {The occurrence of nondescent directions in successive
quadratic programming is studied. It is shown that
simple chemical process examples can be constructed
that exhibit nondescent as a consequence of the
projected indefiniteness of the Hessian matrix of the
Lagrangian function. Moreover, in situations where
multiple Kuhn-Tucker points for the quadratic
programming sub-problems exist, the global optimum
need not necessarily provide a direction of
descent. Thus search for a global solution is
unjustified. To circumvent these difficulties, a
linear programming-based trust region method is
proposed to guarantee descent for any arbitrary merit
function, provided such a direction exists. Geometric
illustrations are used to elucidate the main ideas.},
summary = {Simple chemical process examples are constructed
that exhibit non-descent in successive
quadratic programming as a consequence of the
projected indefiniteness of the Hessian matrix of the
Lagrangian function. Moreover, in situations where
multiple Kuhn-Tucker points for the quadratic
programming sub-problems exist, the global optimum
need not necessarily provide a direction of
descent. Thus search for a global solution is
unjustified. To circumvent these difficulties, a
linear programming-based trust region method is
proposed to guarantee descent for any arbitrary merit
function, provided such a direction exists. Geometric
illustrations are used to elucidate the main ideas.}}
@article{Luci92,
author = {S. Lucidi},
title = {New Results on a Continuously Differentiable Exact Penalty
Function},
journal = SIOPT,
volume = 2, number = 4, pages = {558--574}, year = 1992}
@techreport{LuciPalaRoma94,
author = {S. Lucidi and L. Palagi and M. Roma},
title = {Quadratic programs with a quadratic constraint:
characterisation of {KKT} points and equivalence with an
unconstrained problem},
institution = {University of Rome "La Sapienza"}, address = {Rome},
type = {Technical Report}, number = {24-94}, year = 1994,
abstract = {In this paper, we consider the problem of minimizing a
quadratic function with a quadratic constraint. We point
out some new properties of the problem. In particular, in
the first part of the paper, we show that (i) the number of
values of the objective function at KKT points is bounded
by $3n+1$ where $n$ is the dimension of the problem; (ii)
given a KKT point that is not global minimizer, it is easy
to find a ``better'' feasible point; (iii) strict
complementarity holds at the local-nonglobal minimum point.
In the second part, we show that the original constrained
problem is equivalent to the unconstrained minimization of
a piecewise quartic merit function. Using the unconstrained
formulation, we give, in the nonconvex case, a new second
order necessary condition for global minimum points. A
possible algorithmic application of the preceding results
is briefly outlined.},
summary = {The technical report associated with
\citebb{LuciPalaRoma98}, but containing more technical
detail.}}
@article{LuciPalaRoma98,
author = {S. Lucidi and L. Palagi and M. Roma},
title = {On some properties of quadratic programs with a convex
quadratic constraint},
journal = SIOPT,
volume = 8, number = 1, pages = {105--123}, year = 1998,
abstract = {In this paper, we consider the problem of minimizing a
(possibly nonconvex) quadratic function with a quadratic
constraint. We point out some new properties of the
problem. In particular, in the first part of the paper, we
show that (i) given a KKT point that is not global
minimizer, it is easy to find a ``better'' feasible point;
(ii) strict complementarity holds at the local-nonglobal
minimum point. In the second part, we show that the
original constrained problem is equivalent to the
unconstrained minimization of a piecewise quartic merit
function. Using the unconstrained formulation, we give, in
the nonconvex case, a new second order necessary condition
for global minimum points. In the third part, algorithmic
applications of the preceding results are briefly outlined,
and some preliminary numerical experiments are reported.},
summary = {The problem of minimizing a non-convex quadratic function
with a quadratic constraint is considered, and
properties of the problem identified. In particular, (i)
given a KKT point that is not global minimizer, it is easy
to find a ``better'' feasible point; and (ii) strict
complementarity holds at the local-non-global minimum point.
It is also shown that the original constrained problem is
equivalent to the unconstrained minimization of a piecewise
quartic merit function. Using this
formulation, a second order necessary condition for global
minimum points is given in the non-convex case. Algorithmic
applications are outlined, and preliminary numerical
experiments reported.}}
@article{LuciXuLayn96,
author = {A. Lucia and J. Xu and K. M. Layn},
title = {Nonconvex Process Optimization},
journal = {Computers and Chemical Engineering},
volume = 20, number = 12, pages = {1375--1398}, year = 1996,
abstract = {Difficulties associated with nonconvexity in successive
quadratic programming (SPQ) methods are studied. It is
shown that projected indefiniteness of the Hessian matrix
of the Lagrangian function can (i) place restrictions on
the order in which inequalities can be added or deleted
from the active set, (ii) generate redundant active sets
whose resolution is nontrivial, (iii) give rise to
quadratic programming (QP) subproblems that have multiple
Kuhn-Tucker points, and (iv) produce nondescent directions
in the SQP method that can lead to failure. Related issues
concerned with the use of feasible or infeasible starting
points for the iterative quadratic programs, forcing
positive definiteness to ensure convexity and using
iterative methods to solve the linear Kuhn-Tucker
conditions associated with the QP subproblems are also
studied. A new active set strategy that (i) monitors
projected indefiniteness to guide the addition of
constraints to the active set, (ii) permits line searching
for negative values of the line search parameter, and (iii)
does not necessarily delete active constraints with
incorrect Kuhn-Tucker multipliers is proposed. Constraint
redundancy is circumvented using an algorithm that
identifies all nontrivial redundant subsets of smallest
size and determines which, if any, exchanges are justified.
Nondescent in the NLP's is resolved using a linear
programming (LP)-based trust region method that guarantees
descent regardless of merit function. It is also shown that
there is no justification for using feasible starting
points at the QP level of the calculations, that forcing
positive definiteness to ensure convexity can cause
termination at undesired solutions, and that the use of
iterative methods to solve the linear Kuhn-Tucker equations
for the QP's can cause a deterioration in numerical
performance. Many small chemical process examples are used
to highlight difficulties so that geometric illustrations
can be used while heat exchange network design and
distillation operations examples are used to show that
these same difficulties carry over the larger problems.},
summary = {Difficulties associated with non-convexity in successive
quadratic programming (SPQ) methods are studied. It is
shown that projected indefiniteness of the Hessian matrix
of the Lagrangian function can (i) place restrictions on
the order in which inequalities can be added or deleted
from the active set, (ii) generate redundant active sets
whose resolution is non-trivial, (iii) give rise to
quadratic programming (QP) subproblems that have multiple
Kuhn-Tucker points, and (iv) produce non-descent directions
in the SQP method that can lead to failure. Related issues
concerned with the use of feasible or infeasible starting
points for the iterative quadratic programs, forcing
positive definiteness to ensure convexity and using
iterative methods to solve the linear Kuhn-Tucker
conditions associated with the QP subproblems are also
studied. An active set strategy that (i) monitors
projected indefiniteness to guide the addition of
constraints to the active set, (ii) permits line searching
for negative values of the linesearch parameter, and (iii)
does not necessarily delete active constraints with
incorrect Kuhn-Tucker multipliers is proposed. Constraint
redundancy is circumvented using an algorithm that
identifies all non-trivial redundant subsets of smallest
size and determines which, if any, exchanges are justified.
Non-descent in the NLP's is resolved using a linear
programming (LP)-based trust-region method that guarantees
descent regardless of merit function. It is also shown that
there is no justification for using feasible starting
points at the QP level of the calculations, that forcing
positive definiteness to ensure convexity can cause
termination at undesired solutions, and that the use of
iterative methods to solve the linear Kuhn-Tucker equations
for the QP's can cause a deterioration in numerical
performance. Many small chemical process examples are used
to highlight difficulties so that geometric illustrations
can be used while heat exchange network design and
distillation operations examples are used to show that
these same difficulties carry over the larger problems.}}
@book{Luen69,
author = {D. G. Luenberger},
title = {Optimization by Vector Space Methods},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1969}
@book{Luen84,
author = {D. G. Luenberger},
title = {Linear and Nonlinear Programming},
publisher = ADW, address = ADW-ADDRESS,
edition = {second}, year = 1984}
@article{Luks93,
author = {L. Luk\v{s}an},
title = {Inexact Trust Region Method for Large Sparse Nonlinear
Least-Squares},
journal = {Kybernetica},
volume = 29, number = 4, pages = {305--324}, year = 1993,
abstract = {The main purpose of this paper is to show that linear least
squares methods based on bidiagonalization, namely the LSQR
algorithm, can be used for generation of trust region path.
This property is a basis for an inexact trust region method
which uses the LSQR algorithm for direction determination.
This method is very efficient for large sparse nonlinear
least squares as it is supported by numerical experiments.},
summary = {It is shown that linear least-squares methods based on the
LSQR algorithm can be used for generation of trust-region
path. This property is a basis for an inexact trust-region
method. Numerical experiments suggest that this
method is efficient for large sparse nonlinear
least-squares.}}
@article{Luks94,
author = {L. Luk\v{s}an},
title = {Inexact Trust Region Method for Large Sparse Systems of
Nonlinear Equations},
journal = {Journal of Optimization Theory and Applications},
volume = 81, number = 3, pages = {569--590}, year = 1994,
abstract = {The main purpose of this paper is to prove the global
convergence of the new trust region method based on the
smoothed CGS algorithm. This method is surprisingly
convenient for the numerical solution of large sparse
systems of nonlinear equations, as is demonstrated by
numerical experiments. A modification of the proposed trust
region method does not use matrices, so it can be used for
large dense systems of nonlinear equations.},
summary = {The global convergence of the a trust-region method based
on the smoothed CGS algorithm is proved. Numerical
experiments indicate that the method is surprisingly
convenient for the numerical solution of large sparse
systems of nonlinear equations. A modification of the
method does not use matrices, and can be used for large
dense systems of nonlinear equations.}}
@article{Luks96a,
author = {L. Luk\v{s}an},
title = {Hybrid methods for large sparse nonlinear least-squares},
journal = JOTA,
volume = 89, number = 3, pages = {575--595}, year = 1996,
abstract = {Hybrid methods are developed for improving the Gauss-Newton
method in the case of large residual or ill-conditioned
nonlinear least-square problems. These methods are used
usually in a form suitable for dense problems. But some
standard approaches are unsuitable, and some new
possibilities appear in the sparse case. We propose
efficient hybrid methods for various representations of the
sparse problems. After describing the basic ideas that help
deriving new hybrid methods, we are concerned with
designing hybrid methods for sparse Jacobian and sparse
Hessian representations of the least-square problems. The
efficiency of hybrid methods is demonstrated by extensive
numerical experiments.},
summary = {Hybrid methods are developed for improving the Gauss-Newton
method in the case of large residual or ill-conditioned
nonlinear least-square problems. Hybrid methods for sparse
Jacobian and sparse Hessian representations of the
least-square problems are considered. The efficiency of
these methods is illustrated by extensive numerical
experiments. }}
@article{Luks96b,
author = {L. Luk\v{s}an},
title = {Combined Trust Region Methods for Nonlinear Least-Squares},
journal = {Kybernetica},
volume = 32, number = 2, pages = {121--138}, year = 1996,
abstract = {Trust region realizations of the Gauss-Newton method are
commonly used for obtaining solution of nonlinear least
squares problems. We propose three efficient algorithms
which improve standard trust region techniques : multiple
dog-leg strategy for dense problems and two combined
conjugate gradient Lanczos strategies for sparse problem.
Efficiency of these methods is demonstrated by extensive
numerical experiments.},
summary = {Trust-region realizations of the Gauss-Newton method are
commonly used for obtaining solution of nonlinear
least-squares problems. Three algorithms which improve
standard trust-region techniques are proposed, comprising a
multiple dog-leg strategy for dense problems and two
combined conjugate-gradient Lanczos strategies for sparse
problem. Efficiency of these methods is illustrated by
extensive numerical experiments.}}
@article{Luks96c,
author = {L. Luk\v{s}an},
title = {Efficient Trust Region Method for nonlinear least-squares},
journal = {Kybernetica},
volume = 32, number = 2, pages = {105--120}, year = 1996,
abstract = {The main purpose of this paper is to show that suitable
transformations and decompositions lead to an efficient
trust region method that uses one decomposition in each
iteration only. Convergence properties of the trust region
method with optimal locally constrained step (OLCS) that
uses more than one decomposition in each iteration and,
therefore, that needs a longer time for obtaining results.
This fact is demonstrated by numerical experiments.},
summary = {Suitable transformations and decompositions
lead to an efficient trust-region method that uses a single
factorization at each iteration. This is compared to the
optimal locally constrained step that uses more than one
decomposition per iteration. Numerical experiments suggest
that the former approach is more efficient.}}
@article{LuksVlce96,
author = {L. Luk\v{s}an and J. Vl\v{c}ek},
title = {Optimization of dynamical-systems},
journal = {Kybernetica},
volume = 32, number = 5, pages = {465--482}, year = 1996,
abstract = {Consider an optimization problem where the objective
function is an integral containing the solution of a system
of ordinary differential equations. Suppose we have
efficient optimization methods available as well as
efficient methods for initial value problems for ordinary
differential equations. The main purpose of this paper is
to show how these methods can be efficiently applied to a
considered problem. First, the general procedures for the
evaluation of gradients and Hessian matrices are described.
Furthermore, the new efficient Gauss-Newton-like
approximation of the Hessian matrix is derived for the
special case when the objective function is an integral of
squares. This approximation is used for deriving the
Gauss-Newton-like trust region method, with which global
and superlinear convergence properties are proved. Finally
several optimization methods are proposed and computational
experiments illustrating their efficiency are shown.},
summary = {Optimization problems where the objective function is an
integral containing the solution of a system of ordinary
differential equations are considered. It is shown that
optimization methods and methods for initial value problems
for ordinary differential equations can be efficiently
combined. General procedures for the evaluation of
gradients and Hessian matrices are described. An
efficient Gauss-Newton-like approximation of the Hessian
matrix is derived for the special case when the objective
function is an integral of squares. This approximation is
used to derive a Gauss-Newton-like trust-region method, for
which global and superlinear convergence properties are
proved. Finally several methods are proposed and
illustrated by computational experiments.}}
@article{LuksVlce97,
author = {L. Luk\v{s}an and J. Vl\v{c}ek},
title = {Truncated trust region methods based on preconditioned
iterative subalgorithms for large sparse systems of
nonlinear equations},
journal = JOTA,
volume = 95, number = 3, pages = {637--658}, year = 1997,
abstract = {This paper is devoted to globally convergent methods for
solving large sparse systems of nonlinear equations with an
inexact approximation of the Jacobian matrix. These methods
include difference versions of the Newton method and
various quasi-Newton methods. We propose a class of trust
region methods together with a proof of their global
convergence and describe an implementable globally
convergent algorithm which can be used as a realization of
these methods. Considerable attention is concentrated on
the application of conjugate gradient-type iterative
methods to the solution of linear subproblems. We prove
that both the GMRES and the smoothed CGS
well-preconditioned methods can be used for the
construction of globally convergent trust region methods.
The efficiency of our algorithm is demonstrated
computationally by using a large collection of sparse test
problems.},
summary = {Globally convergent methods for solving large sparse
systems of nonlinear equations with an inexact
approximation of the Jacobian matrix are studied. These
methods include difference versions of the Newton method
and various quasi-Newton methods. A class of trust region
methods is proposed together with a proof of their global
convergence and an implementable globally convergent
algorithm described which can be used as a realization of
these methods. Emphasis is put on the application of
conjugate gradient-type iterative methods to the solution
of linear subproblems. We prove that both the GMRES and the
smoothed CGS well-preconditioned methods can be used for
the construction of globally convergent trust region
methods. The efficiency of our algorithm is demonstrated
computationally by using a large collection of sparse test
problems.}}
@article{LuoTsen93,
author = {Z. Q. Luo and P. Tseng},
title = {Error bounds and convergence analysis of feasible direction
methods: a general approach},
journal = AOR,
volume = 46, pages = {157--178}, year = 1993}
@inproceedings{LuoTsen97,
author = {Z. Q. Luo and P. Tseng},
title = {A New Class of Merit Functions for the Nonlinear
Complemetarity Problem},
crossref = {FerrPang97}, pages = {204--225}}
@article{LyleSzul94,
author = {S. Lyle and M. Szularz},
title = {Local Minima of the Trust Region Problem},
journal = JOTA,
volume = 80, number = 1, pages = {117--134}, year = 1994,
abstract = {We consider the minimization of a quadratic form $z^TVz +
2z^Tq$ subject to the two-norm constraint parallel to
$||z||=\alpha$. The problem received considerable attention
in the literature, notably due to its applications to a
class of trust region methods in nonlinear optimization.
While the previous studies were concerned with just the
global minimum of the problem, we investigate the existence
of all local minima. The problem is approached via the dual
Lagrangian, and the necessary and sufficient conditions for
the existence of all local minima are derived. We also
examine the suitability of the conventional numerical
techniques used to solve the problem to a class of
single-instruction multiple-data computers known as
processor arrays (in our case, AMT DAP 610).
Simultaneously, we introduce certain hardware-oriented
multisection algorithms, showing their efficiency in the
case of small to medium size problems.},
summary = {The minimization of a quadratic form subject to the
two-norm constraint is considered. The existence of local
minima is investigated. The problem is approached via the
dual Lagrangian, and necessary and sufficient conditions
for the existence of all local minima are derived. The
suitability of the conventional numerical techniques used
to solve the problem on processor arrays is
examined. Hardware-oriented multisection algorithms are
considered, and their efficiency demonstrated on small
to medium size problems.}}
%%% M %%%
@article{Maan87,
author = {Z. A. Maany},
title = {A new algorithm for highly curved constrained optimization},
journal = MPS,
volume = 31, pages = {139--154}, year = 1987,
abstract = {This paper describes a new algorithm for highly curved
constrained optimisation. The algorithm under discussion
makes use of the second derivatives of both the objective
function and constraints. At every iteration a subproblem
based on the second order approximation of the objective
and constraints functions is solved. Three strategies to
solve the subproblem are used. Some computational results
are given. Although the performance of the subroutine is
very promising a number of areas are still under
development and further improvement is expected.},
summary = {An algorithm for highly curved constrained optimisation is
considered, which makes use of the second derivatives of
both the objective function and constraints. At every
iteration a subproblem based on the second order
approximation of the objective and constraints functions is
solved. Three strategies to solve the subproblem are used.
Some computational results are given.}}
@article{Mads75,
author = {K. Madsen},
title = {An algorithm for the minimax solution of overdetermined
systems of nonlinear equations},
journal = JIMA,
volume = 16, number = 3, pages = {321--328}, year = 1975,
abstract = {The problem of minimising the maximum residual of a
system of non-linear equations is studied in the
case where the number of equations is larger than
the number of unknowns. It is supposed that the
functions defining the problem have continuous first
derivatives and the algorithm is based on successive
linear approximations to these functions. The
resulting linear systems are solved in the minimax
sense, subject to bounds on the solutions, the
bounds being adjusted automatically, depending on
the goodness of the linear approximations. It is
proved that the method always has sure convergence
properties. Some numerical examples are given.},
summary = {A method for nonlinear minimax in which linear models
are considered subject to an $\ell_{\infty}$ trust
region.}}
@article{MadyAazh94,
author = {R. K. Madyastha and B. Aazhang},
title = {An algorithm for training multilayer perceptrons for
data classification and function interpolation},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
volume = 41, number = 12, pages = {866--875}, year = 1994,
abstract = {This paper addresses the issue of employing a
parametric class of nonlinear models to describe
nonlinear systems. This model class consists of a
subclass of artificial neural networks, multilayer
perceptrons. Specifically, we discuss the application
of a "globally" convergent optimization scheme to the
training of the multilayer perceptron. The algorithm
discussed is termed the conjugate gradients-trust
regions algorithm (CGTR) and combines the merits of
two well known "global" algorithms-the conjugate
gradients and the trust region algorithms. In this
paper we investigate the potential of the multilayer
perceptron, trained using the CGTR algorithm, towards
function approximation in two diverse scenarios: i)
signal classification in a multiuser communication
system, and ii) approximating the inverse kinematics
of a robotic manipulator. Until recently, the most
widely used training algorithm has been the
backpropagation algorithm, which is based on the
linearly convergent steepest descent algorithm. It is
seen that the multilayer perceptron trained with the
CGTR algorithm is able to approximate the desired
functions to a greater accuracy than when trained
using backpropagation. Specifically, in the case of
the multiuser communication problem, we obtain lower
probabilities of error in demodulating a given user's
signal and in the robotics problem, we observe lower
root mean square errors in approximating the inverse
kinematics function.},
summary = {The application of a globally convergent optimization
scheme to the training of the multilayer perceptron is
discussed. The algorithm combines the conjugate-gradients
and the trust-region algorithms. The potential of the
multilayer perceptron, trained using the algorithm, is
considered in signal classification in a multiuser
communication system, and in approximating the inverse
kinematics of a robotic manipulator. It is
seen that the multilayer perceptron trained with the
trust-region algorithm is able to approximate the desired
functions to a greater accuracy than when trained
using backpropagation. Specifically, in the case of
the multiuser communication problem, lower
probabilities of error in demodulating a given user's
signal, and, in the robotics problem, lower
root mean square errors in approximating the inverse
kinematics function, are obtained.}}
@inproceedings{Mall97,
author = {M. K. Mallick},
title = {Applications of Nonlinear Orthogonal Distance Regression in
3D Motion Estimation},
crossref = {VanH97}, pages = {273--282}}
@book{Mang69,
author = {O. L. Mangasarian},
title = {Nonlinear Programming},
publisher = MACGH, address = MACGH-ADDRESS,
year = 1969,
note = {Reprinted as \emph{Classics in Applied Mathematics 10}, SIAM,
Philadelphia, USA, 1994}}
@article{Mang80,
author = {O. L. Mangasarian},
title = {Locally unique solutions of quadratic programs, linear and
non-linear complementarity problems},
journal = MP,
volume = 19, number = 2, pages = {200--212}, year = 1980}
@article{MangFrom67,
author = {O. L. Mangasarian and S. Fromovitz},
title = {The {F}ritz {J}ohn necessary optimality conditions in the
presence of equality and inequality constraints},
journal = {Journal of Mathematical Analysis and Applications},
volume = 17, pages = {37--47}, year = 1967}
@article{MangSolo93,
author = {O. L. Mangasarian and M. V. Solodov},
title = {Nonlinear complementarity as unconstrained and constrained
minimization},
journal = MPB,
volume = 62, number = 2, pages = {277--297}, year = 1993}
@phdthesis{Mara78,
author = {N. Maratos},
title = {Exact penalty function algorithms for finite-dimensional
and control optimization problems},
school = {University of London}, address = {London, England},
year = 1978}
@article{Marq63,
author = {D. Marquardt},
title = {An Algorithm For Least-Squares Estimation Of Nonlinear
Parameters},
journal = {SIAM Journal on Applied Mathematics},
volume = 11, pages = {431--441}, year = 1963,
summary = {Taylor-series and steepest-descent methods are sometimes
ineffective as algorithms for the least-squares estimation
of nonlinear parameters. A maximum neighbourhood method is
developed which, in effect, performs an optimum
interpolation between Taylor-series and steepest-descent
methods. The interpolation is based upon the maximum
neighbourhood in which the truncated Taylor series gives an
adequate representation of the nonlinear model.}}
@article{Mart70,
author = {B. Martinet},
title = {R\'{e}gularisation d'in\'{e}quations variationnelles par
approximations successives},
journal = {Revue Fran\c{c}aise d'Informatique et de Recherche
Op\'{e}rationnelle},
volume = 4, pages = {154--159}, year = 1970}
@article{Mart87,
author = {J. M. Mart\'{\i}nez},
title = {An algorithm for solving sparse nonlinear least squares
problems},
journal = {Computing},
volume = 39, number = 4, pages = {307--325}, year = 1997,
abstract = {We introduce a new method for solving Nonlinear Least
Squares problems when the Jacobian matrix of the system is
large and sparse. The main features of the new method are
the following: 1) the Gauss-Newton equation is
``partially'' solved at each iteration using a
preconditioned Conjugate Gradient algorithm, 2) the new
point is obtained using a two-dimensional trust region
scheme, similar to the one introduced by
\citebb{BultVial87}. We prove global convergence results
and we present some numerical results.},
summary = {A method is given for solving nonlinear least-squares
problems, when the Jacobian matrix of the system is large
and sparse. The main features of the method are that
the Gauss-Newton equation is ``partially'' solved at each
iteration using a preconditioned conjugate-gradient
algorithm, and that the new point is obtained using a
two-dimensional trust region scheme, similar to the one
introduced by \citebb{BultVial87}. Global convergence
results and numerical results are presented.}}
@article{Mart94,
author = {J. M. Mart\'{\i}nez},
title = {Local minimizers of quadratic functions on {E}uclidean
balls and spheres},
journal = SIOPT,
volume = 4, number = 1, pages = {159--176}, year = 1994,
abstract = {In this paper a characterization of the local-nonglobal
minimizer of a quadratic function defined on a Euclidean
ball or sphere is given. It is proven that there exists at
most one local-nonglobal minimizer and that the Lagrange
multiplier that corresponds to this minimizer is the
largest solution of a nonlinear scalar equation. An
algorithm is proposed for computing the local-nonglobal
minimizer.},
summary = {A characterization of the local-non-global minimizers of a
quadratic function defined on a Euclidean ball or sphere is
given. It is proven that there exists at most one
local-non-global minimizer and that the Lagrange multiplier
that corresponds to this minimizer is the largest solution
of a nonlinear scalar equation. An algorithm is proposed
for computing the local-non-global minimizer.}}
@article{Mart95,
author = {J. M. Mart\'{\i}nez},
title = {Discrimination by Means of a Trust Region Method},
journal = {International Journal of Computer Mathematics},
volume = 55, number = {1--2}, pages = {91--103}, year = 1995,
abstract = {Suppose that the individuals of a population are divided
into two groups according to some unknown merit criterion.
It is required to determine weights for a set of variables
which should be positively correlated with merit, in such a
way that scores of the individuals in the superior group
are above some level, and vice-versa. This may be modelled
as an easily-solvable convex optimization problem. Examples
are given.},
summary = {The individuals of a population are divided in two groups
according to some unknown merit criterion and the problem
is considered to determine weights for a set of variables
which should be positively correlated with merit, in such a
way that scores of the individuals in the superior group
are above some level, and vice-versa. This situation is
modeled as an easy convex optimization problem.}}
@article{MartSant97,
author = {J. M. Mart\'{\i}nez and S. A. Santos},
title = {New convergence results on an algorithm for norm
constrained regularization and related problems},
journal = RAIRO-OR,
volume = 31, number = 3, pages = {269--294}, year = 1997,
abstract = {The constrained least-squares regularization of nonlinear
ill-posed problems is a nonlinear programming problem for
which trust-region methods have been developed. In this
paper we complement the convergence theory of one of those
methods showing that, under suitable hypotheses, local
(superlinear or quadratic) convergence holds and every
accumulation point is second-order stationary.},
summary = {The constrained least-squares regularization of nonlinear
ill-posed problems is a nonlinear programming problem for
which trust-region methods have been developed. It is shown
that for one such method, under suitable hypotheses, local
(superlinear or quadratic) convergence occurs and every
accumulation point is second-order stationary.}}
@article{MartMore97,
author = {J. M. Mart\'{\i}nez and A. C. Moretti},
title = {A trust region method for minimization of nonsmooth
functions with linear constraints},
journal = MP,
volume = 76, number = 3, pages = {431--449}, year = 1997,
abstract = {We introduce a trust region algorithm for minimization of
nonsmooth functions with linear constraints. At each
iteration, the objective function is approximated by a
model function that satisfies a set of assumptions stated
recently by \citebb{QiSun94} in the context of
unconstrained nonsmooth optimization. The trust region
iteration begins with the resolution of an ``easy
problem'', as in the recent works of \citebb{MartSant95}
and \citebb{FrieMartSant94}, for smooth constrained
optimization. In practical implementations we use the
infinity norm for defining the trust-region, which fits
well with the domain of the problem. We prove global
convergence and report numerical experiments related to the
parameter estimation problem.},
summary = {A trust-region algorithm for minimization of non-smooth
functions with linear constraints is introduced. At each
iteration, the objective function is approximated by a
model that satisfies assumptions stated by \citebb{QiSun94}
for unconstrained non-smooth optimization. The trust-region
iteration begins with the solution of an ``easy problem'',
as in \citebb{MartSant95} and \citebb{FrieMartSant94}. In
practical implementations, the infinity norm is used to
define the trust region. Global convergence is established,
and numerical experiments for the parameter estimation
problem reported.}}
@article{MartSant95,
author = {J. M. Mart\'{\i}nez and S. A. Santos},
title = {A trust-region strategy for minimization on arbitrary
domains},
journal = MP,
volume = 68, number = 3, pages = {267--301}, year = 1995,
abstract = {We present a trust-region method for minimizing a general
differentiable function restricted to an arbitrary closed
set. We prove a global convergence theorem. The
trust-region method defines difficult subproblems that are
solvable in some particular cases. We analyze in detail the
case where the domain is a Euclidean ball. For this case we
present numerical experiments where we consider different
Hessian approximations.},
summary = {A trust-region method for minimizing a general
differentiable function restricted to an arbitrary closed
set is presented, and global convergence is proved. The
case where the domain is a Euclidean ball is analysed in
detail. For this case, numerical experiments which consider
a variety of Hessian approximations are presented.}}
@article{MaurZowe79,
author = {H. Maurer and J. Zowe},
title = {First and second-order necessary and sufficient optimality
conditions for infinite-dimensional programming problems},
journal = MP,
volume = 16, pages = {98--110}, year = 1979}
@article{MaurMacu97,
author = {D. Mauricio and N. Maculan},
title = {A trust region method for zero-one nonlinear programming},
journal = RAIRO-OR,
volume = 31, number = 4, pages = {331--341}, year = 1997,
abstract = {An $O(n \log n)$ trust region approximation method to solve
$0-1$ non-linear programming is presented. Optimality
conditions and numerical results are reported.},
summary = {An $O(n \log n)$ trust region approximation method to solve
$0-1$ nonlinear programming is presented. Optimality
conditions and numerical results are reported.}}
@article{MaynMara79,
author = {D. Q. Mayne and N. Maratos},
title = {A first-order, exact penalty function algorithm for
equality constrained optimization problems},
journal = MP,
volume = 16, number = 3, pages = {303--324}, year = 1979}
@article{MaynPola82,
author = {D. Q. Mayne and E. Polak},
title = {A superlinearly convergent algorithm for constrained
optimization problems},
journal = MPS,
volume = 16, pages = {45--61}, year = 1982}
@article{McAfGayHozaLaudSchwSund86,
author = {K. B. McAfee and D. M. Gay and R. S. Hozack and
R. A. Laudise and G. Schwartz and W. A. Sunder},
title = {Thermodynamic considerations in the synthesis and crystal
growth of GaSb},
journal = {Journal of Crystal Growth},
volume = 76, number = 2, pages = {263--271}, year = 1986,
abstract = {A newly developed optimization algorithm, the 'sticky
trust region technique', for Gibbs energy
minimization is used to determine the gaseous species
and liquid and solid phases present during the
synthesis and crystal growth of GaSb. The growth
system involves almost thirty species comprising a
gaseous phase and nine condensed species. The system
is modelled as a function of temperature, oxygen and
hydrogen pressure in the presence of an SiO/sub 2/
crucible. Ga/sub 2/O/sub 3/ is identified as the most
stable contaminant compound and is seen as a phase
that floats on the liquid melt during growth. This
oxide often prevents the growth of high quality
crystals.},
summary = {The ``sticky trust region technique'' for Gibbs energy
minimization is used to determine the gaseous species
and liquid and solid phases present during the
synthesis and crystal growth of GaSb. The growth
system involves almost thirty species comprising a
gaseous phase and nine condensed species. }}
@article{McAfGayHozaLaudSund88,
author = {K. B. McAfee and D. M. Gay and R. S. Hozack and
R. A. Laudise and W. A. Sunder},
title = {Thermodynamic stability and reactivity of {A}l{S}b and their
relationship to crystal growth},
journal = {Journal of Crystal Growth},
volume = 88, number = 4, pages = {488--498}, year = 1988,
abstract = {The vapor pressure, species concentration and
reactivity of AlSb is modeled thermodynamically using
the newly-described sticky trust region technique,
STRT, for free energy minimization. The conditions
chosen include oxygen and hydrogen concentrations
appropriate to Czochralski crystal growth in SiO/sub
2/, C, BN, Al/sub 2/O/sub 3/ and BeO crucibles and
growth in the absence of a crucible. Results are
compared with crystal growth experiments where
appropriate. At the melting point the principal vapor
species is Sb/sub 2/ with a total vapor pressure of
approximately 10/sup -3/ atm, which is about 10/sup
2/ larger than for GaSb.},
summary = {The vapor pressure, species concentration and
reactivity of AlSb is modeled thermodynamically using
the newly-described sticky trust-region technique,
STRT, for free energy minimization. The conditions
chosen include oxygen and hydrogen concentrations
appropriate to Czochralski crystal growth.}}
@article{McCa98,
author = {McCartin, B. J.},
title = {A model-trust region algorithm utilizing a quadratic
interpolant},
journal = JCAM,
volume = 91, number = 2, pages = {249--259}, year = 1998,
abstract = {A new model-trust region algorithm for problems in
unconstrained optimization and nonlinear equations
utilizing a quadratic interpolant for step selection is
presented and analyzed. This is offered as an alternative
to the piecewise-linear interpolant employed in the widely
used ''double dogleg'' step selection strategy. After the
new step selection algorithm has been presented, we offer a
summary, with proofs, of its desirable mathematical
properties. Numerical results illustrating the efficacy of
this new approach are presented.},
summary = {A model-trust region algorithm for problems in
unconstrained optimization and nonlinear equations
utilizing a quadratic interpolant is presented and
analyzed. This is offered as an alternative to the
piecewise-linear interpolant employed in the double dogleg
strategy. A step selection algorithm is presented, along
with a summary, with proofs, of its desirable mathematical
properties. Numerical results are presented.}}
@article{McCo69,
author = {G. P. McCormick},
title = {Anti-zig-zagging by bending},
journal = {Management Science},
volume = 15, pages = {315--319}, year = 1969}
@techreport{McCo91,
author = {G. P. McCormick},
title = {The superlinear convergence of a nonlinear primal-dual
algorithm},
institution = {Department of Operations Research, George Washington
University, Washington},
type = {Technical Report}, number = {OR T-550/91}, year = 1991}
@article{McKeMesiZeni95,
author = {M. P. McKenna and J. P. Mesirov and S. A. Zenios},
title = {Data Parallel Quadratic Programming on Box-Constrained
Problems},
journal = SIOPT,
volume = 5, number = 3, pages = {570--589}, year = 1995}
@article{MehrSun91,
author = {S. Mehrotra and J. Sun},
title = {A method of analytic centers for quadratically constrained
convex quadratic programs},
journal = SINUM,
volume = 28, number = {}, pages = {529--544}, year = 1991}
@article{MentAnde91,
author = {J. Mentel and H. Anderson},
title = {A new Kind of Parameter-Estimation of Reactions Under
Dynamic Temperature Program},
journal = {Thermochimica Acta},
volume = 187, number = {SEP}, pages = {121--132}, year = 1991,
abstract = {For kinetic evaluation of TG, DTA and DSC of simple and
complex reactions we tried to apply a parameter adjustment
by means of non- linear optimization (trust-region method
with Marquardt-routine). Including all experimental data
and using the differential equation systems, we found many
advantages over the usual linearizing methods. The problems
of consecutive and competing reactions as well as those
with steady states (enzyme kinetics) were solved
satisfactorily if two or more data sets with different
heating rates are known. Supplementary use of other
analytical methods are recommendable.},
summary = {A trust-region method is applied a parameter estimation
related to kinetic evaluation of TG, DTA and DSC of simple
and complex reactions. When all experimental data is
included and the differential equation systems used, this
proves advantageous compared to usual linearizing methods.
The problems of consecutive and competing reactions as well
as those with steady states (enzyme kinetics) are solved
satisfactorily if two or more data sets with different
heating rates are known. Supplementary use of other
analytical methods are recommended.}}
@article{MentTillMollHabe92,
author = {J. Mentel and V. Tiller and E. Moller and D. Haberland},
title = {Estimation of Parameters in Systems of Ordinary
Differential-Equations to the Determination of
Kinetic-Parameters},
journal = {Chemische Technik},
volume = 44, number = 9, pages = {300--303}, year = 1992,
abstract = {The integral determination of kinetic constants in complex
systems was handled as a special case of parameter
estimation in systems of differential equations. The
program for parameter estimation is written in Turbo-Pasca.
A Trust-Region method with Levenberg- Marquardt routine was
used. Especially with the use of the stable BDF-integration
routine it is possible that the start values differ some
magnitudes from the solution. Simulations of different
models Prove the efficiency of the evaluation also with
non-smooth data. In these cases it is not allowed to
calculate concentrations needed for the evaluation with the
help of stoichiometric relations.},
summary = {The integral determination of kinetic constants in complex
systems is handled as a special case of parameter
estimation in systems of differential equations. A
trust-region method is used. A stable
BDF-integration routine allows the use of bad initial
values. Simulations of
different models prove the efficiency of the method even
with non-smooth data. In these cases it is not possible to
calculate concentrations needed for the evaluation with the
help of stoichiometric relations.}}
@article{Miff75,
author = {R. Mifflin},
title = {A superlinearly convergent algorithm for minimization
without evaluating derivatives},
journal = MP,
volume = 9, number = 1, pages = {100--117}, year = 1975}
@article{Miff75b,
author = {R. Mifflin},
title = {Convergence bounds for nonlinear programming algorithms},
journal = MP,
volume = 8, number = 3, pages = {251--271}, year = 1975}
@article{MineFukuTana84,
author = {H. Mine and M. Fukushima and Y. Tanaka},
title = {On the use of epsilon -most-active constraints in an exact
penalty function method for nonlinear optimization},
journal = {IEEE Transactions on Automatic Control},
volume = {AC-29}, number = 11, pages = {1040--1042}, year = {1984},
abstract = {An algorithm for nonlinear programming problems is
presented which utilizes the epsilon -most-active
constraint strategy in an exact penalty function
method with trust region. The algorithm is
particularly suitable for problems containing a large
number of constraints. The global convergence of the
proposed algorithm is proved. The results of limited
computational experiments on discretized
semi-infinite programming problems are reported to
demonstrate the effectiveness of the present
approach.},
summary = {An globally convergent algorithm for nonlinear
programming problems is presented which utilizes
the epsilon -most-active constraint strategy in an
exact penalty function method with trust region. The
algorithm is particularly suitable for problems
containing a large number of constraints. Some
computational experiments are reported.}}
@article{MongSart95,
author = {M. Mongeau and A. Sartenaer},
title = {Automatic decrease of the penalty parameter in exact
penalty function methods},
journal = {European Journal of Operational Research},
volume = 83, number = 3, pages = {686--699}, year = 1995}
@article{MontAdle89,
author = {R. D. C. Monteiro and I. Adler},
title = {Interior path following primal-dual algorithms. 2. {C}onvex
quadratic programming},
journal = MP,
volume = 44, number = 1, pages = {43--66}, year = 1989}
@article{MontWang98,
author = {R. D. C. Monteiro and Y. Wang},
title = {Trust region affine scaling algorithms for linearly
constrained convex and concave programs},
journal = MP,
volume = 80, number = 3, pages = {283--310}, year = 1998,
abstract = {We study a trust region affine scaling algorithm for
solving the linearly constrained convex or concave
programming problem. Under primal nondegeneracy assumption,
we prove that every accumulation point of the sequence
generated by the algorithm satisfies the first order
necessary condition for optimality of the problem. For a
special class of convex or concave functions satisfying a
certain invariance condition on their Hessians, it is shown
that the sequence of iterates and objective function values
generated by the algorithm converge R-linearly and
Q-linearly, respectively. Moreover, under primal
nondegeneracy and for this class of objective functions, it
is shown that the limit point of the sequence of iterates
satisfies the first and second order necessary conditions
for optimality of the problem.},
summary = {A trust region affine scaling algorithm for solving the
linearly constrained convex or concave programming problem
is presented. Under primal non-degeneracy assumption, every
accumulation point of the sequence generated by the
algorithm satisfies the first order necessary condition.
For a special class of convex or concave
functions satisfying a certain invariance condition on
their Hessians, the sequence of iterates and objective
function values converge R-linearly and Q-linearly,
respectively. Moreover, under primal non-degeneracy and for
this class of objective functions, the limit point of the
sequence of iterates satisfies the first and second-order
necessary conditions.}}
@article{MontTsuc98,
author = {R. D. C. Monteiro and T. Tsuchiya},
title = {Global convergence of the affine scaling algorithm for
convex quadratic programming},
journal = SIOPT,
volume = 8, number = 1, pages = {26--58}, year = 1998,
summary = {A global convergence proof of the second-order affine
scaling algorithm for convex quadratic programming problems
is given, where the new iterate is the point that minimizes
the objective function over the intersection of the
feasible region with the ellipsoid centered at the current
point and whose radius is a fixed fraction $\beta \in
(0,1)]$ of the radius of the largest ``scaled'' ellipsoid
inscribed in the nonnegative orthant. The analysis is based
on the local Karmarkar potential function introduced by
Tsuchiya. For any $\beta \in (0,1)$ and without making any
nondegeneracy assumption on the problem, the sequences of
primal iterates and dual estimates converge to
optimal solutions of the quadratic program and its dual.}}
@inproceedings{More78,
author = {J. J. Mor\'{e}},
title = {The {L}evenberg-{M}arquardt algorithm: implementation and
theory},
crossref = {Wats78}, pages = {105--116},
summary = {A robust implementation of the
Levenberg-Morrison-Marquardt algorithm for nonlinear
least squares is discussed. The proposed method is shown
to have strong convergence properties. In addition to
robustness, the main features are the proper use of
implicitly scaled variables and the choice of the
Levenberg-Morrison-Marquardt parameter via a scheme due to
\citebb{Hebd73}. Numericial results illustrating the
behaviour of this implementation are presented.}}
@inproceedings{More83,
author = {J. J. Mor\'{e}},
title = {Recent developments in algorithms and software for trust
region methods},
crossref = {BachGrotKort83}, pages = {258--287},
abstract = {Trust region methods are an important class of iterative
methods for the solution of systems of nonlinear equations,
nonlinear estimation problems and large-scale optimization.
Interest in trust region methods derives, in part, from the
availability of strong convergence results and from the
development of software for these methods which is
reliable, efficient, and amazingly free of ad-hoc
decisions. In this paper we survey the theoretical and
practical results available for trust region methods and
discuss the relevance of these results to the
implementation of trust region methods},
summary = {The theoretical and practical results available for
trust-region methods in systems of nonlinear equations,
nonlinear estimation problems and large-scale optimization
are surveyed, and their relevance to the implementation
of trust-region methods discussed.}}
@inproceedings{More88,
author = {J. J. Mor\'{e}},
title = {Trust regions and projected gradients},
booktitle = {System Modelling and Optimization},
editor = {M. Iri and K. Yajima},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
volume = 113, pages = {1--13}, year = 1988,
note = {Lecture Notes in Control and Information Sciences},
abstract = {The numerical solution of large scale linearly constrained
problems by algorithms which use the gradient projection
method is a promising research area. Algorithms based on
the gradient projection method are able to drop and add
many constraints at each iteration, and this ability gives
them an important advantage in large scale problems. In
this paper we show how the ideas from the gradient
projection method combine with trust region methods, and we
give an indication of the powerful convergence results that
are available for algorithms of this type.},
summary = {It is shown how the ideas from the gradient projection
method combine with trust-region methods, and an indication
of the powerful convergence results that are available for
gradient-projection algorithms is given.}}
@article{More93,
author = {J. J. Mor\'{e}},
title = {Generalizations of the trust region problem},
journal = OMS,
volume = 2, number = 3, pages = {189--209}, year = 1993,
abstract = {The trust region problem requires the global minimum of a
general quadratic function subject to an ellipsoidal
constraint. The development of algorithms for the solution
of this problem has found applications in nonlinear and
combinatorial optimization. In this paper we generalize the
trust region problem by allowing a general quadratic
constraint. The main results are a characterization of the
global minimizer of the generalized trust region problem,
and the development of an algorithm that finds an
approximate global minimizer in a finite number of
iterations.},
summary = {The trust region subproblem is generalized by allowing a
general quadratic constraint. The main results are a
characterization of the global minimizer of the generalized
trust-region problem, and the development of an algorithm
that finds an approximate global minimizer in a finite
number of iterations.}}
@techreport{MoreGarbHill80,
author = {J. J. Mor\'{e} and B. S. Garbow and K. E. Hillstrom},
title = {User guide for {MINPACK-1}},
institution = ANL, address = ANL-ADDRESS,
number = {80--74}, year = 1980}
@article{MoreGarbHill81,
author = {J. J. Mor\'{e} and B. S. Garbow and K. E. Hillstrom},
title = {Testing Unconstrained Optimization Software},
journal = TOMS,
volume = 7, number = 1, pages = {17--41}, year = 1981}
@article{MoreSore83,
author = {J. J. Mor\'{e} and D. C. Sorensen},
title = {Computing A Trust Region Step},
journal = SISSC,
volume = 4, number = 3, pages = {553--572}, year = 1983,
abstract = {We propose an algorithm for the problem of minimizing a
quadratic function subject to an ellipsoidal constraint and
show that this algorithm is guaranteed to produce a nearly
optimal solution in a finite number of iterations. We also
consider the use of this algorithm in a trust region
Newton's method. In particular, we prove that under
reasonable assumptions the sequence generated by Newton's
method has a limit point which satisfies the first and
second order necessary conditions for a minimizer of the
objective function. Numerical results for GQTPAR, which is
a Fortran implementation of our algorithm, show that GQTPAR
is quite successful in a trust region method. In our tests
a call to GQTPAR only required $1.6$ iterations on the
average.},
summary = {An algorithm for minimizing a quadratic function subject to
an ellipsoidal constraint is proposed. This algorithm is
guaranteed to produce a nearly optimal solution in a finite
number of iterations. The use of this algorithm in a
trust-region Newton's method is also considered. In
particular, it is shown that, under reasonable assumptions
the sequence generated by Newton's method has a limit point
which satisfies first and second order necessary conditions
for a minimizer. Numerical results for GQTPAR, a Fortran
implementation of the algorithm, show that it is quite
successful in a trust-region method. In these tests, a call
to GQTPAR only required $1.6$ iterations on the average.}}
@inproceedings{MoreSore84,
author = {J. J. Mor\'{e} and D. C. Sorensen},
title = {{N}ewton's method},
booktitle = {Studies in Numerical Analysis},
editor = {G. H. Golub},
publisher = AMS, address = AMS-ADDRESS,
series = {MAA Studies in Mathematics}, number = 24, pages = {29--82},
year = 1984,
abstract = {Newton's method plays a central role in the development of
numerical techniques for optimization. In fact, most of the
current practical methods for optimization can be viewed as
variations on {N}ewton's method. It is therefore important
to understand {N}ewton's method as an algorithm in its own
right and as a key introduction to the mot recent ideas in
this area. One of the aims of this expository paper is to
present and analyze two main approaches to {N}ewton's
method for unconstrained optimization: the line search
approach and the trust region approach. The other aim is to
present some of the recent developments in the optimization
field which are related to {N}ewton's method. In
particular, we explore several variations on {N}ewton's
method which are appropriate for large scale problems, and
we also show how quasi-{N}ewton methods can be derived
quite naturally from {N}ewton's method.},
summary = {The linesearch and trust region approaches for unconstrained
optimization are discussed, and some of the recent
developments related to Newton's method are presented.
In particular, several variations on {N}ewton's method
which are appropriate for large-scale problems are explored,
and it is shown how quasi-Newton methods can be derived
quite naturally from Newton's method.}}
@article{MoreTora91,
author = {J. J. Mor\'{e} and G. Toraldo},
title = {On the Solution of Large Quadratic Programming Problems
with Bound Constraints},
journal = SIOPT,
volume = 1, number = 1, pages = {93--113}, year = 1991}
@book{MoreWrig93,
author = {J. J. Mor\'{e} and S. J. Wright},
title = {Optimization Software Guide},
publisher = SIAM, address = SIAM-ADDRESS,
number = 14, series = {Frontiers in Applied Mathematics},
year = 1993}
@article{More62,
author = {J. J. Moreau},
title = {D\'{e}composition orthogonale d'un espace {H}ilbertien
selon deux c\^{o}nes mutuellement polaires},
journal = {Comptes-Rendus de l'Acad\'{e}mie des Sciences (Paris)},
volume = 255, pages = {238--240}, year = 1962}
@inproceedings{Morr60,
author = {D. D. Morrison},
title = {Methods for nonlinear least squares problems and
convergence proofs},
booktitle = {Proceedings of the Seminar on Tracking Programs and Orbit
Determination},
editor = {J. Lorell and F. Yagi},
publisher = {Jet Propulsion Laboratory}, address = {Pasadena, USA},
pages = {1--9}, year = 1960,
abstract = {The STL tracking programs are designed o compute by least
squares the most probable trajectory of a missile, fom
observed radar or optical data. These data can be any
combination of range, azimuth, elevation, range rate, hour
angle and declination, or direction cosines, with respect
to a given location. A differential correction method is
used, starting from an initial estimate, and the iteration
continues until the residuals (observed minus computed
values) are either all within specified limits, or until
there is no further improvement. Errors above a certain
absolute value are automatically eliminated. One program
has been prepared primarily for lunar and interplanetary
flights. In this, Cowell's method of trajectory computation
is used. The partial derivatives used in the least squares
solution are found by solving the related variational
equations. Here, the trajectory elements, to which the
corrections are applied, are the componenets of position
and velocity of the missile at a particular point in the
trajectory (expressed in speherical coordinates). For
earth-satellite tracking, the elliptic elements of the
osculating ellipse are used to specify the trajectory.
Herricks' variation of elements method is used to compute
the trajectory, and the partial derivatives are computed
analytically. For these two programs a special least square
subroutine has been prepared in which convergence can be
assumed by limiting the amount any variable can change in
one iteration. The standard deviation of each variable is
printed out. After a trajectory has been fitted to a
ceratin set of data, additional data can be added without
the necessity of reprocessing the original set, a feature
which is especially valuable in regard to computing time.},
summary = {Least-squares estimation of missile trajectory is
considered in the context of lunar and interplanetary
flights as well as earthbound satellite tracking. A
least-square subroutine is described in which convergence
can be assumed by limiting the amount any variable can
change in one iteration. A method is given that allows the
computation of a quadratic model of the objective function
within a sphere using a single linear system depending on a
parameter. Monotonicity of the optimal model value as a
function of this parameter is proved.}}
@article{MukaPola75,
author = {H. Mukai and E. Polak},
title = {A quadratically convergent primal-dual algorithm with
global convergence properties for solving optimization
problems with inequality constraints},
journal = MP,
volume = 9, number = 3, pages = {336--349}, year = 1975}
@article{MukaTatsFuku98,
author = {K. Mukai and K. Tatsumi and M. Fukushima},
title = {An approximation algorithm for quadratic cost 0-1 mixed
integer programming problems},
journal = {Transactions of the Institute of Electronics, Information
and Communication Engineers A},
volume = {J81-A}, number = 4, pages = {649--657}, year = 1998,
abstract = {In this paper, we focus on the quadratic cost 0-1
mixed integer programming problem. First, we
formulate the problem as a two-level programming
problem that consists of the lower level continuous
quadratic programming problem with 0-1 variables
being fixed and the upper level nonlinear 0-1
programming problem. We propose an approximation
algorithm for solving the upper level 0-1 programming
problem. This algorithm approximately solves a
subproblem obtained by linearizing the objective
function at a current point. To guarantee the descent
property of the generated sequence, we use a trust
region technique that adaptively controls a penalty
constant in the objective function of the
subproblem. To solve subproblems, we apply a Hopfield
network with a new transition rule that allows a
temporary state transition based on the variable
depth method. Some numerical experiments for a
location-transportation problem with quadratic costs
indicate that the proposed algorithm is practically
effective.},
summary = {The quadratic cost 0-1 mixed integer programming problem
is formulated as a two-level programming
problem that consists of the lower level continuous
quadratic programming problem with 0-1 variables
being fixed and the upper level nonlinear 0-1
programming problem. An approximation
algorithm for solving the upper level 0-1 programming
problem is proposed that approximately solves a
subproblem obtained by linearizing the objective
function at a current point. To guarantee the descent
property of the generated sequence, a trust-region
technique adaptively controls a penalty
constant in the objective function of the
subproblem. To solve subproblems, a Hopfield
network is applied with a new transition rule that allows a
temporary state transition based on the variable
depth method. Some numerical experiments for a
location-transportation problem with quadratic costs
indicate that the proposed algorithm is practically
effective.}}
@inproceedings{Murr69,
author = {W. Murray},
title = {An algorithm for constrained minimization},
crossref = {Flet69}, pages = {189--196}}
@article{Murr71,
author = {W. Murray},
title = {Analytical expressions for eigenvalues and eigenvectors of
the {H}essian matrices of barrier and penalty functions},
journal = JOTA,
volume = 7, pages = {189--196}, year = 1971}
@techreport{Murr71b,
author = {W. Murray},
title = {An algorithm for finding a local minimum of an indefinite
quadratic program},
institution = NPL, address = NPL-ADDRESS,
number = {NAC 1}, year = 1971}
@inproceedings{Murr92,
author = {W. Murray},
title = {Ill-conditioning in Barrier Methods},
booktitle = {Advances in numerical partial differential equations and
optimization, Proceedings of the Sixth Mexico-United States
Workshop},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1992}
@techreport{MurrWrig78,
author = {W. Murray and M. H. Wright},
title = {Project {L}agrangian methods based on the trajectories of
penalty and barrier functions},
institution = STANFORD, address = STANFORD-ADDRESS,
type = {Technical Report}, number = {SOL78-23}, year = 1978}
@article{MurrPrie95,
author = {W. Murray and F. J. Prieto},
title = {A Sequential Quadratic Programming Algorithm Using an
Incomplete Solution of the Subproblem},
journal = SIOPT,
volume = 5, number = 3, pages = {590--640}, year = 1995}
@article{MurrWrig82,
author = {W. Murray and M. H. Wright},
title = {Computation of the search direction in constrained
optimization algorithms},
journal = MPS,
volume = 16, number = {MAR}, pages = {62--83}, year = 1982}
@book{Murt81,
author = {B. A. Murtagh},
title = {Advanced Linear Programming},
publisher = MACGH, address = MACGH-ADDRESS,
year = 1981}
@article{MurtKaba87,
author = {K. G. Murty and S. N. Kabadi},
title = {Some {NP}-complete problems in quadratic and nonlinear
programming},
journal = MP,
volume = 39, number = 2, pages = {117--129}, year = 1987}
%%% N %%%
@inproceedings{Nabo87,
author = {N. Nabona},
title = {Computational results of {N}ewton's method with search along
an approximate hook step curve for unconstrained
minimization},
booktitle = {Actas I Seminario Internacional de Investigacion Operativa
del Pais Vasco},
editor = {J. P. Vilaplana and L. F. Escudero},
publisher = {Argitaparen Zerbitzua Euskal Herriko Unibersitatea},
address = {Bilbao, Spain},
pages = {21--54}, year = 1987,
summary = {Numerical experience is presented for an unconstrained
optimization method in which a trust-region step is
computed by minimizing the model along a Bezier curve that
approximates the trajectory of exact minimizers as a
function of the trust-region radius.}}
@manual{NAG98,
author = {{NAG}},
title = {Fortran Library Mark 18},
organization = {NAG Ltd.}, address = {Oxford, England},
year = 1998}
@book{Nagu93,
author = {A. Nagurney},
title = {Network Economics: a Variational Inequality Approach},
publisher = KLUWER, address = KLUWER-ADDRESS,
series = {Advances in Computational Economics}, year = 1993}
@article{Nash84,
author = {S. G. Nash},
title = {{N}ewton-type Minimization via the {L}anczos Method},
journal = SINUM,
volume = 21, number = 4, year = 1984}
@article{Nash85a,
author = {S. G. Nash},
title = {Preconditioning of truncated {N}ewton methods},
journal = SISSC,
volume = 6, number = 3, pages = {599--618}, year = 1985}
@article{NashNoce91,
author = {S. G. Nash and J. Nocedal},
title = {A Numerical Study of the Limited Memory {BFGS} Method and
the Truncated-{N}ewton Method for Large-Scale Optimization},
journal = SIOPT,
volume = 1, number = 3, pages = {358--372}, year = 1991}
@article{NashSofe90,
author = {S. G. Nash and A. Sofer},
title = {Assessing a search direction within a truncated-{N}ewton
method},
journal = {Operations Research Letters},
volume = 9, number = 4, pages = {219--221}, year = 1990}
@article{NashSofe93,
author = {S. G. Nash and A. Sofer},
title = {A barrier method for large-scale constrained optimization},
journal = ORSAC,
volume = 5, number = 1, pages = {40--53}, year = 1993}
@techreport{NashSofe98,
author = {S. G. Nash and A. Sofer},
title = {Why Extrapolation Helps in Barrier Methods},
institution = {Operations Research and Engineering Department, George
Mason University},
address = {Fairfax, USA},
month = {September}, year = 1998}
@article{NeldMead65,
author = {J. A. Nelder and R. Mead},
title = {A simplex method for function minimization},
journal = COMPJ,
volume = 7, pages = {308--313}, year = 1965}
@article{NelsPapa98,
author = {S. A. Nelson and P. Y. Papalambros},
title = {A modified trust region algorithm for hierarchical NLP},
journal = {Structural Optimization},
volume = 16, number = 1, pages = {19--28}, year = 1998,
abstract = {Large-scale optimization problems frequently require the
exploitation of structure in order to obtain efficient and
reliable solutions. Successful algorithms for general
nonlinear programming problems with theoretical
underpinnings do not usually accommodate any additional
properties of the original algorithm.},
summary = {Modifications are made to a trust-region algorithm to take
advantage of the hierarchical structure in large-scale
optimization problems without compromising the theoretical
properties of the original algorithm.}}
@article{NemiSche96,
author = {A. Nemirovskii and K. Scheinberg},
title = {Extension of {K}armarkar's algorithm onto convex
quadratically constrained quadratic problems},
journal = MPA,
volume = 72, number = 3, pages = {273--289}, year = 1996}
@book{NestNemi94,
author = {Y. Nesterov and A. Nemirovskii},
title = {Interior-Point Polynomial Algorithms in Convex Programming},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1994}
@book{NobeDani77,
author = {B. Noble and J. W. Daniel},
title = {Applied Linear Algebra},
publisher = PH, address = PH-ADDRESS,
edition = {second}, year = 1977}
@article{NestTodd98,
author = {Y. Nesterov and M. J. Todd},
title = {Primal-Dual Interior-Point Methods for Self-Scaled Cones},
journal = SIOPT,
volume = 8, number = 2, year = 1998}
@article{Noce80,
author = {J. Nocedal},
title = {Updating quasi-{N}ewton matrices with limited storage},
journal = MC,
volume = 35, pages = {773--782}, year = 1980}
@inproceedings{Noce84,
author = {J. Nocedal},
title = {Trust Region Algorithms for Solving Large Systems of
Nonlinear Equations},
booktitle = {Innovative Methods for Nonlinear Problems},
editor = {W. Liu and T. Belytschko and K. C. Park},
publisher = {Pineridge Press},
pages = {93--102}, year = 1984,
abstract = {We review various algorithms for solving large sparse
systems of nonlinear equations, with emphasis in those
methods based on trust regions. We consider only those
cases when derivatives are available, either analytically
or by finite differences. We describe a new algorithm,
which is based on minimizing the $\ell_1$-norm of the
linearized equations within an $\ell_{\infty}$-norm trust
region. The resulting problems are solved by means of
sparse linear programming techniques},
summary = {Algorithms for solving large sparse systems of
nonlinear equations are reviewed, with emphasis on
trust-region methods. An algorithm is described in which
the $\ell_1$ norm of the linearized
equations is minimized within an $\ell_{\infty}$ norm trust
region. The resulting subproblems are solved by means of
sparse linear programming techniques}}
@inproceedings{Noce86,
author = {J. Nocedal},
title = {Viewing the Conjugate-Gradient Algorithm as a Trust Region
Method},
booktitle = {Numerical Analysis},
editor = {J. P. Hennart},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
series = {Lecture Notes in Mathematics}, volume = 1230,
pages = {118--126}, year = 1986,
summary = {It is shown how a trust-region subproblem, based upon a
memoryless secant-updating formula, may be solved very
efficiently. A connection is made between such subproblems
and the conjugate-gradient method.}}
@article{NoceOver85,
author = {J. Nocedal and M. L. Overton},
title = {Projected {H}essian updating algorithms for nonlinearly
constrained optimization},
journal = SINUM,
volume = 22, pages = {821--850}, year = 1985}
@inproceedings{NoceYuan98,
author = {J. Nocedal and Y. Yuan},
title = {Combining trust region and line search techniques},
crossref = {Yuan98}, pages = {153--176},
abstract = {We propose an algorithm for nonlinear optimization that
employs both trust region techniques and line searches.
Unlike traditional trust region methods, our algorithm does
not resolve the subproblem if the trial step results in an
increase in the objective function, but instead performs a
backtracking line search from the failed point.
Backtracking can be done along a straight line or along a
curved path. We show that the new algorithm preserves the
strong convergence properties of trust region methods.
Numerical results are also presented.},
summary = {An algorithm for nonlinear optimization that employs both
trust-region techniques and linesearches is proposed. This
algorithm does not resolve the subproblem if the trial step
results in an increase in the objective function, but
instead performs a backtracking linesearch from the failed
point. Backtracking can be done along a straight line or
along a curved path. It is shown that the algorithm
preserves the strong convergence properties of trust-region
methods. Numerical results are presented.}}
@article{Nota93,
author = {Y. Notay},
title = {On the convergence rate of the conjugate gradients in
presence of rounding errors},
journal = NUMMATH,
volume = 65, number = 3, pages = {301--317}, year = 1993}
%%% O %%%
@article{OLea80,
author = {D. P. O'Leary},
title = {A generalized conjugate gradient algorithm for solving a
class of quadratic programming problems},
journal = LAA,
volume = 34, pages = {371--399}, year = 1980}
@phdthesis{Omoj89,
author = {E. O. Omojokun},
title = {Trust region algorithms for optimization with nonlinear
equality and inequality constraints},
school = {University of Colorado},
address = {Boulder, Colorado, USA},
year = 1989,
abstract = {We consider the general nonlinear optimization problem
defined as, minimize a nonlinear real-valued function of
several variables, subject to a set of nonlinear equality
and inequality constraints. This class of problems arise
in many real life applications, for example in engineering
design, chemical equilibrium, simulation and data fitting.
In this research, we present algorithms that use the trust
region technique to solve these problems. First, we
develop an algorithm for solving the nonlinear equality
constrained optimization, then we generalize the algorithm
to handle the inclusion of nonlinear inequality constraints
in the problem. The algorithms use the successive
quadratic programming (SQP) approach and trust region
technique. We define a model subproblem which minimizes
a quadratic approximation of the Lagrangian subject to
modified relaxed linearizations of the problem nonlinear
constraints and a trust region constraint. Inequality
constraints are handled by a compromise between an active
set strategy and IQP subproblem solution technique. An
analysis which describes the local convergence properties of
our algorithms is presented. The algorithms are
implemented and the model minimization is done
approximately by using the dogleg approach. Numerical
results are presented and compared with the results of a
popular linesearch method. Some examples are presented in
which the ability of our method to use directions of
negative curvature results in greater reliability. Results
of the numerical experiments indicate that our method is
very robust and reasonably efficient.},
summary = {Trust-region algorithms for the general nonlinear
optimization problem are presented. These handle both the
equality and inequality constrained cases. The algorithms
use the SQP approach combined with the trust-region
technique. A model subproblem is defined in which
a quadratic approximation of the Lagrangian is minimized
subject to modified relaxed linearizations of the problem
nonlinear constraints and a trust-region constraint.
Inequality constraints are handled by a compromise
between an active set strategy and IQP subproblem solution
techniques. A local convergence analysis is presented.
Numerical tests indicate that the proposed methods are
very robust and reasonably efficient.}}
@book{OrteRhei70,
author = {J. M. Ortega and W. C. Rheinboldt},
title = {Iterative Solution of Nonlinear Equations in Several
Variables},
publisher = AP, address = AP-ADDRESS,
year = 1970}
@article{Osbo76,
author = {M. R. Osborne},
title = {Nonlinear least squares---the {L}evenberg-{M}arquardt
algorithm revisited},
journal = {Journal of the Australian Mathematical Society, Series B},
volume = 19, pages = {343--357}, year = 1976,
summary = {A rule for choosing the Levenberg-Morrison-Marquardt
parameter is described, which permits a satisfactory
convergence theorem to be proved, and is capable of
satisfactory computer implementation.}}
@book{Osbo85,
author = {M. R. Osborne},
title = {Finite Algorithms for Optimization and Data Analysis},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1985}
@article{Osbo87,
author = {M. R. Osborne},
title = {Estimating nonlinear models by maximum likelihood for the
exponential family},
journal = SISSC,
volume = 8, number = 3, pages = {446--456}, year = 1987,
abstract = {Many but not all attractive properties of generalized
linear models associated with the exponential family of
distributions are destroyed by nonlinearity. A consequence
is that ensuring the stability of a computational process
for maximizing the likelihood becomes relatively more
important. Here it is shown that trust region methods for
solving nonlinear least squares problems are readily
adapted to maximize likelihoods based on the exponential
family, and that the nice theoretical results available for
the nonlinear least squares problem also generalize.},
summary = {Trust-region methods for solving nonlinear
least-squares problems are adapted to maximize
likelihoods based on the exponential family, while
preserving their theoretical properties.}}
@techreport{Osbo98,
author = {M. R. Osborne},
title = {Variable selection and control in least squares problems},
institution = {Centre for Mathematics and its Applications,
School of Mathematical Sciences,
Australian National University},
address = {Canberra, Australia},
number = {MRR 047-98}, year = 1998,
abstract = {The classical technique of stepwise regression provides a
paradigm for variable selection in the linear least squares
problem. Trust region method which control the size of the
correction to the current solution estimate prove
attractive for nonlinear least squares problems because of
their good global convergence behaviour. Recently there has
been a convergence of these techniques with the realisation
that the $\ell_1$ trust region method also provides a form
of variable selection. These results are reviewed here, and
computational methods discussed.},
summary = {Results are reviewed showing that the $\ell_1$ trust-region
method provides a form of variable selection for
least-squares problems and computational methods are
discussed.}}
@techreport{OutrZoweSchr91,
author = {J. Outrata and J. Zowe and H. Schramm},
title = {Bundle trust methods: {F}ortran codes for nondifferentiable
optimization},
institution = {DFG}, address = {Germany},
number = 269, year = 1991}
%%% P %%%
@phdthesis{Paig71,
author = {C. C. Paige},
title = {The computation of eigenvalues and eigenvectors of very
large sparse matrices},
school = {University of London},
year = 1971}
@article{PaigSaun75,
author = {C. C. Paige and M. A. Saunders},
title = {Solution of sparse indefinite systems of linear equations},
journal = SINUM,
volume = 12, number = 4, pages = {617--629}, year = 1975}
@article{PaigSaun82a,
author = {C. C. Paige and M. A. Saunders},
title = {{LSQR}: an algorithm for sparse linear equations and sparse
least squares},
journal = TOMS,
volume = 8, pages = {43--71}, year = 1982}
@article{Pang81,
author = {J. S. Pang},
title = {An equivalence between two algorithms for quadratic
programming},
journal = MP,
volume = 20, number = 2, pages = {152--165}, year = 1981}
@inproceedings{Pang95,
author = {J. S. Pang},
title = {Complementarity problems},
booktitle = {Handbook of Global Optimization},
editor = {R. Horst and P. Pardalos},
publisher = KLUWER, address = KLUWER-ADDRESS,
pages = {271--338}, year = 1995}
@article{PaniTits91,
author = {E. Panier and A. L. Tits},
title = {Avoiding the {M}aratos effect by means of a nonmonotone
linesearch {I}. General constrained problems},
journal = SINUM,
volume = 28, pages = {1183--1195}, year = 1991}
@article{Pant88,
author = {J. F. A. Pantoja},
title = {Differential dynamic-programming and {N}ewton method},
journal = {International Journal of Control},
volume = 47, number = 5, pages = {1539-1553}, year = 1988}
@article{PantMayn91,
author = {J. F. A. Pantoja and D. Q. Mayne},
title = {Exact penalty functions with simple updating of the penalty
parameter},
journal = JOTA,
volume = 69, pages = {441--467}, year = 1991}
@book{PapaStei82,
author = {C. H. Papadimitriou and K. Steiglitz},
title = {Combinatorial Optimization},
publisher = PH, address = PH-ADDRESS,
year = 1982}
@book{Parl80,
author = {B. N. Parlett},
title = {The Symmetric Eigenvalue Problem},
publisher = PH, address = PH-ADDRESS,
year = 1980,
note = {Reprinted as \emph{Classics in Applied Mathematics 20}, SIAM,
Philadelphia, USA, 1998}}
@article{ParlReid81,
author = {B. N. Parlett and J. K. Reid},
title = {Tracking the progress of the {L}anczos algorithm for large
symmetric eigenproblems},
journal = JIMA,
volume = 1, pages = {135--155}, year = 1981}
@book{Patr94,
author = {M. Patriksson},
title = {The Traffic Assignment Problem: Models and Methods},
publisher = {VSP}, address = {Utrecht, The Netherlands},
year = 1994}
@book{Patr98,
author = {M. Patriksson},
title = {Nonlinear Programming and Variational Inequality Problems,
a Unified Approach},
publisher = KLUWER, address = KLUWER-ADDRESS,
year = 1998}
@article{Peng96,
author = {J. Peng},
title = {Unconstrained Methods for generalized nonlinear
complementarity and variational inequality problems},
journal = JCM,
volume = 14, number = 2, pages = {99--107}, year = 1996,
abstract = {In this paper, we construct unconstrained methods for the
generalized nonlinear complementarity problem and
variational inequalities. Properties of the correspondent
unconstrained optimization problem are studied. We apply
these methods to the subproblems in trust region method,
and study their interrelationships. Numerical results are
also presented.},
summary = {Unconstrained methods for the generalized nonlinear
complementarity problem and variational inequalities are
constructed. Properties of the corresponding unconstrained
optimization problem are studied. These methods are applied
to the subproblems in trust-region methods, and their
interrelationships studied. Numerical results are
presented.}}
@inproceedings{Peng98,
author = {J. Peng},
title = {A Smoothing Function and Its Applications},
crossref = {FukuQi98}, pages = {293--316}}
@article{PengYuan97,
author = {J. Peng and Y. Yuan},
title = {Optimality conditions for the minimization of a quadratic
with two quadratic constraints},
journal = SIOPT,
volume = 7, number = 3, pages = {579--594}, year = 1997,
abstract = {The trust region method has been proven to be very
successful in both unconstrained and constrained
optimization. It requires the global minimum of a general
quadratic function subject to ellipsoid constraints. In
this paper, we generalize the trust region subproblem by
allowing two general quadratic constraints. Conditions and
properties of its solution are discussed.},
summary = {The trust region subproblem is generalized by allowing two
general quadratic constraints. Conditions and properties of
its solution are discussed.}}
@techreport{Perr76,
author = {A. Perry},
title = {A modified conjugate gradient algorithm},
institution = {Center for Mathematical Studies in Economics and Management
Science, Northwestern University},
address = NWU-ADDRESS,
number = 229, year = 1976}
@article{PetzRenMaly97,
author = {L. R. Petzold and Y. H. Ren and T. Maly},
title = {Regularization of higher-index differential-algebraic
equations with rank-deficient constraints},
journal = SISC,
volume = 18, number = 3, pages = {753--774}, year = 1997,
abstract = {In this paper we present several regularizations for
higher-index differential-algebraic equations with
rank-deficient or singular constraints. These types of
problems arise, for example, in the solution of constrained
mechanical systems, when a mechanism's trajectory passes
through or near a kinematic singularity. We derive a class
of regularizations for these problems which is based on
minimization of the norm of the constraints. The new
regularizations are analogous to trust-region methods of
numerical optimization. We give convergence results for the
regularizations and present some numerical experiments
which illustrate their effectiveness.},
summary = {Several regularizations for higher-index
differential-algebraic equations with rank-deficient or
singular constraints are presented. These types of problems
arise in the solution of constrained mechanical systems,
when a mechanism's trajectory passes through or near a
kinematic singularity. Regularizations for these
problems are derived which are based on minimization of the
norm of the constraints. They are analogous to trust-region
methods. Convergence results are given and numerical
experiments are presented.}}
@article{PhamWangYass90,
author = {Pham Dinh, T. and S. Wang},
title = {Training multi-layered neural network with a trust-region
based algorithm},
journal = RAIRO-MM,
volume = 24, number = 4, pages = {523--553}, year = 1990,
abstract = {In this paper, we first show how the problem of training a
neural network is modelized as an optimization problem; and
the generally used training algorithm. Then we propose a
new algorithm based on a trust-region technique which is
very efficient for non-convex optimization problems.
Experimental results show that the new algorithm is much
faster and robust compared with GBP. It makes the design of
neural net architecture much less problem-dependent.},
summary = {The problem of training a neural network is modelled as an
optimization problem and gradient backpropagation, the most
commonly used training method, is described. A
trust-region algorithm is then proposed. Experimental
results show that the algorithm is much faster and
robust compared with gradient backpropagation. It makes the
design of neural net architecture much less
problem-dependent.}}
@article{PhamHoai95,
author = {Pham Dinh, T. and Le Thi, H. A.},
title = {{L}agrangian stability and global optimality on nonconvex
quadratic minimization over {E}uclidean balls and spheres},
journal = {Journal of Convex Analysis},
volume = 2, number = {1--2}, pages = {263--276}, year = 1995,
abstract = {We prove in this paper the stability of the Lagrangian
duality in nonconvex quadratic minimization over Euclidean
balls and spheres. As direct consequences we state both
global optimality conditions in these problems and detailed
descriptions of the structure of their solution sets. These
results are essential for devising solution algorithms.},
summary = {Stability of the Lagrangian duality in non-convex quadratic
minimization over Euclidean balls and spheres is proved.
Global optimality conditions for these problems are deduced
together with the detailed descriptions of the structure of
their solution sets.}}
@article{PhamHoai98,
author = {Pham Dinh, T. and Le Thi, H. A.},
title = {{D.C.} optimization algorithm for solving the trust-region
subproblem},
journal = SIOPT,
volume = 8, number = 2, pages = {476--505}, year = 1998,
abstract = {This paper is devoted to the framework of d.c. (difference
of convex functions) optimization: d.c. duality, local and
global optimalities in d.c. programming, the d.c. algorithm
(DCA) and its application to solving the trust-region
problem. The DCA is an iterative method which is quite
different from well-known related algorithms. Thanks to the
particular structure of the problem, the DCA becomes very
simple (it requires only matrix-vector products) and, in
practice, converges to a global solution. For checking the
global optimality of solutions provided by the DCA the
quite inexpensive Implicitly Restarted Lanczos method of
Sorensen has been used. A simple numerical procedure has
been introduced (in the case of nonglobal solutions) in
order to find a feasible point having a smaller objective
value and to restart the DCA with this point. It has been
stated that in the nonconvex case (problem ($Q_1$) with $A$
being nonpositive semidefinite) the DCA with at most $2m+2$
restartings ($m$ is the number of distinct negative
eigenvalues of $A$) requires only matrix-vector products
too and converges to a global solution. Numerical
simulations proved the robustness and the efficiency of the
DCA with respect to related standard methods, especially in
large scale problems.},
summary = {The framework of d.c. (difference of convex functions)
optimization is presented, including d.c. duality, local
and global optimality, the d.c. algorithm (DCA) and its
application to solving the trust-region problem, which only
requires matrix-vector products. In practice, it converges
to a global solution, which is checked by using the
Implicitly Restarted Lanczos method of Sorensen. If a
nonglobal solution is found, a procedure is proposed, that
finds a feasible point having a smaller objective value at
which the DCA may then be restarted. It is proved that, in
the non-convex case, the DCA needs at most $2m+2$ restarts
to converge to a global solution, where $m$ is the number
of distinct negative eigenvalues of the Hessian. The
robustness and efficiency of the DCA is illustrated by
numerical experiments.}}
@article{PhamPhonHoraQuan97,
author = {Pham Dinh, T. and T. Q. Phong and R. Horaud and L. Quan},
title = {Stability of {L}agrangian duality for nonconvex quadratic
programming. Solution methods and applications in computer
vision},
journal = RAIRO-MM,
volume = 31, number = 1, pages = {57--90}, year = 1997,
abstract = {The problem of minimizing a quadratic form over a ball
centered at the origin is considered. The stability of
Lagrangian duality is established and complete
characterizations of a global optimal solution are given.
On the basis of this theoretical study, two principal
solution methods are presented. An important application of
nonconvex quadratic programming is the computation of rite
step to a new iterate in the Trust Region (TR) approach
methods which are known to be efficient for nonlinear
optimization problems. Also, we discuss the mathematical
models of some important problems encountered in Computer
Vision. Most of them can be formulated as a minimization of
a sum of squares of nonlinear functions. A practical
TR-based algorithm is proposed for nonlinear least squares
problem which seems to be well suited for our applications.},
summary = {The problem of minimizing a quadratic form over a ball is
considered. The stability of Lagrangian duality is
established and complete characterizations of a global
optimal solution are given. Two solution methods
are deduced, with application to the trust-region
subproblem. Mathematical models of some important problems
encountered in computer vision are discussed, which
can be formulated as a minimization of a sum of squares of
nonlinear functions. A practical trust-region based
algorithm is proposed for the nonlinear least-squares problem
which seems to be well suited to the computer vision
applications.}}
@article{Phan82,
author = {Phan huy Hao, E.},
title = {Quadratically constrained quadratic programming: some
applications and a method for solution},
journal = {Zeitschrift f\"{u}r Operations Research},
volume = 26, number = 3, pages = {105--119}, year = 1982,
abstract = {(no abstract)},
summary = {A method is proposed for the solution of quadratically
constrained quadratic programming.}}
@inproceedings{PhonHoraYassPham93,
author = {T. Q. Phong and R. Horaud and A. Yassine and Pham Dinh, T.},
title = {Optimal estimation of object pose from a single perspective
view},
booktitle = {1993 Proceedings Fourth International Conference on
Computer Vision},
publisher = {IEEE Computer Society Press},
address = {Los Alamitos, CA, USA},
pages = {534--539}, year = 1993,
abstract = {The authors present a method for robustly and
accurately estimating the rotation and translation
between a camera and a 3-D object from point and line
correspondences. First they devise an error function
and then show how to minimize this error
function. The quadratic nature of this function is
made possible by representing rotation and
translation with a dual number quaternion. A detailed
account is provided of the computational aspects of a
trust-region optimization method. This method
compares favourably with Newton's method, which has
extensively been used to solve the problem, and with
Faugeras-Toscani's linear method (1986) for
calibrating a camera. Some experimental results are
presented which demonstrate the robustness of the
method with respect to image noise and matching
errors.},
summary = {A method for robustly and
accurately estimating the rotation and translation
between a camera and a 3-D object from point and line
correspondences. An error function is defined
and then minimized. The quadratic nature of this function is
made possible by representing rotation and
translation with a dual number quaternion. A detailed
account is provided of the computational aspects of a
trust-region optimization method. This method
compares favourably with Newton's method, which has
extensively been used to solve the problem, and with
Faugeras-Toscani's linear method (1986) for
calibrating a camera. Experimental results are
presented which demonstrate the robustness of the
method with respect to image noise and matching
errors.}}
@article{PhonHoraYassPham95,
author = {T. Q. Phong and R. Horaud and A. Yassine and Pham Dinh, T.},
title = {Object pose from 2-D to 3-D Point and Line Correspondences},
journal = {International Journal of Computer Vision},
volume = 15, number = 3, pages = {225--243}, year = 1995,
abstract = {In this paper we present a method for optimally estimating
the rotation and translation between a camera and a 3-D
object from point and/or line correspondences. First we
devise an error function and second we show how to minimize
this error function. The quadratic nature of this function
is made possible by representing rotation and translation
with a dual number quaternion. We provide a detailed
account of the computational aspects of a trust-region
optimization method. This method compares favourably with
Newton's method which has extensively been used to solve
the problem at hand, with the Faugeras-Toscani linear
method for calibrating a camera, and with the
Levenberg-Marquardt non-linear optimization method. Finally
we present some experimental results which demonstrate the
robustness of our method with respect to image noise and
matching errors.},
summary = {A method for optimally estimating the rotation and
translation between a camera and a 3-D object from point
and/or line correspondences is presented. An error function
is devised. The quadratic nature of the error function is
obatined by representing rotation and translation with a
dual number quaternion. We provide a detailed account of
the computational aspects of a trust-region optimization
method to minimize the error. This method compares
favourably with Newton's method, with the Faugeras-Toscani
linear method for calibrating a camera, and with the
Levenberg-Morrison-Marquardt nonlinear optimization method.
Experimental results demonstrate the robustness of the
method with respect to image noise and matching errors.}}
@inproceedings{PiepMcMuLipk98,
author = {J. A. Piepmeier and G. V. McMurray and H. Lipkin},
title = {Tracking a moving target with model independent visual
servoing: a predictive estimation approach},
booktitle = {Proceedings 1998 IEEE International Conference on Robotics
and Automation},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 3, pages = {2652--2657}, year = 1998,
abstract = {Target tracking by model independent visual servo
control is achieved by augmenting quasi-Newton trust
region control with target prediction. Model
independent visual servo control is defined using
visual feedback to control the robot without precise
kinematic and camera models. While a majority of the
research assumes a known robot and camera model,
there is a paucity of literature addressing model
independent control. In addition, most researches
have focused primarily on static targets. The work
presented here demonstrates the use of predictive
filters to improve the performance of the control
algorithm for linear and circular target motions. The
results show a performance of the same order of
magnitude as compared to some model based visual
servo control research. Certain limitations to the
algorithm are also discussed.},
summary = {Target tracking by model independent visual servo
control is achieved by augmenting quasi-Newton trust-region
control with target prediction. Model independent visual
servo control is defined using visual feedback to control
the robot without precise kinematic and camera models. The
use of predictive filters to improve the performance of the
control algorithm for linear and circular target motions is
demonstrated. The results show a performance of the same
order of magnitude as compared to some model based visual
servo control research. Certain limitations to the
algorithm are also discussed.}}
@article{Piet69,
author = {T. Pietrzykowski},
title = {An Exact Potential Method for Constrained Maxima},
journal = SINUM,
volume = 6, number = 2, pages = {299--304}, year = 1969}
%abstract = {The main result of the paper consists of the theorem that
% under certain, natural assumptions the local conditional
% maximum $x_0$ of the function $f$ on the set \[A = \{x\in
% \Re^n | \phi_i(x)\geq 0, \psi_j(x) = 0, i=1,\ldots,
% k,j=1,\ldots,l\}\] is identical with the unconditional
% maximum of the potential function \[p(x,\mu) = \mu f(x) +
% \sum_{i=1}^k \neg(\phi_i(x))- \sum_{j=1}^l | \psi(x)|,
% x\in \Re^n, \mu \geq 0,\] for $\mu$ sufficiently small.
% There is also provided a draft of a modified gradient
% procedure for maximizing the potential $p(x,\mu)$ since
% it is generally nonsmooth even for differentiable
% $f, \phi_i$ and $\psi_j$.},
%summary = {This paper presents a theorem that under certain, natural
% assumptions the local conditional maximum $x_0$ of the
% function $f$ on the set $A = \{x\in \Re^n | \phi_i(x)\geq 0,
% \psi_j(x) = 0, i=1,\ldots, k,j=1,\ldots,l\}$ is identical
% with the unconditional maximum of the potential function
% $p(x,\mu) = \mu f(x) + \sum_{i=1}^k \neg(\phi_i(x))-
% \sum_{j=1}^l | \psi(x)|, x\in \Re^n, \mu \geq 0,$ for $\mu$
% sufficiently small. In addition an outline of a modified
% gradient procedure for maximizing the potential $p(x,\mu)$
% is given. The modification is necessary since the
% potential function is generally non-smooth even for
% differentiable $f, \phi_i$ and $\psi_j$.}}
@article{Plan99,
author = {T. D. Plantenga},
title = {A trust-region method for nonlinear programming based on
primal interior point techniques},
journal = SISC,
volume = 20, number = 1, pages = {282--305}, year = 1999,
abstract = {This paper describes a new trust region method for solving
large-scale optimization problems with nonlinear equality
and inequality constraints. The new algorithm employs
interior-point techniques from linear programming, adapting
them for more general nonlinear problems. A software
implementation based entirely on sparse matrix methods is
described. The software handles infeasible start points,
identifies the active set of constraints at a solution, and
can use second derivative information to solve problems.
Numerical results are reported for large and small
problems, and a comparison made with other large-scale
codes.},
summary = {A trust region method for large-scale optimization
problems with nonlinear equality and inequality constraints
is described. The algorithm employs interior-point
techniques from linear programming, adapting them for more
general nonlinear problems. A software implementation based
entirely on sparse matrix methods is described. The
software handles infeasible start points, identifies the
active set of constraints at a solution, and can use second
derivative information. Numerical results are reported for
large and small problems, and a comparison made with other
large-scale codes.}}
@book{Pola97,
author = {E. Polak},
title = {Optimization. Algorithms and Consistent Approximations},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
series = {Applied Mathematical Sciences, Volume 124},
year = 1997}
@article{PolaTits80,
author = {E. Polak and A. L. Tits},
title = {A globally convergent implementable multiplier method with
automatic penalty limitation},
journal = {Applied Mathematics and Optimization},
volume = 6, pages = {335--360}, year = 1980}
@article{PoliQi95,
author = {R. Poliquin and L. Qi},
title = {Iteration Functions in some nonsmooth optimization
algorithms},
journal = MOR,
volume = 20, number = 2, pages = {479--496}, year = 1995,
abstract = {Recently, several globally convergent model algorithms
based on iteration functions have been proposed for solving
nonsmooth optimization problems. In particular,
\citebb{PangHanRang91} proposed such an algorithm for
minimizing a locally Lipschitzian function. We determine
properties of iteration functions (calculus, existence); we
also identify characteristics of functions that possess
iteration functions. We show that a locally Lipschitzian
function has a Pang-Han-Rangaraj iteration function only
when the function is pseudo-regular (in the sense of
Borwein), and that a subsmooth (lower-C-1)function always
has a Pang-Han-Rangaraj iteration function.},
summary = {Properties of iteration functions (calculus, existence)
arising in model trust-region algorithms for non-smooth
problems are analyzed. It is shown that a locally
Lipschitzian function has a Pang-Han-Rangaraj iteration
function only when the function is pseudo-regular (in the
sense of Borwein), and that a sub-smooth (lower-C-1)
function always has a Pang-Han-Rangaraj iteration function.}}
@article{PoljWolk95,
author = {S. Poljack and H. Wolkowicz},
title = {Convex relaxations of (0, 1)--Quadratic Programming},
journal = MOR,
volume = 20, number = 3, pages = {550--561}, year = 1995}
@article{PoljRendWolk95,
author = {S. Poljack and F. Rendl and H. Wolkowicz},
title = {A recipe for semidefinite relaxation for (0,1)--quadratic
programming},
journal = {Journal of Global Optimization},
volume = 7, number = 1, pages = {51--73}, year = 1995,
abstract = {We review various relaxations of (0,1)-quadratic
programming problems. These include semidefinite programs,
parametric trust region problems and concave quadratic
maximization. All relaxations that we consider lead to
efficiently solvable problems. The main contributions of
the paper are the following. Using Lagrangian duality, we
prove equivalence of the relaxations in a unified and
simple way. Some of these equivalences have been known
previously, but our approach leads to short and transparent
proofs. Moreover we extend the approach to the case of
equality constraints into the objective function. We show
how this technique can be applied to the Quadratic
Assignment Problem, the Graph Partition Problem and the
Max-Clique Problem. Finally, we show our relaxation to be
best possible among all quadratic majorants with zero
trace.},
summary = {Various relaxations of (0,1)-quadratic programming problems
are reviewed, including semidefinite programs, parametric
trust-region problems and concave quadratic maximization.
All lead to efficiently solvable problems. Using Lagrangian
duality, equivalence of the relaxations is proved in a
unified way. The approach is extended to the case where
equality constraints are present. It is shown how this
technique can be applied to the Quadratic Assignment
Problem, the Graph Partition Problem and the Max-Clique
Problem. The relaxation is the best possible
among all quadratic majorants with zero trace.}}
@article{Poly69,
author = {B. T. Polyak},
title = {The conjugate gradient method in extremal problems},
journal = {U.S.S.R. Computational Mathematics and Mathematical Physics},
volume = 9, pages = {94--112}, year = 1969}
@misc{Poly82,
author = {R. Polyak},
title = {Smooth optimization methods for solving nonlinear extremal
and equilibrium problems with constraints},
howpublished = {Presentation at the IXth International Symposium on
Mathematical Programming, Bonn},
month = {August}, year = 1982}
@article{Poly90,
author = {R. Polyak},
title = {Modified barrier functions (theory and methods)},
journal = MP,
volume = 54, number = 2, pages = {177--222}, year = 1992}
@phdthesis{Ponc90,
author = {D. B. Poncele\'{o}n},
title = {Barrier methods for large-scale quadratic programming},
school = STANFORD, address = STANFORD-ADDRESS,
year = 1990}
@inproceedings{PornFichMullZapf90,
author = {F. Pornbacher and U. Fichter and G. Muller-Liebler
and H. Zapf},
title = {A new method for an efficient optimization of MOS
transistor models},
booktitle = {1990 IEEE International Symposium on Circuits
and Systems},
publisher = {IEEE}, address = {New York, USA},
volume = 1, pages = {81--84}, year = 1990,
abstract = {Two methods that are especially useful for an
accurate optimization of complex transistor models
are presented. The first method focuses on sample
reduction before the optimization process. An
algorithm is described which allows a reduction of
the number of samples by a factor of 10 to 20 in an
efficient way. The second method is a
trust-region-type optimization algorithm which is
especially designed for this application. A
substantial part of it is a new algorithm for the
calculation of the step length. Industrial examples
demonstrating the quality of the algorithms are
given.},
summary = {Two methods are described for accurate
optimization of complex transistor models.
The first focuses on sample reduction before the
optimization process. An algorithm is described which
allows a reduction of the number of samples by a factor
of 10 to 20. The second method is a trust-region-type
optimization algorithm which is especially designed for
this application. Industrial examples
demonstrate the quality of the algorithms.}}
@inproceedings{Powe69,
author = {M. J. D. Powell},
title = {A method for nonlinear constraints in minimization problems},
crossref = {Flet69}, pages = {283--298}}
@inproceedings{Powe70a,
author = {M. J. D. Powell},
title = {A New Algorithm for Unconstrained Optimization},
crossref = {RoseMangRitt70}, pages = {31--65},
abstract = {A new algorithm is described for calculating the least
value of a given differentiable function of several
variables. The user must program the evaluation of the
function and its first derivatives. Some convergence
theorems are given that impose very mild conditions on the
objective function. These theorems, together with some
numerical results, indicate that the new method may be
preferable to current algorithms for solving many
unconstrained minimization problems.},
summary = {A trust-region algorithm is described for unconstrained
smooth minimization. Convergence theorems are given that
impose very mild conditions on the objective function.
These theorems, together with some numerical results,
indicate that the method may be preferable to then current
algorithms for solving unconstrained minimization problems.}}
@techreport{Powe70b,
author = {M. J. D. Powell},
title = {A {F}ortran Subroutine for Unconstrained Minimization
Requiring First Derivatives of The Objective Function},
institution = HARWELL, address = HARWELL-ADDRESS,
number = {R-6469}, year = 1970,
summary = {Details of the implementation of the algorithm described in
\citebb{Powe70a} are provided and numerical experiments are
discussed. The Fortran code is given in appendix.}}
@inproceedings{Powe70c,
author = {M. J. D. Powell},
title = {A hybrid method for nonlinear equations},
crossref = {Rabi70}, pages = {87--114},
summary = {An algorithm for solving systems of nonlinear equations is
described that does not require the evaluation of the
Jacobian of the system. Instead, the derivatives are
approximated by Broyden's quasi-Newton formula. The
algorithm uses a Levenberg-Morrison-Marquardt procedure for
computing a new iterate, together with a safeguarding
technique that prevents any tendencies towards linear
independence of steps. Convergence of the algorithm is
proved to either a solution of the system or a local
minimum of the norm of its residual. }}
@inproceedings{Powe70d,
author = {M. J. D. Powell},
title = {A {F}ortran subroutine for solving systems of nonlinear
algebraic equations},
crossref = {Rabi70}, pages = {115--161},
summary = {Details of the implementation of the algorithm described in
\citebb{Powe70c} are provided and numerical experiments are
discussed. The Fortran code is given in appendix.}}
@inproceedings{Powe75,
author = {M. J. D. Powell},
title = {Convergence Properties of a Class of Minimization
Algorithms},
crossref = {MangMeyeRobi75}, pages = {1--27}}
abstract = {Many iterative algorithms for minimizing a function
$F(x)=F(x_1,x_2\ldots,x_n)$ require first derivatives of
$F(x)$ to be calculated, but they maintain an approximation
to the second derivative matrix automatically. In order
that the approximation is useful, the change in $x$ made by
each iteration is subject to a bound that is also revised
automatically. Some convergence theorems for a class of
minimization algorithms of this type are presented, which
apply to methods proposed by \citebb{Powe70a} and
\citebb{Flet72}. This theory has the following three
valuable features which are rather uncommon. There is no
need for the starting vector $x_1$ to be close to the
solution. The function $F(x)$ need not be convex.
Superlinear convergence is proved even though the second
derivative approximations may not converge to the true
second derivatives at the solution.},
summary = {Some convergence theorems are presented for a class of
minimization algorithms including methods proposed by
\citebb{Powe70a} and \citebb{Flet72}.
There is no need for the starting vector $x_1$ to
be close to the solution. The function $f$ need not be
convex. Superlinear convergence is proved even though the
second derivative approximations may not converge to the
true second derivatives at the solution.}}
@inproceedings{Powe78,
author = {M. J. D. Powell},
title = {A Fast Algorithm For Nonlinearly Constrained Optimization
Calculations},
crossref = {Wats78}, pages = {144--157}}
@inproceedings{Powe81,
author = {M. J. D. Powell},
title = {An upper triangular matrix method for quadratic programming},
editor = {O. L. Mangasarian and R. R. Meyer and S. M. Robinson},
booktitle = {Nonlinear Programming, 2},
publisher = AP, address = AP-ADDRESS,
year = 1981}
@book{Powe81b,
author = {M. J. D. Powell},
title = {Approximation Theory and Methods},
publisher = CUP, address = CUP-ADDRESS,
year = 1981}
@techreport{Powe83,
author = {M. J. D. Powell},
title = {General algorithms for discrete nonlinear approximation
calculations},
institution = DAMTP, address = DAMTP-ADDRESS,
number = {DAMTP/NA2}, year = 1983}
@article{Powe84,
author = {M. J. D. Powell},
title = {On the global convergence of trust region algorithms for
unconstrained minimization},
journal = MP,
volume = 29, number = 3, pages = {297--303}, year = 1984,
abstract = {Many trust region algorithms for unconstrained optimization
have excellent global convergence properties if their
second derivative approximations are not too large
(\bciteb{Powe75}). We consider how large these
approximations have to be, if they prevent convergence when
the objective function is bounded below and continuously
differentiable. Thus we obtain a useful convergence result
in the case when there is a bound on the second derivative
approximations that depends linearly on the iteration
number.},
summary = {Global convergence for trust-region methods in
unconstrained optimization is obtained in the case when
there is a bound on the Hessian approximations that depends
linearly on the iteration number.}}
@article{Powe85,
author = {M. J. D. Powell},
title = {On the quadratic-programming algorithm of
{G}oldfarb and {I}dnani},
journal = MPS,
volume = 25, pages = {46--61}, year = 1985}
@inproceedings{Powe87,
author = {M. J. D. Powell},
title = {Methods for Nonlinear Constraints in Optimization
Calculations},
crossref = {IserPowe87}, pages = {325--358}}
@inproceedings{Powe93,
author = {M. J. D. Powell},
title = {Log barrier methods for semi-infinite programming
calculations},
booktitle = {Advances on Computer Mathematics and its Applications},
editor = {E. A. Lipitakis},
publisher = WSP, address = WSP-ADDRESS,
pages = {1--21}, year = 1993}
@inproceedings{Powe94a,
author = {M. J. D. Powell},
title = {A direct search optimization method that models the
objective and constraint functions by linear interpolation},
crossref = {GomeHenn94}, pages = {51--67}}
@misc{Powe94b,
author = {M. J. D. Powell},
title = {A direct search optimization method that models the
objective by quadratic interpolation},
howpublished = {Presentation at the 5th Stockholm Optimization Days,
Stockholm},
year = 1994}
@misc{Powe96,
author = {M. J. D. Powell},
title = {Trust region methods that employ quadratic interpolation to
the objective function},
howpublished = {Presentation at the 5th SIAM Conference on Optimization,
Victoria},
year = 1996}
@inproceedings{Powe98,
author = {M. J. D. Powell},
title = {The use of band matrices for second derivative
approximations in trust region algorithms},
crossref = {Yuan98}, pages = {3--28},
abstract = {In many trust region algorithms for optimization
calculations, each iteration seeks a vector $d \in \Re^n$
that solves the linear system of equations $(B+\lambda I)
d=- g$, where $B$ is a symmetric estimate of a second
derivative matrix, $I$ is the unit matrix, $g$ is a known
gradient vector, and $\lambda$ is a parameter that controls
the length of $d$. Several values of $\lambda$ may be tried
on each iteration, and, when there is no helpful sparsity
in $B$, it is usual for each solution to require $O(n^3)$
operations. However, if an orthogonal matrix $\Omega$ is
available such that $M=\Omega^T B \Omega$ is an $n \times
n$ matrix of bandwidth $2s+1$, then $\Omega^Td$ can be
calculated in only $O(ns^2)$ operations for each new
$\lambda$ by writing the system in the form $(M + \lambda
I)(\Omega^Td=- \Omega^Tg$. We find unfortunately, that the
construction of $M$ and $\Omega$ from $B$ is usually more
expensive than the solution of the original system, but in
variable metric and quasi-Newton algorithms for
unconstrained optimization, each iteration changes $B$ by a
matrix whose rank is at most two, and then updating
techniques can be applied to $\Omega$. Thus it is possible
to reduce the average work per iteration from $O(n^3)$ to
$O(n^{7/3})$ operations. Here the elements of each
orthogonal matrix are calculated explicitly, but instead
one can express the orthogonal matrix updates as products
of Givens rotations, which allows the average work per
iteration to be only $O(n^{11/5})$ operations. Details of
procedures that achieve these savings are described, and
the $O(n^{7/3})$ complexity is confirmed by numerical
results.},
summary = {In many trust-region algorithms, each iteration seeks a
vector $d \in \Re^n$ that solves the linear system of
equations $(B + \lambda I) d=- g$, where $B$ is a symmetric
estimate of a second derivative matrix, $g$ is a known
gradient vector, and $\lambda$ is a parameter that controls
the length of $d$. Several values of $\lambda$ may be tried
on each iteration, and, when there is no helpful sparsity
in $B$, it is usual for each solution to require $O(n^3)$
operations. However, if an orthogonal matrix $\Omega$ is
available such that $M=\Omega^T B \Omega$ is an $n \times
n$ matrix of bandwidth $2s+1$, then $\Omega^Td$ can be
calculated in only $O(ns^2)$ operations for each new
$\lambda$ by writing the system in the form $(M + \lambda
I)(\Omega^Td=- \Omega^Tg$. Unfortunately, it is found that
the construction of $M$ and $\Omega$ from $B$ is usually
more expensive than the solution of the original system,
but in variable metric and quasi-Newton algorithms for
unconstrained optimization, each iteration changes $B$ by a
matrix whose rank is at most two, and then updating
techniques can be applied to $\Omega$. Thus it is possible
to reduce the average work per iteration from $O(n^3)$ to
$O(n^{7/3})$ operations. Here the elements of each
orthogonal matrix are calculated explicitly, but instead
one can express the orthogonal matrix updates as products
of Givens rotations, which allows the average work per
iteration to be only $O(n^{11/5})$ operations. Details of
procedures that achieve these savings are described, and
the $O(n^{7/3})$ complexity is confirmed by numerical
results.}}
@article{Powe98b,
author = {M. J. D. Powell},
title = {Direct search algorithms for optimization calculations},
journal = {Acta Numerica},
volume = 7, pages = {287--336}, year = 1998}
@misc{Powe98c,
author = {M. J. D. Powell},
title = {A quadratic model trust region method for unconstrained
minimization without derivatives},
howpublished = {Presentation at the International Conference on Nonlinear
Programming and Variational Inequalities, Hong Kong},
year = 1998,
abstract = {A trust region method for unconstrained minimization
calculates a trial change in the variables by minimizing
an approximation to the objective function subject to a
bound on the length of the trial step. The author has
developed an easy-to-use algorithm of this kind by letting
each approximation be a linear polynomial, but the rate of
convergence is usually very slow because second derivatives
are ignored. Therefore we will consider the use of
quadratic approximations that are quadratic polynomials.
They are constructed by interpolation to values of the
objective function, so one has to ensure that the positions
of the interpolation points are suitable. recent work on
this important question will be reviewed. Attention will
be given to the adjustment of the trust region radius,
including the idea of having a separate radius for
controlling the distance between interpolation points.},
summary = {A derivative-free trust-region method for unconstrained
optimization is described that uses quadratic
interpolation models. These are chosen to be linear
combination of the Lagrange fundamental polynomials
associated with the interpolation problem. It is shown
that the coefficients of these polynomials can be updated
from iteration to iteration in a numerically stable way.
The method also uses a separate radius for controlling the
distance between interpolation points.}}
@article{PoweToin79,
author = {M. J. D. Powell and Ph. L. Toint},
title = {On The Estimation of Sparse {H}essian Matrices},
journal = SINUM,
volume = 16, number = 6, pages = {1060--1074}, year = 1979}
@article{PoweYuan86,
author = {M. J. D. Powell and Y. Yuan},
title = {A recursive quadratic-programming algorithm that uses
differentiable exact penalty-functions},
journal = MP,
volume = 35, number = 3, pages = {265--278}, year = 1986}
@article{PoweYuan90,
author = {M. J. D. Powell and Y. Yuan},
title = {A trust region algorithm for equality constrained
optimization},
journal = MP,
volume = 49, number = 2, pages = {189--213}, year = 1990,
abstract = {A trust region algorithm for equality constrained
optimization is proposed that employs a differentiable
exact penalty function. Under certain conditions global
convergence and local superlinear convergence results are
proved.},
summary = {A trust-region algorithm for equality constrained
optimization is proposed that employs a differentiable
exact penalty function. Under certain conditions global
convergence and local superlinear convergence results are
proved.}}
@misc{PrieMogu99,
author = {F. Prieto and J. M. Moguerza},
title = {Interior-Point Methods using Negative Curvature},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999,
abstract = {Interior-point methods are very promising for the solution
of large-scale nonlinear problems. In the nonconvex case,
the use of negative curvature information may allow
additional improvements in the behaviour of these methods
without any significant increase in their computational
costs. However, the efficient implementation of such a
method requires addressing several issues; for example,
how to guarantee the convergence of the method, and how
to compute and use the negative-curvature information
without incussring in excessive costs. We describe the
implementation of two variants of an interior-point method
using negative curvature. Particular attention is paid to
the conditions under which the negative curvature
information is used, the way in which the descent and
negative curvature directions are combined, and to the
updating of the barrier parameter. The convergence
properties of the method are analyzed, and the conditions
under which the methods are globally convergent and may
attain quadratic convergence are discussed. Finally,
computational results on a set of small and medium-sized
problems are presented and compared with those obtained
using other interior-point implementations, and with the
results obtained by the proposed method when negative
curvature information is not used.},
summary = {The implementation of two variants of an interior-point
method using negative curvature is discussed. Particular
attention is paid to the conditions under which the negative
curvature information is used, the way in which the descent
and negative curvature directions are combined, and to the
updating of the barrier parameter. The conditions under
which the methods are globally convergent and may attain
quadratic convergence are discussed. Computational results
on a set of small and medium-sized problems are compared
with those obtained using other interior-point
implementations, and with those obtained by the proposed
method when negative curvature information is not used.}}
@article{PropPukh93,
author = {A. I. Propoi and A. V. Pukhlikov},
title = {{N}ewton Stochastic Method in nonlinear extremal problems},
journal = {Automation and Remote Control},
volume = 54, number = {4--Part 1}, pages = {605--613}, year = 1993,
abstract = {This paper is concerned with the construction of
consistent model-trust region pairs for optimization
procedures of the Newtonian type. A problem of random
extremal search in the class of normal distributions
is studied. Two "training" and "improvement"
components of optimization motion are singled out and
analyzed.},
summary = {A random search procedure are explored in the context
of trust-region methods.}}
@article{Psch70,
author = {B. N. Pschenichny},
title = {Algorithms for general problems of mathematical programming},
journal = {Kibernetica},
volume = 6, pages = {120--125}, year = 1970}
@article{PsiaPark95,
author = {M. Psiaki and K. Park},
title = {Augmented {L}agrangian Nonlinear-Programming algorithm that
uses {SQP} and trust region techniques},
journal = JOTA,
volume = 86, number = 2, pages = {311--325}, year = 1995,
abstract = {An augmented Lagrangian nonlinear programming algorithm has
been developed. Its goals are to achieve robust global
convergence and fast local convergence. Several unique
strategies help the algorithm achieve these dual goals. The
algorithm consists of three nested loops. The outer loop
estimates the Kuhn-Tucker multipliers at a rapid linear
rate of convergence. The middle loop minimizes the
augmented Lagrangian function for fixed multipliers. This
loop uses the sequential quadratic programming technique
with a box trust region stepsize restriction. The inner
loop solves a single quadratic program. Slack variables and
a constrained form of the fixed-multiplier middle-loop
problem work together with curved line searches in the
inner-loop problem to allow large penalty weights for rapid
outer-loop convergence. The inner-loop quadratic programs
include quadratic constraint terms, which complicate the
inner loop, but speed the middle-loop progress when the
constraint curvature is large. The new algorithm compares
favorably with a commercial sequential quadratic
programming algorithm on five low-order test problems. Its
convergence is more robust, and its speed is not much
slower.},
summary = {An augmented Lagrangian nonlinear programming algorithm is
developed. The algorithm consists of three nested loops.
The outer loop estimates the Kuhn-Tucker multipliers at a
rapid linear rate of convergence. The middle loop minimizes
the augmented Lagrangian function for fixed multipliers.
This loop uses the sequential quadratic programming
technique with a box trust-region stepsize restriction. The
inner loop solves a single quadratic program. Slack
variables and a constrained form of the fixed-multiplier
middle-loop problem work together with curved linesearches
in the inner-loop problem to allow large penalty weights
for rapid outer-loop convergence. The inner-loop quadratic
programs include quadratic constraint terms, which
complicate the inner loop, but speed the middle-loop
progress when the constraint curvature is large.}}
%%% Q %%%
@article{QiSun94,
author = {L. Qi and J. Sun},
title = {A trust region algorithm for minimization of locally
{L}ipschitzian functions},
journal = MP,
volume = 66, number = 1, pages = {25--43}, year = 1994,
abstract = {The classical trust region algorithm for smooth nonlinear
programs is extended to the nonsmooth case where the
objective function is only locally Lipschitzian. At each
iteration, an objective function that carries both first
and second order information is minimized over a trust
region. The term that carries the first order information
is an iteration function that may not explicitly depend on
subgradients or directional derivatives. We prove that the
algorithm is globally convergent. This convergence result
extends the result of \citebb{Powe84} for minimization of
smooth functions, the result of \citebb{Yuan85b} for
minimization of composite convex functions, and the result
of \citebb{DennLiTapi95} for minimization of regular
functions. In addition, compared with the recent model of
\citebb{PangHanRang91} for minimization of locally
Lipschitzian functions using a linesearch, this algorithm
has the same convergence property without assuming positive
definiteness and uniform boundedness of the second order
term. Applications of the algorithm to various nonsmooth
optimization problems are discussed.},
summary = {The classical trust-region algorithm for smooth nonlinear
programs is extended to the non-smooth case where the
objective function is only locally Lipschitzian. At each
iteration, an objective function that uses both first and
second order information is minimized over a trust region.
The term that carries the first order information is an
iteration function that may not explicitly depend on
subgradients or directional derivatives. It is proved that
the algorithm is globally convergent. Applications of the
algorithm to various non-smooth optimization problems are
discussed.}}
@article{Qi95,
author = {L. Qi},
title = {Trust Region Algorithms for Solving Nonsmooth Equations},
journal = SIOPT,
volume = 5, number = 1, pages = {219--230}, year = 1995,
abstract = {Two globally convergent trust region algorithms are
presented for solving nonsmooth equations, where the
functions are only locally Lipschitzian. The first
algorithm is an extension of the classic
Levenberg-Marquardt method by approximating the locally
Lipschitzian function with a smooth function and using the
derivative of the smooth function in the algorithm wherever
a derivative is needed. Global convergence for this
algorithm is established under a regular condition. In the
second algorithm, successive smooth approximation functions
and their derivatives are used. Global convergence for the
second algorithm is established under mild assumptions.
Both objective functions of subproblems of these two
algorithms are quadratic functions.},
summary = {Two globally convergent trust-region algorithms are
presented for solving non-smooth equations, for the case
where the functions are only locally Lipschitzian. The
first algorithm is an extension of the classic
Levenberg-Morrison-Marquardt method, obtained by
approximating the locally Lipschitzian function with a
smooth function and using the derivative of the smooth
function in the algorithm wherever a derivative is needed.
Global convergence is established under a regularity
condition. In the second algorithm, successive smooth
approximation functions and their derivatives are used.
Global convergence for the second algorithm is established
under mild assumptions. Both objective functions of
subproblems of these two algorithms are quadratic
functions.}}
@techreport{QiQiSun99,
author = {H. Qi and L. Qi and D. Sun},
title = {Solving {KKT} Systems via the Trust Region and the
Conjugate Gradient Methods},
institution = {School of Mathematics, The university of New South Wales},
address = {Sydney, Australia},
number = {15-09-99}, year = 1999,
abstract = {In this paper, we propose a trust region method for
solving KKT systems arising from the variational inequality
problem and the constrained optimization problem. The
trust region subproblem is derived from the reformulation
of the KKT system as a constrained optimization problem and
is solved by the truncated conjugate gradient method;
meanwhile the variables remain feasible with respect to
the constrained optimization problem. Global and
superlinear convergence are established. Some preliminary
numerical experiments show that the method is quite
promising.},
summary = {A trust-region method is proposed for solving KKT systems
arising from the variational inequality and the constrained
optimization problems. The trust-region subproblem is
derived from the reformulation of the KKT system as a
constrained optimization problem and is solved by the
truncated conjugate gradient method while maintaining
feasibility. Global and superlinear convergence is
established. Preliminary numerical experiments illustrate
the method.}}
%%% R %%%
@inproceedings{Reid71,
author = {J. K. Reid},
title = {On the method of conjugate gradients for the solution of
large sparse linear equations},
booktitle = {Large sparse sets of linear equations},
editor = {J. K. Reid},
publisher = AP, address = AP-ADDRESS,
pages = {231--254}, year = 1971}
@inproceedings{Reid73,
author = {J. K. Reid},
title = {Least squares solution of sparse systems of non-linear
equations by a modified {M}arquardt algorithm},
booktitle = {Decomposition of Large-Scale Problems},
editor = {D. M. Himmelblau},
publisher = NH, address = NH-ADDRESS,
pages = {437--445}, year = 1973,
abstract = {Marquardt's algorithm is adapted to exploit sparsity in
non-linear least squares problem that may or may not be
overdetermined. Consideration is given to cases when
derivatives are available analytically and where they have
to be estimated. The results of numerical experiments are
presented.},
summary = {The Levenberg-Morrison-Marquardt algorithm is compared to
the dogleg method for sparse least-squares problems.
Approximation scheme for the Jacobian are also considered.}}
@article{Rein71,
author = {C. Reinsch},
title = {Smoothing by spline functions {II}},
journal = NUMMATH,
volume = 16, pages = {451--454}, year = 1971}
@article{RendWolk97,
author = {F. Rendl and H. Wolkowicz},
title = {A Semidefinite Framework for Trust Region Subproblems with
Applications to Large Scale Minimization},
journal = MP,
volume = 77, number = 2, pages = {273--299}, year = 1997,
abstract = {A primal-dual pair of semidefinite programs provides a
general framework for the theory and algorithms for the
trust region subproblem (TRS). This problem consists in
minimizing a general quadratic function subject to a convex
quadratic constraint and, therefore, it is a generalization
of the minimum eigenvalue problem. The importance of TRS is
due to the fact that it provides the step in trust region
minimization algorithms. The semidefinite framework is
studied as an interesting instance of semidefinite
programming as well as a tool for viewing known algorithms
and deriving new algorithms for TRS. In particular, a dual
simplex type method is studied that solves TRS as a
parametric eigenvalue problem. This method uses the Lanczos
algorithm for the smallest eigenvalue as a black box.
Therefore, the essential cost of the algorithm is the
matrix-vector multiplication and, thus, sparsity can be
exploited. A primal simplex type method provides steps for
the so-called hard case. Extensive numerical tests for
large sparse problems are discussed. These tests show that
the cost of the algorithm is $1+\alpha(n)$ times the cost
of finding a minimum eigenvalue using the Lanczos
algorithm, where $0<\alpha(n)<1$ is a fraction which
decreases as the dimension increases.},
summary = {A primal-dual pair of semidefinite programs provides a
general framework for the theory and algorithms for the
trust region subproblem (TRS). This problem is a
generalization of the minimum eigenvalue problem. The
semidefinite framework is studied as an instance of
semidefinite programming as well as a tool for viewing
known algorithms and deriving new algorithms for TRS. In
particular, a dual simplex type method is studied that
solves TRS as a parametric eigenvalue problem. This method
uses the Lanczos algorithm for the smallest eigenvalue as a
black box. Therefore, the essential cost of the algorithm
is the matrix-vector multiplication and, thus, sparsity can
be exploited. A primal simplex type method provides steps
for the so-called hard case. Numerical tests for large
sparse problems show that the cost of the algorithm is
$1+\alpha(n)$ times the cost of finding a minimum
eigenvalue using the Lanczos algorithm, where
$0<\alpha(n)<1$ is a fraction which decreases as the
dimension increases.}}
@article{RendVandWolk95,
author = {F. Rendl and R. J. Vanderbei and H. Wolkowicz},
title = {Max-min eigenvalue problems, primal-dual interior point
algorithms, and trust region subproblems},
journal = OMS,
volume = 5, number = 1, pages = {1--16}, year = 1995,
abstract = {Two Primal-dual interior point algorithms are presented for
the problem of maximizing the smallest eigenvalue of a
symmetric matrix over diagonal perturbations. These
algorithms prove to be simple, robust, and efficient. Both
algorithms are based on transforming the problem to one
with constraints over the cone of positive semidefinite
matrices, i.e. L\"{o}wner order constraints. One of the
algorithms does this transformation through an intermediate
transformation to a trust region subproblem. This allows
the removal of a dense row.},
summary = {Two primal-dual interior-point algorithms are presented for
maximizing the smallest eigenvalue of a symmetric matrix
over diagonal perturbations. These algorithms prove to be
simple, robust, and efficient. Both algorithms are based on
transforming the problem to one with constraints over the
cone of positive semidefinite matrices. One of the
algorithms does this transformation through an intermediate
transformation to a trust-region subproblem. This allows
the removal of a dense row.}}
@article{Robi74,
author = {S. M. Robinson},
title = {Perturbed {K}uhn-{T}ucker points and rates of convergence
for a class of nonlinear programming algorithms},
journal = MP,
volume = 7, number = 1, pages = {1--16}, year = 1974}
@inproceedings{Robi83,
author = {S. M. Robinson},
title = {Generalized equations},
crossref = {BachGrotKort83}, pages = {346--367}}
@book{Rock70,
author = {R. T. Rockafellar},
title = {Convex Analysis},
publisher = {Princeton University Press}, address = {Princeton, USA},
year = 1970}
@article{Rock74,
author = {R. T. Rockafellar},
title = {Augmented {L}agrangian multiplier functions and duality in
nonconvex programming},
journal = SICON,
volume = 12, number = 2, pages = {268--285}, year = 1974}
@article{Rock76,
author = {R. T. Rockafellar},
title = {Augmented {L}agrangians and applications of the proximal
point algorithm in convex programming},
journal = MOR,
volume = 1, pages = {97--116}, year = 1976}
@article{Rock76b,
author = {R. T. Rockafellar},
title = {Monotone Operators and the Proximal Point Algorithm},
journal = SICON,
volume = 14, pages = {877--898}, year = 1976}
@inproceedings{Rock83,
author = {R. T. Rockafellar},
title = {Generalized subgradients in mathematical programming},
crossref = {BachGrotKort83}, pages = {368--390}}
@article{RodrRenaWats99,
author = {J. F. Rodr\`{\i}guez and J. E. Renaud and L. T. Watson},
title = {Convergence of Trust Region Augmented {L}agrangian Methods
Using Variable Fidelity Approximation Data},
journal = {Structural Optimization},
volume = 15, number = {3--4}, pages = {141--156}, year = 1999,
abstract = {To date the primary focus of most constrained approximate
optimization strategies is that application of the method
should lead to improved designs. Few researchers have
focused on the development of constrained approximate
optimization strategies that are assured of converging to a
Karush-Kuhn-Tucker (KKT) point for the problem. Recent work
by the authors based on a trust region model management
strategy has shown promise in managing the convergence of
constrained approximate optimization in application to a
suite of single level optimization test problems. Using a
trust-region model management strategy, coupled with an
augmented Lagrangian approach for constrained approximate
optimization, the authors have shown in application studies
that the approximate optimization process converges to a
KKT point for the problem. The approximate optimization
strategy sequentially builds a cumulative response surface
approximation of the augmented Lagrangian which is then
optimized subject to a trust region constraint. In this
research the authors develop a formal proof of convergence
for the response surface approximation based optimization
algorithm. Previous application studies were conducted on
single level optimization problems for which response
surface approximations were developed using conventional
statistical response sampling techniques such as central
composite design to query a high fidelity model over the
design space. In this research the authors extend the scope
of application studies to include the class of
multidisciplinary design optimization (MDO) test problems.
More importantly the authors show that response surface
approximations constructed from variable fidelity data
generated during concurrent subspace optimizations (CSSOs)
can be effectively managed by the trust region model
management strategy.},
summary = {An augmented Lagrangian trust-region method is given, that
converges to a Karush-Kuhn-Tucker point for constrained
optimization. The method behaves well on single level
optimization test problems. Applications include
multidisciplinary design optimization test problems. It is
shown that response surface approximations constructed from
variable fidelity data generated during concurrent subspace
optimizations can be effectively managed by the
trust-region model management strategy.}}
@article{RodrRenaWats98b,
author = {J. F. Rodr\`{\i}guez and J. E. Renaud and L. T. Watson},
title = {Trust-region augmented {L}agrangian methods for sequential
response surface approximation and optimization},
journal = {Journal of Mechanical Design},
volume = 120, number = 1, pages = {58--66}, year = 1998,
abstract = {A common engineering practice is the use of approximation
models in place of expensive computer simulations to drive
a multidisciplinary design process based on nonlinear
programming techniques. The use of approximation strategies
is designed to reduce the number of detailed, costly
computer simulations required during optimization while
maintaining the pertinent features of the design problem.
To date the primary focus of most approximate optimization
strategies is that application of the method should lead to
improved designs. This is a laudable attribute and
certainly relevant for practicing designers. However to
date few researchers have focused on the development of
approximate optimization strategies that are assured of
converging to a solution of the original problem. Recent
works based on trust region model management strategies
have shown promise in managing convergence in unconstrained
approximate minimization. In this research we extend these
well established notions from the literature on
trust-region methods to manage the convergence of the more
general approximate optimization problem where equality,
inequality and variable bound constraints are present. The
primary concern addressed in this study is how to manage
the interaction between the optimization and the fidelity
of the approximation models to ensure that the process
converges to a solution of the original constrained design
problem. Using a trust-region model management strategy
coupled with an augmented Lagrangian approach for
constrained approximate optimization, one can show that the
optimization process converges to a solution of the
original problem. In this research an approximate
optimization strategy is developed in which a cumulative
response surface approximation of the augmented Lagrangian
is sequentially optimized subject to a trust region
constraint. Results for several test problems are presented
in which convergence to a Karush-Kuhn-Tucker (KKT) point is
observed.},
summary = {A common engineering practice is the use of approximation
models in place of expensive computer simulations to drive
a multidisciplinary design process based on nonlinear
programming techniques. Well established notions on
trust-region methods are extended to manage the convergence
of the general approximate problem where equality,
inequality and variable bound constraints are present. The
primary concern is to manage the interaction between the
optimization and the fidelity of the approximation models
to ensure that the process converges to a solution of the
original constrained design problem. This is achieved by
using a trust-region model management strategy coupled with
an augmented Lagrangian approach for constrained
approximate optimization. An approximate optimization
strategy is developed in which a cumulative response
surface approximation of the augmented Lagrangian is
sequentially optimized subject to a trust region
constraint. Results for several test problems are presented
in which convergence to a Karush-Kuhn-Tucker (KKT) point is
observed.}}
@inproceedings{RogeTerpGoss88,
author = {A. Roger and P. Terpolilli and O. Gosselin},
title = {Trust region methods for seismic inverse problems},
booktitle = {Proceedings of the XVIth Workshop on Interdisciplinary
Study of Inverse Problems: Some Topics on Inverse Problems},
editor = {P. C. Sabatier},
publisher = {World Scientific}, address = {Singapore},
pages = {93--103}, year = 1988,
abstract = {The authors present a class of optimization
algorithms, called trust region algorithms which
exhibit attractive theoretical convergence
properties. They outline the main features of trust
region algorithms, give a few classical properties
without demonstration, and present numerical results
which show that this class of algorithm actually fits
the required features.},
summary = {Trust-region methods are presented in the context of
seismic inversion problems.}}
@phdthesis{Roja98,
author = {M. Rojas},
title = {A large-scale trust-region approach to the regularization
of discrete ill-posed problems},
school = {Rice University}, address = RICE-ADDRESS,
year = 1998,
abstract = {We consider the problem of computing the solution of
large-scale discrete ill-posed problems when there is noise
in the data. These problems arise in important areas such
as seismic inversion, medical imaging and signal
processing. We pose the problem as a quadratically
constrained least squares problem and develop a method for
the solution of such problem. Our method does not require
factorization of the coefficient matrix, it has very low
storage requirements and handles the high degree of
singularities arising in discrete ill-posed problems. We
present numerical results on test problems and an
application of the method to a practical problem with real
data.},
summary = {A algorithm based on the proposals by \citebb{Sore97} and
\citebb{SantSore95} is described for the solution of
large-scale linear least-squares problems subject to a
quadratic constraint. It is applied to regularize
large discrete ill-posed problems. Numerical experiments
on test cases and a real inverse interpolation problem
illustrate the method.}}
@techreport{RojaSantSore99,
author = {M. Rojas and S. A. Santos and D. C. Sorensen},
title = {A new matrix-free algorithm for the large-scale
trust-region subproblem},
institution = CAAM, address = RICE-ADDRESS,
number = {TR99-19}, year = 1999,
abstract = {We present a matrix-free algorithm for the large-scale
trust-region subproblem. Our algorithm relies on
matrix-vector products only and does not require
matrix factorizations. We recast the trust-region
subproblem as a parametrized eigenvalue problem and
compute an optimal value for the parameter. We then
find the optimal solution of the trust-region
subproblem from the eigenvectors associated with two
of the smallest eigenvalues of the parameterized
eigenvalue problem corresponding to the optimal
parameter. The new algorithm uses a different
interpolating scheme than existent methods and
introduces a unified iteration that naturally
includes the so-called hard case. We show that the
new iteration is well defined and convergent at a
superlinear rate. We present computational results to
illustrate convergence properties and robustness of
the method.},
summary = {A matrix-free algorithm for the large-scale
trust-region subproblem is presented, that relies on
matrix-vector products only and does not require
matrix factorizations. the trust-region subproblem
is recast as a parametrized eigenvalue problem and
an optimal value for the parameter computed.
The optimal solution of the trust-region subproblem
is then found from the eigenvectors associated with two
of the smallest eigenvalues of the parameterized
eigenvalue problem corresponding to the optimal
parameter. The new algorithm uses a unified iteration
that naturally includes the hard case, and is superlinearly
convergent. Computational results illustrate convergence
properties and robustness of the method.}}
@article{Rose60,
author = {H. H. Rosenbrock},
title = {An automatic method for finding the greatest or least value
of a function},
journal = COMPJ,
volume = 3, pages = {175--184}, year = 1960}
@article{RoyServ99,
author = {R. Roy and Sevick Muraca, E. M.},
title = {Truncated {N}ewton's optimization scheme for absorption
and fluorescence optical tomography: {P}art {I}
theory and formulation },
journal = {Optics Express},
volume = 4, number = 10, pages = {353--371}, year = 1999,
abstract = {The development of non-invasive, biomedical optical
imaging from time-dependent measurements of
near-infrared (NIR) light propagation in tissues
depends upon two crucial advances: (i) the
instrumental tools to enable photon
''time-of-flight'' measurement within rapid and
clinically realistic times, and (ii) the
computational tools enabling the reconstruction of
interior tissue optical property maps from exterior
measurements of photon ''time-of-flight'' or photon
migration. In this contribution, the image
reconstruction algorithm is formulated as an
optimization problem in which an interior map of
tissue optical properties of absorption and
fluorescence lifetime is reconstructed from
synthetically generated exterior measurements of
frequency-domain photon migration (FDPM). The inverse
solution is accomplished using a truncated Newton's
method with trust region to match synthetic
fluorescence FDPM measurements with that predicted by
the finite element prediction. The computational
overhead and error associated with computing the
gradient numerically is minimized upon using modified
techniques of reverse automatic differentiation.},
summary = {The development of non-invasive, biomedical optical
imaging from time-dependent measurements of
near-infrared (NIR) light propagation in tissues
depends upon two crucial advances: (i) the
instrumental tools to enable photon
''time-of-flight'' measurement within rapid and
clinically realistic times, and (ii) the
computational tools enabling the reconstruction of
interior tissue optical property maps from exterior
measurements of photon ''time-of-flight'' or photon
migration. The image reconstruction algorithm is
formulated as an optimization problem in which an
interior map of tissue optical properties of
absorption and fluorescence lifetime is reconstructed
from synthetically generated exterior measurements of
frequency-domain photon migration (FDPM). The inverse
solution is accomplished using a truncated Newton's
method with trust region to match synthetic
fluorescence FDPM measurements with that predicted by
the finite element prediction. The computational
overhead and error associated with computing the
gradient numerically is minimized upon using modified
techniques of reverse automatic differentiation.}}
@article{Rudn94,
author = {M. Rudnicki},
title = {Smoothing strategies in solving inverse electromagnetic
problems},
journal = {International Journal of Applied Electromagnetics in
Materials},
volume = 4, number = 3, pages = {239-264}, year = 1994,
abstract = {Many inverse and optimal design problems of
electrical engineering are formulated in terms of
least squares methodology (linear and nonlinear least
squares). The objective function used makes a
square-root deviation between prescribed and actual
field distribution in a controlled subregion. In the
paper some numerical methods that proved to be
particularly useful in solving such problems of
electrical engineering are presented. The common
feature of these methods is the need of choosing a
smoothing parameter controlling the solution
quality. It is shown how this parameter may be chosen
when using the linear least squares (regularization)
approach and the trust-region approach in the
nonlinear least squares method. For completeness the
author presents the zeroth-order stochastic methods
of optimization as well. As opposed to the least
squares approach, the latter do not require
restrictive a priori assumptions about convexity and
smoothness. Some electromagnetic optimal design
problems solved by means of the above techniques are
referred.},
summary = {It is shown how the regularization parameter may
be chosen when using the linear least-squares
approach and the trust-region approach
in the nonlinear least squares method for solving inverse
and optimal design problems of electrical engineering. The
zeroth-order stochastic methods of optimization is also
discussed, because it does not require
restrictive a priori assumptions about convexity and
smoothness. Some electromagnetic optimal design
problems solved by means of these techniques.}}
%%% S %%%
@book{Saad91,
author = {Y. Saad},
title = {Numerical Methods for Large Eigenvalue Problems},
publisher = {Manchester University Press},
year = 1991}
@book{Saad96,
author = {Y. Saad},
title = {Iterative Methods for Sparse Linear Systems},
publisher = {PWS Publishing Company}, address = {Boston, USA},
year = 1996}
@techreport{SachSart99,
author = {E. W. Sachs and A. Sartenaer},
title = {A Class of Augmented {L}agrangian Algorithms for Infinite
Dimensional Optimization with Equality Constraints},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {(in preparation)}, year = 1999}
@article{SadjPonn99,
author = {S. J. Sadjadi and K. Ponnambalam},
title = {Advances in trust region algorithms for constrained
optimization},
journal = {Applied Numerical Mathematics},
volume = 29, number = 3, pages = {423--443}, year = 1999,
abstract = {Constrained optimization problems occur in many applications
of engineering, science and medicine. Much attention
has recently been devoted to solving this class of
problems using trust region algorithms with strong
convergence properties, in part because of the
availability of reliable software. This paper
presents a survey of recent advances in trust region
algorithms. We then explain the different choices of
penalty function, Lagrange function and expanded
Lagrangian function used for modeling constrained
optimization problems and solving these equations
using trust region algorithms. Finally, some
numerical results for the implementation of our
proposed method on different test problems with
various sizes are presented.},
summary = {A survey of recent advances in trust-region methods
for constrained minimization is given. Different
choices for the penalty function, Lagrange function
and expanded Lagrangian functions which are used are
compared. Some numerical results for an
implementation of a recommended method on different
test problems with various sizes are presented.}}
@article{Sahb87,
author = {M. Sahba},
title = {Globally convergent algorithm for nonlinearly constrained
optimization problems},
journal = JOTA,
volume = 52, number = 2, pages = {291--309}, year = 1987}
@misc{SachSchuFrom98,
author = {E. W. Sachs and M. Schulze and S. Fromme},
title = {Neural Networks---An application of numerical optimization
in the financial markets},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {We present results from a joint research project between a
research group of a bank and an optimization group of a
university. The goal was to develop an efficient code to
train neural networks and to use this software too to
forecast some economic indicators. We show how fully
interactive methods exploit the structure of the neural
networks and give theoretical and numerical results. The
software is applied to forecasting the German stock index
DAX using various input parameters. The success of the
forecast is measured in the development of the value of a
portfolio.},
summary = {The development of an efficient code to train neural
networks and the use of this software to forecast some
economic indicators is discussed. The proposed method
exploits the underdetermined nature of the application and
uses a Steihaug-Toint truncated conjugate gradient
trust-region method. The software is applied to forecasting
the German stock DAX index. The success of the forecast is
measured in the development of the value of a portfolio.}}
@article{SagaFuku91,
author = {N. Sagara and M. Fukushima},
title = {A Hybrid Method for the nonlinear least-squares problem
with simple bounds},
journal = JCAM,
volume = 36, number = 2, pages = {149--157}, year = 1991,
abstract = {This paper presents a new method with trust region
technique for solving the nonlinear least-squares problem
with lower and upper bounds on the variables. The proposed
method constructs trust region constraints that are
ellipses centered at the iterative points in such a way
that they lie in the interior of the feasible region. Thus
the method belongs to the class of interior point methods,
and hence we may expect that the generated sequence
approaches a solution smoothly without the combinatorial
complications inherent to traditional active set methods.
We establish a convergence theorem for the proposed method
and show its practical efficiency by numerical experiments.},
summary = {An interior trust-region method is presented for solving
the nonlinear least-squares problem with lower and upper
bounds on the variables. Convergence is established and the
practical efficiency of the method is illustrated by
numerical experiments.}}
@article{SagaFuku95,
author = {N. Sagara and M. Fukushima},
title = {A Hybrid Method for solving the nonlinear least-squares
problem with linear inequality constraints},
journal = {Journal of the Operations Research Society of Japan},
volume = 38, number = 1, pages = {55--69}, year = 1995,
abstract = {This paper presents a new method with trust region
technique for solving the nonlinear least squares problem
with linear inequality constraints. The method proposed in
this paper stems from the one presented in a recent paper
by the authors. The method successively constructs trust
region constraints, which are ellipsoids centered at the
iterative points, in such a way that they lie in the
relative interior of the feasible region. Thus the method
belongs to the class of interior point methods, and hence
we may expect that the generated sequences approaches a
solution smoothly without the combinatorial complications
inherent to traditional active set methods. We establish a
convergence theorem for the proposed method and show its
practical efficiency by numerical experiments.},
summary = {A trust-region method is presented for solving the
nonlinear least-squares problem with linear inequality
constraints. It is an adaptation of the method by
\citebb{SagaFuku91} to this more general case, and enjoy
similar properties.}}
@article{Sala87,
author = {D. E. Salane},
title = {A continuation approach for solving large-residual
nonlinear least squares problems},
journal = SISSC,
volume = 8, number = 4, pages = {655--671}, year = {1987},
abstract = {This paper is concerned with the solution of the
nonlinear least squares problem. A continuation
method is used to develop a new framework for the
model trust region approach for solving nonlinear
least squares problem. This framework gives a
motivation for the direct selection of the trust
region parameter. It also provides a natural
safeguard for trust region methods and leads to a
very robust algorithm. A class of algorithms based on
the continuation method is presented. In addition,
the implementation details for one member of the new
class are examined. The convergence and descent
properties of this algorithm are also
discussed. Numerical evidence is given showing that
the new algorithms are competitive with existing
model trust region algorithms. For large-residual
problems or problems in which a good initial starting
guess is not available, the performance of the new
algorithm is very promising.},
summary = {A continuation method is used to develop a new framework
for the model trust-region approach for solving nonlinear
least squares problem, which motivates
the direct selection of the trust-region parameter. It
also provides a safeguard for trust-region methods
and leads to a very robust algorithm. A class of algorithms
based on the continuation method is presented. In addition,
the implementation details for one member of the new
class are examined. Convergence and descent
properties of this algorithm are also
discussed. Numerical evidence is given showing that
the new algorithms are competitive with existing
trust-region algorithms. }}
@article{Salz60,
author = {H. E. Salzer},
title = {A note on the solution of quartic equations},
journal = MC,
volume = 14, number = 71, pages = {279--281}, year = 1960}
@techreport{SantSore95,
author = {S. A. Santos and D. C. Sorensen},
title = {A new matrix-free algorithm for the large-scale
trust-region subproblem},
institution = CAAM, address = RICE-ADDRESS,
number = {TR95-20}, year = 1995,
abstract = {The trust-region subproblem arises frequently in linear
algebra and optimization applications. Recently,
matrix-free methods have been introduced to solve
large-scale trust-region subproblems. These methods only
require a matrix-vector product and do not rely on matrix
factorizations (\bciteb{RendWolk97}, \bciteb{Sore97}).
These approaches recast the trust-region subproblem in
terms of a parameterized eigenvalue problem and then adjust
the parameter to find the optimal solution from the
eigenvector corresponding to the smallest eigenvalue of the
parameterized eigenvalue problem. This paper presents a new
matrix-free algorithm for the large-scale trust-region
subproblem. The new algorithm improves upon the previous
algorithms by introducing a unified iteration that
naturally includes the so called hard case. The new
iteration is shown to be superlinearly convergent in all
cases. Computational results are presented to illustrate
convergence properties and robustness of the method.},
summary = {A matrix-free algorithm for the large-scale trust-region
subproblem is presented, which improves upon the previous
algorithms by introducing a unified iteration that
naturally includes the hard case. The iteration is
superlinearly convergent in all cases. Computational
results are presented to illustrate its convergence
properties and robustness.}}
@inproceedings{Sarg74,
author = {R. W. H. Sargent},
title = {Reduced-gradient and projection methods for nonlinear
programming},
crossref = {GillMurr74a}, pages = {149--174}}
@misc{SargZhan98,
author = {R. W. H. Sargent and X. Zhang},
title = {An Interior-point Algorithm for Solving General Variational
Inequalities and Nonlinear Programs},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998}
% abstract = {The paper describes a new algorithm for finding local
% solutions of general variational inequalities or
% nonlinear programs. It applies the interior-point
% approach directly to the necessary conditions for the
% nonlinear problem, which can then be degenerate. It
% is proved that the algorithm always has finite
% termination, and that the rate of convergence is
% globally linear, and under a certain regularity condition
% ultimately Q-subquadratic. The regularity condition
% does not necessarily imply an isolated solution point.
% Numerical results are given for a selection of nonlinear
% programming problems from the CUTE package.},
% abstract = {An algorithm for finding local solutions of general
% variational inequalities or nonlinear programs is described.
% It applies the interior-point approach directly to the
% necessary conditions for the nonlinear problem, which can
% then be degenerate. It is proved that the algorithm always
% has finite termination, and that the rate of convergence
% is globally linear, and under a certain regularity
% condition ultimately Q-subquadratic. The regularity
% condition does not necessarily imply an isolated solution
% point. Numerical results are given for a selection of
% nonlinear programming problems from the {\sf CUTE} package.}}
@phdthesis{Sart91,
author = {A. Sartenaer},
title = {On some strategies for handling constraints in nonlinear
optimization},
school = FUNDP, address = FUNDP-ADDRESS,
year = 1991}
@article{Sart93b,
author = {A. Sartenaer},
title = {Armijo-type condition for the determination of a
generalized {C}auchy point in trust region algorithms using
exact or inexact projections on convex constraints},
journal = {Belgian Journal of Operations Research, Statistics and
Computer Science},
volume = 33, number = 4, pages = {61--75}, year = 1993,
abstract = {This paper considers some aspects of two classes of trust
region methods for solving constrained optimization
problems. The first class proposed by \citebb{Toin88} uses
techniques based on the explicitly calculated projected
gradient, while the second class proposed by
\citebb{ConnGoulSartToin96a} allows for inexact projections
on the constraints. We propose and analyze for each class a
step-size rule in the spirit of the Armijo rule for the
determination of a Generalized Cauchy Point. We then prove
under mild assumptions that, in both cases, the classes
preserve their theoretical properties of global convergence
and identification of the correct active set in a finite
number of iterations. Numerical issues are also discussed
for both classes.},
summary = {An Armijo step-size rule is proposed for the determination
of a Generalized Cauchy Point and analyzed for the
trust-region methods proposed by \citebb{Toin88} and
\citebb{ConnGoulSartToin96a}. It is proved under mild
assumptions that both classes preserve their theoretical
properties of global convergence and identification of the
correct active set in a finite number of iterations.
Numerical issues are also discussed for both classes.}}
@article{Sart95,
author = {A. Sartenaer},
title = {A class of trust region methods for nonlinear network
optimization problems},
journal = SIOPT,
volume = 5, number = 2, pages = {379--407}, year = 1995,
abstract = {We describe the results of a series of tests upon a class
of new methods of trust region type for solving the
nonlinear network optimization problem. The trust region
technique considered is characterized by the use of the
infinity norm and of inexact projections on the network
constraints. The results are encouraging and show that this
approach is particularly useful in solving large-scale
nonlinear network optimization problems, especially when
many bound constraints are expected to be active at the
solution.},
summary = {The results of a series of tests upon a class of
methods of trust-region type for solving the nonlinear
network optimization problem are described. The
trust-region technique considered is characterized by the
use of the infinity norm and of inexact projections on the
network constraints. The results are encouraging and show
that this approach is particularly useful in solving
large-scale nonlinear network optimization problems,
especially when many bound constraints are expected to be
active at the solution.}}
@article{Sart97,
author = {A. Sartenaer},
title = {Automatic determination of an initial trust region in
nonlinear programming},
journal = SISC,
volume = 18, number = 6, pages = {1788--1803}, year = 1997,
abstract = {This paper presents a simple but efficient way to find a
good initial trust region radius in trust region methods
for nonlinear optimization. The method consists of
monitoring the agreement between the model and the
objective function along the steepest descent direction,
computed at the starting point. Further improvements for
the starting point are also derived from the information
gleaned during the initializing phase. Numerical results
on a large set of problems show the impact the initial
trust region radius may have on trust region methods
behaviour and the usefulness of the proposed strategy.},
summary = {A simple but efficient way to find a good initial trust
region radius in trust-region methods for nonlinear
optimization consists of monitoring the agreement between
the model and the objective function along the
steepest-descent direction at the starting point. Further
improvements for the starting point are also derived from
the information gleaned during the initializing phase.
Numerical results on a large set of problems show the
impact the initial trust region radius may have on
trust-region methods behaviour.}}
@article{Saue95,
author = {Th. Sauer},
title = {Computational aspects of multivariate polynomial
interpolation},
journal = {Advances in Computational Mathematics},
volume = 3, pages = {219--238}, year = 1995}
@misc{Saue96,
author = {Th. Sauer},
title = {Notes on polynomial interpolation},
howpublished = {Private communication},
month = {February}, year = 1996}
@article{SaueXu95,
author = {Th. Sauer and Y. Xu},
title = {On multivariate {L}agrange interpolation},
journal = MC,
volume = 64, pages = {1147--1170}, year = 1995}
@inproceedings{SchaLude94,
author = {J. Schaepperle and E. Luder},
title = {Optimization of distributed parameter systems with a
combined statistical-deterministic method},
booktitle = {1994 IEEE International Symposium on Circuits and Systems},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 6, pages = {141--144}, year = 1994,
abstract = {This paper describes an optimization method for
nonlinear systems with properties that depend on
functions instead of discrete parameters. The method
is applied to the design of systems with spatially
distributed parameters. The underlying mathematical
problem of the calculus of variations is approximated
by a finite dimensional constrained nonlinear
minimax-problem. This is solved with a method that
combines a deterministic algorithm for local
optimisation with a statistical method for global
optimization. The former is based on linearization
and linear programming with adaptive trust region,
while the latter uses elements from genetic methods
and pattern recognition. An example with a
nonlinearly loaded nonuniform transmission line shows
the capability of the algorithm to determine the
unknown optimum function with high precision.},
summary = {An optimization method is described for
nonlinear systems with properties that depend on
functions instead of discrete parameters. The method
is applied to the design of systems with spatially
distributed parameters. The underlying mathematical
problem of the calculus of variations is approximated
by a finite-dimensional constrained nonlinear
minimax-problem. This is solved with a method that
combines a deterministic algorithm for local
optimization with a statistical method for global
optimization. The former is based on linearization
and linear programming with adaptive trust region,
while the latter uses elements from genetic methods
and pattern recognition. An example with a
nonlinearly loaded non-uniform transmission line shows
the capability of the algorithm to determine the
unknown optimum function with high precision.}}
@article{Shii99,
author = {T. Shiina},
title = {Numerical solution technique for joint chance-constrained
programming problem---an application to electric power
capacity expansion },
journal = {Journal of the Operations Research Society of Japan},
volume = 42, number = 2, pages = {128--140}, year = 1999,
abstract = {We consider a joint chance-constrained linear programming
problem with random right hand side vector. The
deterministic equivalent of the joint
chance-constraint is already known in the case that
the right hand side vector is statistically
independent. But if the right hand side vector is
correlative, it is difficult to derive the
deterministic equivalent of the joint
chance-constraint. We discuss two methods for
calculating the joint chance-constraint. For the case
of uncorrelated right hand side, we try a direct
method different from the usual deterministic
equivalent, for the correlative right hand side case,
we apply numerical integration. In this paper a
chance-constrained programming problem is developed
for electric power capacity expansion, where the
error of forecasted electricity demand is defined by
a random variable. Finally we show that this problem
can be solved numerically using the trust region
method and numerical integration, and we present the
results of our computational experiments.},
summary = {A joint chance-constrained linear programming
problem with random right hand side vector is
considered. The deterministic equivalent of the
joint chance-constraint is already known in the case
that the right hand side vector is statistically
independent. But if the right hand side vector is
correlated, it is difficult to derive the
deterministic equivalent of the joint
chance-constraint. Two methods for calculating the
joint chance-constraint are discussed. For the case
of uncorrelated right hand side, a direct method
different from the usual deterministic equivalent is
tried, while for the correlated right hand side
case, numerical integration is applied. A
chance-constrained programming problem is developed
for electric power capacity expansion, where the
error of forecasted electricity demand is defined by
a random variable. This problem is solved numerically
using a trust region method, and numerical
integration, and results of computational experiments
are presented.}}
@article{Schi81,
author = {K. Schittkowski},
title = {The nonlinear programming method of {W}ilson, {H}an and
{P}owell with an augmented {L}agrangian type line search
function},
journal = NUMMATH,
volume = 38, pages = {83--114}, year = 1981}
@phdthesis{Schl95,
author = {S. Schleiff},
title = {Parametersch\"{a}tzung in nichtlinearen {M}odellen unter
besonderer {B}er\-\"{u}cks\-ichtigung kritischer {P}unkte},
school = {Martin-Luther-Universit\"{a}t},
address = {Halle-Wittenberg, Germany},
year = 1995}
@article{Schl93,
author = {T. Schlick},
title = {Modified {C}holesky factorizations for sparse
preconditioners},
journal = SISC,
volume = 14, number = 2, pages = {424--445}, year = 1993}
@article{SchnEsko91,
author = {R. B. Schnabel and E. Eskow},
title = {A new modified {C}holesky factorization},
journal = SISC,
volume = 11, number = 6, pages = {1136--1158}, year = 1991}
@article{SchnEsko99,
author = {R. B. Schnabel and E. Eskow},
title = {A revised modified {C}holesky factorization},
journal = SIOPT,
number = 9, volume = 4, pages = {1064--1081}, year = 1999}
@article{SchnKoonWeis85,
author = {R. B. Schnabel and J. E. Koontz and B. E. Weiss},
title = {A modular system of algorithms for unconstrained
minimization},
journal = {ACM Transactions on Mathematical Software},
volume = 11, number = 4, pages = {419--440}, year = 1985,
abstract = {We describe a new package, UNCMIN, for finding a local
minimizer of a real valued function of more than one
variable. The novel feature of UNCMIN is that it is a
modular system of algorithms, containing three different
step selection strategies (line search, dogleg, and optimal
step) that may be combined with either analytic or finite
difference gradient evaluation and with either analytic,
finite difference, or BFGS Hessian approximation. We
present the results of a comparison of the three step
selection strategies on the problems in
\citebb{MoreGarbHill81} in two separate cases: using finite
difference gradients and Hessians, and using finite
difference gradients with BFGS Hessian approximations. We
also describe a second package, REVMIN, that uses
optimization algorithms identical to UNCMIN but obtains
values of user-supplied functions by reverse communication.},
summary = {UNCMIN is a modular system of algorithms for optimization
containing three different step selection strategies
(linesearch, dogleg, and optimal step) that may be combined
with either analytic or finite difference gradient
evaluation and with either analytic, finite difference, or
BFGS Hessian approximation. The results of a comparison of
the three step selection strategies on the problems in
\citebb{MoreGarbHill81} are compared when using finite
difference gradients and Hessians, or finite difference
gradients with BFGS Hessian approximations. A second
package, REVMIN, that uses optimization algorithms
identical to UNCMIN but obtains values of user-supplied
functions by reverse communication is also described.}}
@article{SchoStoh99,
author = {S. Scholtes and M. St\"{o}hr},
title = {Exact penalization of mathematical programs with
equilibrium constraints},
journal = SICON,
volume = 37, number = 2, pages = {617--652}, year = 1999,
abstract = {We study theoretical and computational aspects of an exact
penalization approach to mathematical programs with
equilibrium constraints (MPEC). In the first part, we prove
that a Mangasarian-Fromowitz type condition ensures the
existence of a stable local error bound at the root of a
real-valued nonnegative piecewise smooth function. A
specification to nonsmooth formulations of equilibrium
constraints, e.g. complementarity conditions or normal
equations, provides conditions which guarantee the
existence of a nonsmooth exact penalty function for MPECs.
In the second part, we study a trust region minimization
method for a class of composite nonsmooth functions which
comprises exact penalty functions arising from MPECs. We
prove a global convergence result for the general method
and incorporate a penalty update rule. A further
specification results in an SQP trust region method for
MPECs based on an $\ell_1$ penalty function.},
summary = {Theoretical and computational aspects of an exact
penalization approach to mathematical programs with
equilibrium constraints (MPEC) are studied. It is
shown that a Mangasarian-Fromowitz type condition ensures
the existence of a stable local error bound at the root of
a real-valued non-negative piecewise smooth function. A
specification to non-smooth formulations of equilibrium
constraints, e.g. complementarity conditions or normal
equations, then provides conditions which guarantee the
existence of a non-smooth exact penalty function for MPECs.
A trust-region minimization method for a class of composite
non-smooth functions is then presented, which comprises
exact penalty functions arising from MPECs. Global
convergence is proved and a penalty update rule is
described. A further specification results in an SQP
trust-region method for MPECs based on an $\ell_1$
penalty function.}}
@article{SchrZowe92,
author = {H. Schramm and J. Zowe},
title = {A version of the bundle idea for minimizing a nonsmooth
function: conceptual idea, convergence analysis, numerical
results},
journal = SIOPT,
volume = 2, number = 1, pages = {121--152}, year = 1992,
abstract = {During recent years various proposals for the minimization
of a nonsmooth functional have been made. Amongst these,
the bundle concept turned out to be an especially fruitful
idea. Based on this concept, a number of authors have
developed codes that can successfully deal with nonsmooth
problems. The aim of the paper is to show that, by adding
some features of the trust-region philosophy to the bundle
concept, the end result is a distinguished member of the
bundle family with a more stable behaviour than some other
versions. The reliability and efficiency of this code is
demonstrated on the standard academic examples and on some
real-life problems},
summary = {An stable algorithm of the bundle family is obtained for
solving non-smooth minimization problems by adding some
features of the trust-region philosophy to the bundle
concept. The reliability and efficiency of the
corresponding code is demonstrated on the standard academic
examples and on some real-life problems.}}
@article{SchwTill89,
author = {H. Schwetlick and V. Tiller},
title = {Nonstandard Scaling Matrices for Trust Region
{G}auss-{N}ewton Methods},
journal = SISSC,
volume = 10, number = 4, pages = {654--670}, year = 1989,
abstract = {For solving large nonlinear problems via the trust-region
Gauss-Newton methods, nonstandard scaling matrices are
proposed for scaling the norm of the step. The scaling
matrices are rectangular, of full rank, and contain a block
of the Jacobian matrix of the residual function. Three
types of such matrices are investigated. The corresponding
trust region methods are shown to have qualitatively the
same convergence properties as the standard method.
Nonstandard scaling matrices are especially intended for
solving large and structured problems such as orthogonal
distance regression or surface fitting. Initial
computational experience suggests that for such problems
the proposed scaling implies sometimes a modest increase in
the number of iterations but reduces overall computational
cost.},
summary = {Non-standard scaling matrices are proposed for scaling the
norm of the step in the solution of large nonlinear
problems via the trust-region Gauss-Newton methods. The
scaling matrices are rectangular, of full rank, and contain
a block of the Jacobian matrix of the residual function.
Three types of such matrices are investigated. The
corresponding trust-region methods have
qualitatively the same convergence properties as the
standard method. Non-standard scaling matrices are
especially intended for solving large and structured
problems such as orthogonal distance regression or surface
fitting. Initial computational experience suggests that for
such problems the proposed scaling sometimes implies a
modest increase in the number of iterations but reduces
overall computational cost.}}
@phdthesis{Sebu92b,
author = {Ch. Sebudandi},
title = {Algorithmic developments in seismic tomography},
school = FUNDP, address = FUNDP-ADDRESS,
year = 1992}
@article{SebuToin92,
author = {Ch. Sebudandi and Ph. L. Toint},
title = {Nonlinear optimization for seismic travel time tomography},
journal = {Geophysical Journal International},
volume = 115, pages = {929--940}, year = 1993,
abstract = {This paper presents a non-linear algorithmic approach for
seismic traveltime. It is based on large-scale optimization
using non-linear least-squares and trust-region methods.
These methods provide a natural way to stabilize algorithms
based on Newton's iteration for non-linear minimization.
They also correspond to an alternative (and often more
efficient) view of the Levenberg-Marquardt method.
Numerical experience on synthetic data and on real
borehole-to-borehole problems are presented. In particular,
results produced by the algorithms are compared with those
of Ivansson (1985) for the Kr\aa m\aa la experiment.},
summary = {A nonlinear algorithm for seismic traveltime analysis is
presented, based on large-scale nonlinear least-squares
and trust-region methods. Numerical experience on synthetic
data and on real borehole-to-borehole problems is
presented. Results produced by the algorithms are compared
with those of Ivansson (1985) for the Kr\aa m\aa la
experiment.}}
@article{Semp97,
author = {J. Semple},
title = {Optimality conditions and solution procedures for
nondegenerate dual-response systems},
journal = {IIE Transactions},
volume = 29, number = 9, pages = {743-752}, year = 1997,
abstract = {This paper investigates the dual-response problem in the
case where the response functions are nonconvex
(nonconcave) quadratics and the independent variables
satisfy a radial bound. Sufficient conditions for a global
optimum are established and shown to generalize to the
multi-response case. It is then demonstrated that the
sufficient conditions may not hold if the problem is
'degenerate'. However, if the problem is nondegenerate, it
is shown that the sufficient conditions are necessarily
satisfied by some stationary point. In this case, a
specialized algorithm (DRSALG) is shown to locate the
global optimum in a finite number of steps. DRSALG will
also identify the degenerate case and pinpoint the location
where degeneracy occurs. The algorithm is easy to implement
from the standpoint of code development, and we illustrate
our elementary version on a well-studied dual-response
example from quality control.},
summary = {The dual-response problem in the case where the response
functions are non-convex (non-concave) quadratics and the
independent variables satisfy a radial bound is
investigated. Sufficient conditions for a global optimum
are established and generalized to the multi-response case.
It is demonstrated that the sufficient conditions may not
hold if the problem is ``degenerate''. However, if the
problem is non-degenerate, the sufficient conditions are
necessarily satisfied by some stationary point. In this
case, a specialized algorithm (DRSALG) locates
the global optimum in a finite number of steps. DRSALG also
identifies the degenerate case and pinpoints the location
where degeneracy occurs. The algorithm is easy to implement,
and is illustrated on a well-studied dual-response
example from quality control.}}
@phdthesis{Shah96,
author = {J. S. Shahabuddin},
title = {Structured trust-region algorithms for the minimization of
nonlinear functions},
school = CS-CORNELL, address = CORNELL-ADDRESS,
year = 1996,
summary = {Trust-region algorithms are a popular and successful class
of tools for the solution of nonlinear, non-convex
optimization problems. The basic trust-region algorithm is
extended so that it takes advantage of partial separability
to solve such large-scale problems in an efficient way. It
aims at simplifying the proposal of
\citebb{ConnGoulSartToin96a} in the case where no
constraints are imposed in the problem. Three related
approaches of ``multiple''or ``structured'' trust regions
are proposed. Convergence results are presented for them in
the unconstrained case. Some computational results are also
discussed, in which the three approaches are compared.}}
@article{Shan70,
author = {D. F. Shanno},
title = {Conditioning of Quasi-{N}ewton Methods For Function
Minimization},
journal = MC,
volume = 24, pages = {647--657}, year = 1970}
@article{Shan78b,
author = {D. F. Shanno},
title = {Conjugate gradient methods with inexact searches},
journal = MOR,
volume = 3, pages = {244--256}, year = 1978}
@misc{Shan99,
author = {D. F. Shanno},
title = {Topics in Implementing an Interior Point Method for
Nonconvex Nonlinear Programming},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999}
@book{Shef85,
author = {Y. Sheffi},
title = {Urban Transportation Networks},
publisher = PH, address = PH-ADDRESS,
year = 1985}
@article{ShulSchnByrd85,
author = {G. A. Shultz and R. B. Schnabel and R. H. Byrd},
title = {A family of trust-region-based algorithms for unconstrained
minimization with strong global convergence properties},
journal = SINUM,
volume = 22, number = 1, pages = {47--67}, year = 1985,
abstract = {This paper has two aims: to exhibit very general conditions
under which members of a broad class of unconstrained
minimization algorithms are globally convergent in a strong
sense, and to propose several new algorithms that use
second derivative information and achieve such convergence.
In the first part of the paper, we present a general
trust-region-based algorithm schema that includes an
undefined step selection strategy. We give general
conditions on the step selection strategy under which limit
points of the algorithm will satisfy first and second order
necessary conditions for unconstrained minimization. Our
algorithm schema is sufficiently broad to include line
search algorithms as well. Next, we show that a wide range
of step selection strategies satisfy the requirements of
our convergence theory. This leads us to propose several
new algorithms that use second derivative information and
achieve strong global convergence, including an indefinite
line search algorithm, several indefinite dogleg algorithms
and a modified ``optimal-step'' algorithm. Finally, we
propose an implementation of one such indefinite dogleg
algorithm.},
summary = {A general trust-region-based algorithm schema that includes
an undefined step selection strategy is presented. General
conditions on the step selection strategy under which limit
points will satisfy first and second order
necessary conditions are
given. The algorithm schema is sufficiently broad to
include linesearch methods as well. It is shown that a wide
range of step selection strategies satisfy the requirements
of the convergence theory, and several algorithms that use
second derivative information and achieve strong global
convergence are proposed. These include an indefinite
linesearch algorithm, several indefinite dogleg algorithms
and a modified ``optimal-step'' algorithm. An
implementation of one such indefinite dogleg algorithm is
proposed.}}
@inproceedings{SimaShan97,
author = {E. M. Simantiraki and D. F. Shanno},
title = {An Infeasible-Interior-Point Method for Linear
Complementarity Problems},
crossref = {DuffWats97}, pages = {339--362}}
@article{Smal83,
author = {S. Smale},
title = {On the average number of steps of the simplex method of
linear programming},
journal = MP,
volume = 27, number = 3, pages = {241--262}, year = 1983}
@article{SmitBowe93,
author = {R. C. Smith and K. L. Bowers},
title = {{S}inc-{G}alerkin Estimation of Diffusivity in Parabolic
Problems},
journal = {Inverse Problems},
volume = 9, number = 1, pages = {113--135}, year = 1993,
abstract = {A fully Sinc-Galerkin method for the numerical recovery of
spatially varying diffusion coefficients in linear
parabolic partial differential equations is presented.
Because the parameter recovery problems are inherently
ill-posed, an output error criterion in' conjunction with
Tikhonov regularization is used to formulate them as
infinite-dimensional minimization problems. The forward
problems are discretized with a sinc basis in both the
spatial and temporal domains thus yielding an approximate
solution which displays an exponential convergence rate and
is valid on the infinite time interval. The minimization
problems are then solved via a quasi-Newton/trust region
algorithm. The L-curve technique for determining an
appropriate value of the regularization parameter is
briefly discussed, and numerical examples are given which
demonstrate the applicability of the method both for
problems with noise-free data as well as for those whose
data contain white noise.},
summary = {A fully Sinc-Galerkin method for the numerical recovery of
spatially varying diffusion coefficients in linear
parabolic partial differential equations is presented.
Because the parameter recovery problems are inherently
ill-posed, an output error criterion in conjunction with
Tikhonov regularization is used to formulate them as
infinite-dimensional minimization problems. The forward
problems are discretized with a sinc basis in both the
spatial and temporal domains thus yielding an approximate
solution which displays an exponential convergence rate and
is valid on the infinite time interval. The minimization
problems are then solved via a quasi-Newton/trust-region
algorithm. The L-curve technique for determining an
appropriate value of the regularization parameter is
briefly discussed, and numerical examples illustrate the
applicability of the method both for problems with
noise-free data as well as for those whose data contain
white noise.}}
@article{Sore82,
author = {D. C. Sorensen},
title = {{N}ewton's Method with a Model Trust-Region Modification},
journal = SINUM,
volume = 19, number = 2, pages = {409--426}, year = 1982,
abstract = {A modified Netwon method for unconstrained minimization is
presented and analyzed. The modification is based upon the
model trust region approach. This report contains a
thorough analysis of the locally constrained quadratic
minimizations that arise as subproblems in the modified
Newton iteration. Several promising alternatives are
presented for solving these subproblems in ways that
overcome certain theoretical difficulties exposed by the
analysis. Very strong convergence results are presented
concerning the minimization algorithm. In particular, the
explicit use of second order information is justified by
demonstrating that the iterates converge to a point which
satisfies the second order necessary conditions for
minimization. With the exception of very pathological cases
this occurs whenever the algorithm is applied to problems
with continuous second partial derivatives.},
summary = {A modified Netwon method for unconstrained minimization is
presented and analyzed. The modification is based upon the
model trust-region approach. A thorough analysis of the
locally constrained quadratic minimizations that arise as
subproblems in the modified Newton iteration is given.
Several promising alternatives are presented for solving
these subproblems in ways that overcome certain theoretical
difficulties exposed by the analysis. Very strong
convergence results are presented concerning the
minimization algorithm. In particular, the explicit use of
second order information is justified by demonstrating that
the iterates converge to a point which satisfies the second
order necessary conditions for minimization. With the
exception of very pathological cases this occurs whenever
the algorithm is applied to problems with continuous second
partial derivatives.}}
@inproceedings{Sore82b,
author = {D. C. Sorensen},
title = {Trust Region Methods for Unconstrained Optimization},
crossref = {Powe82}, pages = {29--39},
summary = {The basic trust-region approach to safeguarding Newton-like
methods for unconstrained optimization is discussed.}}
@article{Sore97,
author = {D. C. Sorensen},
title = {Minimization of a Large-Scale Quadratic Function subject to
a Spherical Constraint},
journal = SIOPT,
volume = 7, number = 1, pages = {141--161}, year = 1997,
abstract = {An important problem in linear algebra and optimization is
the Trust-Region Subproblem: Minimize a quadratic function
subject to an ellipsoidal or spherical constraint. This
basic problem has several important large scale
applications including seismic inversion and forcing
convergence in optimization methods. Existing methods to
solve the trust-region subproblem require matrix
factorizations, which are not feasible in large scale
setting. This paper presents an algorithm for solving the
large scale trust-region subproblem that requires a fixed
size limited storage proportional to order of the quadratic
and that relies only on matrix-vector products. The
algorithm recasts the trust-region subproblem in terms of a
parametrized eigenvalue problem and adjusts the parameter
with a superlinearly convergent iteration to find the
optimal solution from the eigenvector of the parametrized
problem. Only the smallest eigenvalue and corresponding
eigenvector of the parametrized problem needs to be
computed. The Implicitly Restarted Lanczos Method is
well-suited to this subproblem.},
summary = {An algorithm is presented for solving the large scale
trust-region subproblem that requires a fixed size limited
storage proportional to order of the quadratic and that
relies only on matrix-vector products. The algorithm
recasts the trust-region subproblem in terms of a
parametrized eigenvalue problem and adjusts the parameter
with a superlinearly convergent iteration to find the
optimal solution from the eigenvector of the parametrized
problem. Only the smallest eigenvalue and corresponding
eigenvector of the parametrized problem needs to be
computed. The Implicitly Restarted Lanczos Method is
well-suited to this subproblem.}}
@article{StanSnym93,
author = {N. Stander and J. A. Snyman},
title = {A new first-order interior feasible direction method for
structural optimization},
journal = {International Journal for Numerical Methods in Engineering},
volume = 36, number = 23, pages = {4009--4025}, year = 1993}
@article{Stei83a,
author = {T. Steihaug},
title = {The conjugate gradient method and trust regions in large
scale optimization},
journal = SINUM,
volume = 20, number = 3, pages = {626--637}, year = 1983,
abstract = {Algorithms based on trust regions have been shown to be
robust methods for unconstrained optimization problems. All
existing methods, either based on the dogleg strategy or
\citebb{Hebd73}--\citebb{More78} iterations, require the
solution of system of linear equations. In large scale
optimization this may be prohibitively expensive. It is
shown in this paper that an approximate solution of the
trust region problem may be found by the preconditioned
conjugate gradient method. This may be regarded as a
generalized dogleg technique where we asymptotically take
the inexact quasi-Newton step. We also show that we have
the same properties as existing methods based on the dogleg
strategy using an approximate Hessian.},
summary = {It is shown that an approximate solution of the
trust-region problem may be found by the preconditioned
conjugate gradient method. This may be regarded as a
generalized dogleg technique where asymptotically the
inexact quasi-Newton step is taken. The resulting algorithm
has the same properties as existing methods
based on the dogleg strategy using an approximate Hessian.}}
@article{SterWolk94,
author = {R. J. Stern and H. Wolkowicz},
title = {Trust region problems and nonsymetric eigenvalue
perturbations},
journal = SIMAA,
volume = 15, number = 3, pages = {775--778}, year = 1994,
abstract = {A characterization is given for the spectrum of a symmetric
matrix to remain real after a nonsymmetric sign-restricted
border perturbation, including the case where the
perturbation is skew-symmetric. The characterization is in
terms of the stationary points of a quadratic function on
the unit sphere. This yields interlacing relationships
between the eigenvalues of the original matrix and those of
the perturbed matrix. As a result of the linkage between
the perturbation and stationarity problems, new theoretical
insights are gained for each. Applications of the main
results include a characterization of those matrices that
are exponentially nonnegative with respect to the
n-dimensional ice-cream cone, which in turn leads to a
decomposition theorem for such matrices. In addition,
results are obtained for nonsymmetric matrices regarding
interlacing and majorization.},
summary = {A characterization is given for the spectrum of a symmetric
matrix to remain real after a non-symmetric sign-restricted
border perturbation, including the case where the
perturbation is skew-symmetric. The characterization is in
terms of the stationary points of a quadratic function on
the unit sphere. This yields interlacing relationships
between the eigenvalues of the original matrix and those of
the perturbed matrix. Applications include a
characterization of matrices that are exponentially
non-negative with respect to the $n$-dimensional ice-cream
cone, which leads to a decomposition theorem for
such matrices. Results are obtained for
non-symmetric matrices regarding interlacing and
majorization.}}
@article{SterWolk95,
author = {R. J. Stern and H. Wolkowicz},
title = {Indefinite Trust Region Subproblems and Nonsymmetric
Eigenvalue Perturbations},
journal = SIOPT,
volume = 5, number = 2, pages = {286--313}, year = 1995,
abstract = {This paper extends the theory of trust region subproblems
in two ways: (i) it allows indefinite inner products in the
quadratic constraint, and (ii) it uses a two-sided (upper
and lower bound) quadratic constraint. Characterizations of
optimality are presented that have no gap between necessity
and sufficiency. Conditions for the existence of solutions
are given in terms of the definiteness of a matrix pencil.
A simple dual program is introduced that involves the
maximization of a strictly concave function on an interval.
The dual program simplifies the theory and algorithms for
trust region subproblems. It also illustrates that the
trust region subproblems are implicit convex programming
problems, and thus explains why they are so tractable. The
duality theory also provides connections to eigenvalue
perturbation theory. Trust region subproblems with zero
linear term in the objective function correspond to
eigenvalue problems, and adding a linear term in the
objective function is seen to correspond to a perturbed
eigenvalue problem. Some eigenvalue interlacing results are
presented.},
summary = {The theory of trust-region subproblems is extended in two
ways: (i) indefinite inner products in the quadratic
constraint are allowed, and (ii) a two-sided (upper and
lower bound) quadratic constraint is used.
Characterizations of optimality are presented that have no
gap between necessity and sufficiency. Conditions for the
existence of solutions are given in terms of the
definiteness of a matrix pencil. A simple dual program is
introduced that involves the maximization of a strictly
concave function on an interval. The dual program
simplifies the theory and algorithms for trust-region
subproblems. It also illustrates that they are implicit
convex programming problems, and thus explains why they are
so tractable. The duality theory provides connections to
eigenvalue perturbation theory. Trust-region subproblems
with zero linear term in the objective function correspond
to eigenvalue problems, and adding a linear term in the
objective function is seen to correspond to a perturbed
eigenvalue problem. Some eigenvalue interlacing results are
presented.}}
@article{Stew67,
author = {G. W. Stewart},
title = {A Modification of {D}avidon's Minimization Method to Accept
Difference Approximations of Derivatives},
journal = {Journal of the ACM},
volume = 14, year = 1967}
@inproceedings{Stoe83,
author = {J. Stoer},
title = {Solution of Large Linear Systems of Equations by Conjugate
Gradient Type Methods},
crossref = {BachGrotKort83}, pages = {540--565}}
@phdthesis{Stoh99,
author = {M. St\"{o}hr},
title = {Nonsmooth trust-region methods and their applications to
mathematical programs with equilibrium constraints},
school = {Faculty of Mathematics, University of Karlsruhe},
address = {Karlsruhe, Germany},
year = 1999,
abstract = {(none)},
summary = {A trust-region method for the solution of mathematical
programs with equilibrium constraints (MPEC) is proposed
and analyzed. It makes use of exact penalty functions
arising from the MPECs formulation. A variant of the
algorithm by \citebb{SchoStoh99} is discussed that
uses the concept of Cauchy point rather than requiring
a model decrease proportional to that obtained at the
global solution of the trust-region subproblem. Global
convergence is proved and some numerical tests illustrate
the method.}}
@inproceedings{StudLuth97,
author = {G. Studer and H.-J. L\"{u}thi},
title = {Maximum loss for risk measurement of portfolios},
booktitle = {Operations Research Proceedings 1996: Selected Papers of
the Symposium on Operations Research (SOR 96)},
editor = {U. Zimmermann and U. Derigs and W. Gaul and
R. H. Mohring and K. P. Schuster},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
pages = {386--391}, year = 1997,
abstract = {Effective risk management requires adequate risk
measurement. A basic problem herein is the quantification
of market risks: what is the overall effect on a portfolio
if market rates change? The first chapter gives a brief
review of the standard risk measure "Value-At-Risk" (VAR)
and introduces the concept of "Maximum Loss" (ML) as a
method for identifying the worst case in a given scenario
space, called "Trust Region". Next, a technique for
calculating efficiently ML for quadratic functions is
described; the algorithm is based on the
Levenberg-Marquardt theorem, which reduces the high
dimensional optimization problem to a one dimensional root
finding. Following this, the idea of the "Maximum Loss
Path" is presented: repetitive calculation of ML for a
growing trust region leads to a sequence of worst cases,
which form a complete path. Similarly, the paths of
"Maximum Profit" (MP) and "Expected Value" (EV) can be
determined; the comparison of them permits judgments on
the quality of portfolios. These concepts are also
applicable to non-quadratic portfolios by using "Dynamic
Approximations", which replace arbitrary profit and loss
functions with a sequence of quadratic functions. Finally,
the idea of "Maximum Loss Distribution" is explained. The
distributions of ML and MP can be obtained directly from
the ML and MP paths. They lead to lower and upper bounds of
VAR and allow statements about the spread of ML and MP.},
summary = {A brief review of the standard risk measure "Value-At-Risk"
(VAR) is given and the concept of "Maximum Loss" (ML) for
identifying the worst case in a given scenario space, a
trust region, introduced. A technique for calculating
efficiently ML for quadratic functions is described; the
algorithm is based on the Levenberg-Morrison-Marquardt
theorem. The idea of the "Maximum Loss Path" is presented.
Repetitive calculation of ML for a growing trust region
leads to a sequence of worst cases, which form a complete
path. Similarly, the paths of "Maximum Profit" (MP) and
"Expected Value" (EV) can be determined; the comparison of
them permits judgments on the quality of portfolios. These
concepts are applicable to non-quadratic portfolios by
using "Dynamic Approximations", which replace arbitrary
profit and loss functions with a sequence of quadratic
functions. The idea of "Maximum Loss Distribution" is
explained. The distributions of ML and MP can be obtained
directly from the ML and MP paths, lead to lower and upper
bounds of VAR and allow statements about the spread of ML
and MP.}}
@article{Stra91,
author = {Z. Strako\v{s}},
title = {On the real convergence rate of the conjugate gradient
method},
journal = LAA,
volume = {154-156}, pages = {535--549}, year = 1991}
@article{Sun96,
author = {L. P. Sun},
title = {A restricted trust region method with supermemory for
unconstrained optimization},
journal = JCM,
volume = 14, number = 3, pages = {195--202}, year = 1996,
abstract = {A new method for unconstrained optimization problems is
presented. It belongs to the class of trust-region method,
in which the descent direction is sought by using the trust
region steps within the restricted subspace. Because this
subspace can be specified to include information about
previous steps, the method is also related to a supermemory
descent method without performing multiple dimensional
searches. Trust region methods have attractive global
convergence property. Since the method possesses the
characteristics of both the trust region methods and the
supermemory descent methods, it is endowed with rapidly
convergence. Numerical tests illustrate this point.},
summary = {A trust-region method for unconstrained optimization
problems is presented, in which the descent direction is
sought by using the trust-region steps within a
restricted subspace. Because this subspace can be specified
to include information about previous steps, the method is
also related to a supermemory descent methods. It is endowed
with rapidly convergence, as illustrated by numerical
tests.}}
@article{Sun96b,
author = {W. Sun},
title = {Optimization methods for nonquadratic model},
journal = {Asia-Pacific Journal of Operational Research},
volume = 13, number = 1, pages = {43--63}, year = 1996}
% abstract = {In this paper we review the state of the art and future
% development on non-quadratic model optimization (NQMO).
% Optimization methods based on the non-quadratic model are
% more interesting than ones based on the quadratic model
% because they have more interpolation information and better
% approximation effect, especially for ill-conditioned
% functions or functions with strong non-quadratic behaviour.
% The main non-quadratic model methods include conic model
% methods, homogeneous model methods, nonlinear scaling model
% methods, tensor model methods and trust region model
% methods. We think that further study of NQMO is necessary,
% the area shows great potential and it is interesting.},
% summary = {The state of the art and future development on
% non-quadratic model optimization (NQMO) is reviewed.
% Optimization methods based on the non-quadratic model are
% more interesting than ones based on the quadratic model
% because they have more interpolation information and better
% approximation effect, especially for ill-conditioned
% functions or functions with strong non-quadratic behaviour.
% The main non-quadratic models include conic models,
% homogeneous models, nonlinear scaling models,
% tensor models and trust-region models.}
@article{Sun93,
author = {J. Sun},
title = {A convergence proof for an affine-scaling algorithm for
convex quadratic programming without nondegeneracy
assumptions},
journal = MP,
volume = 60, number = 1, pages = {69--79}, year = 1993}
@article{Sun97,
author = {J. Sun},
title = {On piecewise quadratic {N}ewton and trust-region problems},
journal = MP,
volume = 76, number = 3, pages = {451--468}, year = 1997,
abstract = {Some recent algorithms for nonsmooth optimization require
solutions to certain piecewise quadratic programming
subproblems. Two types of subproblems are considered in
this paper. The first type seeks the minimization of a
continuously differentiable and strictly convex piecewise
quadratic function subject to linear equality constraints.
We prove that a nonsmooth version of Newton's method is
globally and finitely convergent in this case. The second
type involves the minimization of a possibly nonconvex and
nondifferentiable piecewise quadratic function over a
Euclidean ball. Characterizations of the global minimizer
are studied under various conditions. The results extend a
classical result on the trust region problem.},
summary = {Some algorithms for non-smooth optimization require the
solutions to certain piecewise quadratic programming
subproblems. Two types of subproblems are considered. The
first uses the minimization of a continuously
differentiable and strictly convex piecewise quadratic
function subject to linear equality constraints.
A non-smooth version of Newton's method is
globally and finitely convergent in this case. The second
type involves the minimization of a possibly non-convex and
non-differentiable piecewise quadratic function over a
Euclidean ball. Characterizations of the global minimizer
are studied under various conditions.}}
@article{SunRued93,
author = {J.-Q. Sun and K. Ruedenberg},
title = {Quadratic steepest descent on potential energy surfaces.
{I}. {B}asic formalism and quantitative assessment},
journal = {Journal of Chemical Physics},
volume = 99, number = 7, pages = {5257--5268}, year = 1993,
abstract = {A novel second-order algorithm is formulated for
determining steepest-descent lines on potential
energy surfaces. The reaction path is deduced from
successive exact steepest-descent lines of local
quadratic approximations to the surface. At each
step, a distinction is made between three points: the
center for the local quadratic Taylor expansion of
the surface, the junction of the two adjacent local
steepest-descent line approximations, and the
predicted approximation to the true steepest-descent
line. This flexibility returns a more efficient yield
from the calculated information and increases the
accuracy of the local quadratic approximations by
almost an order of magnitude. In addition, the step
size is varied with the curvature and, if desired,
can be readjusted by a trust region
assessment. Applications to the Gonzalez-Schlegel and
the Muller-Brown surfaces show the method to compare
favorably with existing methods. Several measures are
given for assessing the accuracy achieved without
knowledge of the exact steepest-descent line. The
optimal evaluation of the predicted gradient and
curvature for dynamical applications is discussed.},
summary = {A second-order algorithm is formulated for
determining steepest-descent lines on potential
energy surfaces, in which the step size is varied with the
curvature and, if desired, readjusted by a trust region
assessment. Applications to the Gonzalez-Schlegel and
the Muller-Brown surfaces show the method to behave well.
Several measures are given for assessing the accuracy
achieved without knowledge of the exact steepest-descent
line. The optimal evaluation of the predicted gradient and
curvature for dynamical applications is discussed.}}
@misc{SunYuan98,
author = {W. Sun and Y. Yuan},
title = {A Conic Model Trust Region Method for Nonlinearly
Constrained Optimization},
howpublished = {Presentation at the International Conference on Nonlinear
Programming and Variational Inequalities, Hong Kong},
year = 1998,
abstract = {In this paper we present conic trust region methods for
constrained optimization problems. We give necessary and
sufficient conditions for the solution of the associated
trust region subproblems. Several equivalent variations
and their properties are discussed. Some conic trust
region algorithms are constructed. Finally, we establish
the global and local convergence of our algorithms},
summary = {Trust-region methods for constrained optimization using
conic models are considered, and necessary and sufficient
conditions given for the solution of several equivalent
formulations of the associated subproblems. Convergence
properties of the resulting algorithm are established.}}
@article{SunaBele91,
author = {M. Sunar and A. D. Belegundu},
title = {Trust Region Methods for Structural Optimization Using
Exact 2nd-Order Sensitivity},
journal = {International Journal for Numerical Methods in Engineering},
volume = 32, number = 2, pages = {275--293}, year = 1991,
abstract = {The performance of multiplier algorithms for structural
optimization has been significantly improved by using trust
regions. The trust regions are constructed using analytical
second order sensitivity, and within this region, the
augmented Lagrangian $\phi$ is minimized subject to bounds.
Evaluation of first and second derivatives of $\phi$ by the
adjoint method does not require derivations of individual
(implicit) constraint functions, which makes the method
economical. Eight test problems are considered and a vast
improvement over previously used multiplier algorithms has
been noted. Also, the algorithm is robust with respect to
scaling, input parameters and starting designs.},
summary = {A trust-region method for structural
optimization is constructed using analytical
second order sensitivity. The augmented Lagrangian $\phi$
is minimized in this region subject to bounds.
Evaluation of first and second derivatives of $\phi$ by the
adjoint method does not require derivations of individual
(implicit) constraint functions, which makes the method
economical. The algorithm is robust with respect to
scaling, input parameters and starting designs.}}
@book{Suth75,
author = {W. A. Sutherland},
title = {Introduction to Metric and Topological Spaces},
publisher = OUP, address = OUP-ADDRESS,
year = 1975}
%%% T %%%
@article{Tapi77,
author = {R. A. Tapia},
title = {Diagonalized multiplier methods and quasi-{N}ewton methods
for constrained optimization},
journal = JOTA,
volume = 22, pages = {135--194}, year = 1977}
@inproceedings{TappJala90,
author = {C. Tappayuthpijarn and J. Jalali},
title = {Loadflow solution by applying hybrid algorithm to the
{N}ewton-{R}aphson method},
booktitle = {Proceedings of the American Power Conference,
Illinois Institute of Technology, Chicago, IL, USA},
pages = {234--238}, year = 1990,
abstract = {The purpose of the hybrid method in solving power
flow problems is to improve the efficiency in
convergence of the existing Newton-Raphson method
(NR) when its close initial estimates are not
available. The method is based on interpolating
between the fast convergence standard Newton-Raphson
iteration and the method of steepest descent applied
to the sum of the square of mismatch $f_i(x)$. The
balance between these two methods is governed by
introducing the concept of the trust region to
restrict the step predicted by the classical method
to be in the quadratic region and to switch to the
steepest decent method that is better when the
initial starts are far from the solution. The concept
of trust radius and switching policies are given by
the author. Digital computer results and their
comparison of the 10 bus system, with different
initial starts, by the proposed method and by the
classical method are also given.},
summary = {A trust-region based method is described for enforcing
global convergence in solving power flow problems.}}
@article{Tana99,
author = {Y. Tanaka},
title = {A trust region method for semi-infinite programming
problems },
journal = {International Journal of Systems Science},
volume = 30, number = 2, pages = {199--204}, year = 1999,
abstract = {We present a new successive quadratic programming (SQP)
approach for semi-infinite programming problems with
a trust region technique. Numerical methods for
solving semi-infinite programming problems can be
divided into continuous methods and discretization
methods. We begin with a trust region method for
nonlinear programming problems which possesses a fast
and global convergence property and obviates the
Maratos effect which is an unfavourable phenomenon
that sometimes occurs for general SQP-type
approaches. Then we apply the method to discretized
semi-infinite programming problems by utilizing an
$L_{\infty}$-exact penalty function and
$\epsilon$-most-active constraints. The
$L_{\infty}$-exact penalty function is, in fact,
essential for continuity methods for semi-infinite
programming problems so as to maintain continuity, of
the exact penalty function, and enables the use of
epsilon-most-active constraints in discretized
semi-infinite programming problems. The results of
preliminary computational experiments demonstrate the
effectiveness of our approach for discretized
semi-infinite programming problems.},
summary = {A trust region SQP method, using second-order
corrections, is applied to discretized
semi-infinite programming problems, using an
$L_{\infty}$-exact penalty function and
$\epsilon$-most-active constraints.
Preliminary computational experiments demonstrate the
viability of this approach.}}
@article{TanaFukuHase87,
author = {Y. Tanaka and M. Fukushima and T. Hasegawa},
title = {Implementable $L_{\infty}$ penalty-function method
for semi-infinite optimization},
journal = {International Journal of Systems Science},
volume = 18, number = 8, pages = {1563--1568}, year = 1987,
abstract = {This paper considers general nonlinear semi-infinite
programming problems and presents an implementable
method which employs an exact $L_{\infty}$
penalty function. Since the $L_{\infty}$ penalty
function is continuous even if the number of
representative constraints changes, trust-region
techniques may effectively be adopted to obtain
global convergence. Numerical results are given to
show the efficiency of the proposed algorithm.},
summary = {An implementable method for general nonlinear
semi-infinite programming problems is described, which
employs an exact $L_{\infty}$ penalty function. Since
this function is continuous even if the number of
representative constraints changes, trust-region
techniques may effectively be adopted to obtain
global convergence. Numerical results are given to
show the efficiency of the proposed algorithm.}}
@article{TanaFukuIbar88,
author = {Y. Tanaka and M. Fukushima and T. Ibaraki},
title = {A globally convergent SQP method for semi-infinite
nonlinear optimization},
journal = JCAM,
volume = 23, number = 2, pages = {141--153}, year = 1988,
abstract = {A new approach for semi-infinite programming problems
is presented, which belongs to the class of
successive quadratic programming (SQP) methods with
trust region technique. The proposed algorithm
employs the exact L/sub infinity / penalty function
as a criterion function and incorporates an
appropriate scheme for estimating active
constraints. It is proved that the algorithm is
globally convergent under some assumptions. Numerical
experiments show that the algorithm is very promising
in practice.},
summary = {A trust-region SQP method for semi-infinite programming
is presented. The proposed algorithm
employs the exact $L_{\infty}$ penalty function
and incorporates a scheme for estimating active
constraints. It is proved to be
globally convergent. Numerical
experiments show that the algorithm is promising.}}
@article{Tebo97,
author = {M. Teboulle},
title = {Convergence of proximal-like algorithms},
journal = SIOPT,
volume = 6, number = 3, pages = {617--625}, year = 1997}
@article{Terp95,
author = {P. Terpolilli},
title = {Trust region method in nonsmooth optimization},
journal = {Comptes rendus de l'{A}cad\'{e}mie des {S}ciences,
s\'{e}rie {M}ath\'{e}matique},
volume = 321, number = 7, pages = {945--948}, year = 1995,
abstract = {In this note, a new framework for nonsmooth optimization is
introduced. We consider then an algorithm using trust
region stategy and prove some global convergence results.
We pay particular attention to the use of inexact local
models : actually, we give a convergence result in a
situation where local models are computed by a numerical
procedure.},
summary = {A framework for non-smooth optimization is introduced.
An algorithm using trust-region stategy is considered and
global convergence results are established. Particular
attention is paid to the use of inexact local models. A
convergence result is given in the situation where local
models are computed by a numerical procedure.}}
@phdthesis{Tern94,
author = {D. J. Ternet},
title = {A trust region algorithm for reduced {H}essian successive
quadratic programming},
school = {Department of Chemical Engineering, Carnegie Mellon
University},
address = {Pittsburgh, USA},
year = 1994,
abstract = {Successive Quadratic Programming (SQP) has become a powerful
tool for solving large-scale process optimization problems.
An SQP code has recently been developed at CMU to solve
nonlinear programming problems. Some areas of improvement
in the current algorithm include the introduction of a
barrier function to reduce the constraints, the development
of a sparse matrix version of the code for larger problems,
and the solution of a trust region constraint to the
solution of the quadratic programming (QP) subproblem. Of
these possible improvements, the trust region
implementation was chosen as the focus of this paper. The
Successive Quadratic Programming method of solving
nonlinear programming problems is reviewed, as well as the
role of line search and trust region methods within this
framework. A trust region was added to the current
algorithm to increase the algorithms robustness while
maintaining its superlinear convergence properties. The
benefits of combining a line search method with a trust
region method are also explained. A set of test problems
for the new algorithm are explained which motivates the need
for the addition of the trust region.},
summary = {An SQP algorithm for solving nonlinear programming problems
is reviewed, as well as the role of linesearch and
trust-region methods within this framework. A trust region
is added to the current algorithm to increase the
algorithms robustness while maintaining its superlinear
convergence properties. The benefits of combining a
linesearch method with a trust-region method are also
explained. A set of test problems are described which
motivate the need for the trust region.}}
@phdthesis{Thom75,
author = {S. Thomas},
title = {Sequential estimation techniques for quasi-{N}ewton
algorithms},
school = {Cornell University}, address = CORNELL-ADDRESS,
year = 1975}
@article{Tibs96,
author = {R. Tibshirani},
title = {Regression shrinkage and selection via the lasso},
journal = {Journal of the Royal Statistical Society B},
volume = 58, number = 1, pages = {267--288}, year = 1996}
@article{TinnWalk67,
author = {W. F. Tinney and J. W. Walker},
title = {Direct solution of sparse network equations by optimally
ordered triangular factorization},
journal = {Proceedings of the IEEE},
volume = 55, pages = {1801--1809}, year = 1967}
@article{Toin77a,
author = {Ph. L. Toint},
title = {On sparse and symmetric matrix updating subject to a linear
equation},
journal = MC,
volume = 31, number = 140, pages = {954--961}, year = 1977}
@article{Toin78,
author = {Ph. L. Toint},
title = {Some Numerical Result Using a Sparse Matrix Updating
Formula in Unconstrained Optimization},
journal = MC,
volume = 32, number = 143, pages = {839--851}, year = 1978}
%abstract = {This paper presents a numerical comparison between
% algorithms for unconstrained optimization that take account
% of sparsity in the second derivative matrix of the
% objective function. Some of the methods
% included in the comparison use difference approximation
% schemes to evaluate the second derivative matrix and
% other use an approximation to it which is updated regularly
% using the changes in the gradient. These results show what
% method to use in what circumstances and also suggest
% interesting future developments.}}
@article{Toin79,
author = {Ph. L. Toint},
title = {On the Superlinear Convergence of an Algorithm for Solving
a Sparse Minimization Problem},
journal = SINUM,
volume = 16, pages = {1036--1045}, year = 1979}
@inproceedings{Toin80a,
author = {Ph. L. Toint},
title = {Sparsity Exploiting Quasi-{N}ewton Methods for
Unconstrained Optimization},
crossref = {DixoSpedSzeg80}, pages = {65--90}}
@inproceedings{Toin81b,
author = {Ph. L. Toint},
title = {Towards an Efficient Sparsity Exploiting {N}ewton Method
for Minimization},
booktitle = {Sparse Matrices and Their Uses},
editor = {I. S. Duff},
publisher = AP, address = AP-ADDRESS,
pages = {57--88}, year = 1981,
abstract = {The paper surveys some recently proposed algorithms for
unconstrained minimization when second derivative of the
objective function is sparse. Updating and estimation
procedures are considered from the efficiency point of
view. Special attention is given to the case where the
Hessian has a band structure. A new strategy for the choice
of the step is also discussed and some numerical results on
a specially designed test function are presented.},
summary = {Recently proposed algorithms for unconstrained minimization
when second derivative of the objective function is sparse
are surveyed. Updating and estimation procedures are
considered from the efficiency point of view. Special
attention is given to the case where the Hessian has a band
structure. A strategy for the choice of the step using
truncated conjugate-gradients is also discussed and some
numerical results on a specially designed test function
are presented.}}
@techreport{Toin81f,
author = {Ph. L. Toint},
title = {Convergence properties of a class of minimization
algorithms that use a possibly unbounded sequence of
quadratic approximations},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {81/1}, year = 1981,
abstract = {Global convergence results are established for a
trust-region like algorithm. At variance with previous
contributions, this theory does not make assumptions the
norm of the Hessian approximations, but rather on the
Rayleigh quotients of this approximation in certain
specific directions. This allows for the case where Hessian
approximations may become arbitrarily large provided they
remain reasonable in these directions.},
summary = {Global convergence results are established for a
trust-region algorithm without assumptions on the
norm of the Hessian approximations, but rather on the
Rayleigh quotients of this approximation in certain
directions. This allows for the case where Hessian
approximations may become arbitrarily large provided they
remain reasonable in these directions.}}
@techreport{Toin83b,
author = {Ph. L. Toint},
title = {User's Guide to the Routine {PSPMIN} for Solving Partially
Separable Bounded Optimization Problems},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {83/1}, year = 1983}
@article{Toin83e,
author = {Ph. L. Toint},
title = {{VE08AD}, a routine for partially separable optimization
with bounded variables},
journal = {Harwell Subroutine Library},
volume = 2, year = 1983}
@article{Toin86b,
author = {Ph. L. Toint},
title = {On large scale nonlinear least squares calculations},
journal = SISSC,
volume = 8, number = 3, pages = {416--435}, year = 1987,
abstract = {The nonlinear model fitting problem is analyzed in this
paper, with special emphasis on the practical solution
techniques when the number of parameters in the model is
large. Classical approaches to small dimensional least
squares are reviewed and an extension of them to problems
involving many variables is proposed. This extension uses
the concept of partially separable structures, which has
already proved its applicability for large scale
optimization. An adaptable algorithm is discussed, which
chooses between various possible models of the objective
function. Preliminary numerical experience is also
presented, which shows that actual solution of a large
class of fitting problems involving several hundreds of
nonlinear parameters is possible at a reasonable cost.},
summary = {The nonlinear model fitting problem is analyzed, with
special emphasis on the practical solution techniques when
the number of parameters in the model is large. An
extension of classical approaches to problems involving
many variables is proposed, that uses the concept of
partially separable structures. An adaptable algorithm is
discussed, which chooses between various possible models of
the objective function. Preliminary numerical experience
shows that the solution of a large class of fitting
problems involving several hundreds of nonlinear parameters
is possible at a reasonable cost.}}
@article{Toin87d,
author = {Ph. L. Toint},
title = {{VE10AD}, a routine for large scale nonlinear least squares},
journal = {Harwell Subroutine Library},
volume = 2, year = 1987}
@article{Toin88,
author = {Ph. L. Toint},
title = {Global convergence of a class of trust region methods for
nonconvex minimization in {H}ilbert space},
journal = IMAJNA,
volume = 8, number = 2, pages = {231--252}, year = 1988,
abstract = {A class of trust-region methods for solving constrained
optimization problems in Hilbert space is described. The
algorithms of the class use, at every iteration, a local
model of the objective, on which very weak conditions are
imposed. Global convergence results are then derived for
the class without assuming convexity of the objective
functional. It is also shown that convergence of the
classical projected-gradient method can be viewed as a
special case of this theory. An example is finally given
that points out some difficulties appearing when using
active-set strategies in infinite-dimensional spaces.},
summary = {A trust-region method for solving constrained optimization
problems in Hilbert space is described. Global convergence
results are derived without assuming convexity of the
objective functional. It is also shown that convergence of
the classical projected-gradient method can be viewed as a
special case of this theory. An example is given that
points out some difficulties appearing when using
active-set strategies in infinite-dimensional spaces.}}
@techreport{Toin94d,
author = {Ph. L. Toint},
title = {A non-monotone trust-region algorithm for nonlinear
optimization subject to convex constraints: the complete
numerical results},
institution = FUNDP, address = FUNDP-ADDRESS,
number = {94/26}, year = 1994}
%abstract = {The purpose of this paper is to detail the complete
% results of all test runs reported on in the companion
% paper \citebb{Toin96b}.},
@article{Toin96a,
author = {Ph. L. Toint},
title = {An assessment of non-monotone linesearch techniques for
unconstrained optimization},
journal = SISSC,
volume = 17, number = 3, pages = {725--739}, year = 1996}
%abstract = {The purpose of this paper is to discuss the potential of
% nonmonotone techniques for enforcing convergence of
% unconstrained minimization algorithms from starting points
% distant from the solution. Linesearch-based algorithms are
% considered for both small and large problems, and
% extensive numerical experiments show that this potential
% is sometimes considerable. A new variant is introduced in
% order to limit some of the identified drawbacks of the
% existing techniques. This variant is again numerically
% tested and appears to be competitive. Finally, the impact
% of preconditioning on the considered methods is examined.},
@article{Toin96b,
author = {Ph. L. Toint},
title = {A non-monotone trust-region algorithm for nonlinear
optimization subject to convex constraints},
journal = MP,
volume = 77, number = 1, pages = {69--94}, year = 1997,
abstract = {This paper presents two new trust-region methods for
solving nonlinear optimization problems over convex
feasible domains. These methods are distinguished by the
fact that they do not enforce strict monotonicity of the
objective function values at successive iterates. The
algorithms are proved to be convergent to critical points
of the problem from any starting point. Extensive numerical
experiments show that this approach is competitive with the
{\sf LANCELOT} package.},
summary = {Two trust-region methods for nonlinear optimization over
convex feasible domains are presented. These methods are
distinguished by the fact that they do not enforce strict
monotonicity of the objective function values at successive
iterates. The algorithms are proved to be convergent to
critical points of the problem from any starting point.
Extensive numerical experiments show that this approach is
competitive with {\sf LANCELOT}.}}
@techreport{TongZhou99,
author = {X. Tong and S. Zhou},
title = {A trust-region algorithm for nonlinear inequality
constrained optimization},
institution = {Department of Mathematics, Hunan University},
address = {Changsha, China},
number = {July}, year = 1999,
abstract = {This paper presents a new trust-region algorithm for
nonlinear optimization subject to nonlinear
inequality constraints. An equivalent KKT condition
is derived, which is the base of constructing the new
algorithm. A global convergence of the algorithm to a
first-order KKT point is established under mild
conditions on the trial step, and a local quadratic
convergence theorem is proved for nondegenerate
minimizer point.},
summary = {A trust-region algorithm for
nonlinear optimization subject to nonlinear
inequality constraints, based on an equivalent
reformulations of the KKT conditions, is presented.
Global convergence of the algorithm to a
first-order KKT point is established under mild
conditions on the trial step, and a local Q-quadratic
convergence rate is attainable at a nondegenerate
minimizer.}}
@misc{Tsen98,
author = {P. Tseng},
howpublished = {(private communication)},
month = {September}, year = 1998}
@techreport{Tsen99,
author = {P. Tseng},
title = {A Convergent Infeasible Interior-Point Trust-Region Method
for Constrained Optimization},
institution = {Department of Mathematics, University of Washington},
address = {Seattle, USA},
month = {May}, year = 1999,
abstract = {We study an infeasible interior-point trust-region method
for constrained optimization. This method uses a
logarithmic-barrier function for the slack variables and
updates the slack variables using a second-order correction.
We show that if a certain set containing the iterates is
bounded and the origin is not in the convex hull of the
nearly active constraint gradients everywhere on this set,
then any cluster point of the iterates is a 1st-order
stationary point. If the cluster point satisfies an
additional assumption (which holds when the constraints
are linear or when the cluster point satisfies strict
complementarity and a local error bound holds), then it is
a 2nd-order stationary point.},
summary = {A primal interior-point method is presented for the
inequality constrained nonlinear programming problem,
that allows for infeasible points. The method uses a
logarithmic barrier term for the slack variables and uses a
trust-region to find a step. The associated subproblem is
solved exactly. Convergence to first-order critical points
is proved, as well as convergence to second-order ones under
additional assumptions.}}
@article{TsenYamaFuku96,
author = {P. Tseng and N. Yamashita and M. Fukushima},
title = {Equivalence of complemetarity problems to differentiable
minimization: a unified approach},
journal = SIOPT,
volume = 6, number = 2, pages = {446--460}, year = 1996}
@inproceedings{TsioMjol96,
author = {D. I. Tsioutsias and E. Mjolsness},
title = {A multiscale attentional framework for relaxation neural
networks},
booktitle = {Advances in Neural Information Processing Systems.
Proceedings of the 1995 Conference},
editor = {D. S. Touretzky and M. C. Mozer and M. E. Hasselmo},
publisher = {MIT Press}, address = {Cambridge, MA, USA},
volume = 8, pages = {631--639}, year = 1996,
abstract = {We investigate the optimization of neural networks
governed by general objective functions. Practical
formulations of such objectives are notoriously
difficult to solve and a common problem is the poor
local extrema that result by any of the applied
methods. In this paper, a novel framework is
introduced for the solution of large-scale
optimization problems. It assumes little about the
objective function and can be applied to general
nonlinear, non-convex functions and objectives in
thousand of variables are thus efficiently minimized
by a combination of techniques-deterministic
annealing, multiscale optimization, attention
mechanisms and trust region optimization methods.},
summary = {The optimization of neural networks
governed by general objective functions is investigated.
A novel framework is introduced for the solution of
large-scale such problems. It assumes little about the
objective function and can be applied to general
nonlinear, non-convex functions and objectives in
thousand of variables are thus efficiently minimized
by a combination of techniques-deterministic
annealing, multiscale optimization, attention
mechanisms and trust-region optimization methods.}}
@article{Tsuc93,
author = {T. Tsuchiya},
title = {Global convergence of the affine scaling algorithm for
primal degenerate strictly convex quadratic programming
problems},
journal = AOR,
volume = 47, pages = {509--539}, year = 1993}
@book{Turn39,
author = {H. W. Turnbull},
title = {Theory of Equations},
publisher = {Oliver and Boyd}, address = {Edinburgh and London},
year = 1939}
%%% U %%%
@techreport{Ulbr99,
author = {M. Ulbrich},
title = {Non-monotone Trust-Region Methods for Bound-Constrained
Semismooth Equations with Applications to Nonlinear Mixed
Complementarity Problems},
institution = {Faculty of Mathematics, Technische Universit\"{a}t
M\"{u}nchen},
number = {TUM-M9906}, year = 1999,
abstract = {We develop and analyze a class of trust-region methods
for bound-constrained semismooth systems of equations.
The algorithm is based on a simply constrained
differentiable minimization reformulation. Our global
convergence results are developed in a very general
setting that allows for non-monotonicity of the function
values at subsequent iterates. we propose a way of
computing trial steps by a semismooth Newton-like method
that is augmented by a projection onto the feasible set.
Under a Dennis-Mor\'{e}-type condition we prove that close
to a BD-regular solution the trust-region algorithm turns
into this projected Newton method, which is shown to
converge locally q-superlinearly or quadratically,
respectively, depending on the quality of the
approximate BD-subdifferentials used. As an important
application we discuss in detail how the developed
algorithm can be used to solve nonlinear mixed
complementarity problems (MCPs). Hereby, the MCP is
converted into a bound-constrained semismooth equation
by means of an MCP function. We propose and investigate
a new class of MCP-functions that are motivated by
affine-scaling techniques for nonlinear programming.
These functions have attractive theoretical properties
and prove to be efficient in practice. This is
documented by our numerical results for a subset of
the MCPLIB problem collection.},
summary = {A class of trust-region methods for bound-constrained
semismooth systems of equations is developed, which is
based on a simply constrained differentiable minimization
reformulation. Global convergence results are proved
that allow for non-monotonicity of the function values
at successive iterates. Trial steps are computed by a
semismooth Newton-like method augmented by a projection
onto the feasible set. Under a suitable condition and close
to a regular solution, this technique turns into a projected
Newton method, which converges locally $Q$-superlinearly
or quadratically, depending on the quality of the
approximate subdifferentials used. An application of this
method to the solution of nonlinear mixed complementarity
problems (MCPs) is then discussed, where the MCP is
converted into a bound-constrained semismooth equation by
means of an MCP function. A new class of MCP-functions is
introduced, that is motivated by affine-scaling techniques
for nonlinear programming. Numerical results for a subset of
the MCPLIB problem collection illustrate the efficiency of
this approach.}}
@techreport{UlbrUlbr97,
author = {M. Ulbrich and S. Ulbrich},
title = {Superlinear Convergence of Affine-scaling Interior-point
{N}ewton Methods for Infinite-dimensional Problems with
Pointwise Bounds.},
institution = CAAM, address = RICE-ADDRESS,
number = {TR97-05}, year = 1997,
abstract = {We develop and analyze a superlinearly convergent
affine-scaling interior-point Newton method for
infinite-dimensional problems with pointwise bounds in
$L^p$-space. The problem formulation is motivated by
optimal control problems with $L^p$-controls and pointwise
control constraints. The finite-dimensional convergence
theory by \citebb{ColeLi96b} makes essential use of the
equivalence of norms and the exact identifiability of the
active constraints close to an optimizer with strict
complementarity. Since these features are not available in
our infinite-dimensional framework, algorithmic changes are
necessary to ensure fast local convergence. The main
building block is a Newton-like iteration for an
affine-scaling formulation of the KKT-condition. We
demonstrate in an example that a stepsize rule to obtain an
interior iterate may require very small stepsizes even
arbitrarily close to a nondegenerate solution. Using a
pointwise projection instead we prove superlinear
convergence under a weak strict complementarity condition
and convergence with $Q-$rate $>1$ under a slightly
stronger condition if a smoothing step is available. We
discuss how the algorithm can be embedded in the class of
globally convergent trust-region interior-point methods
recently developed by \citebb{UlbrUlbrHein99}. Numerical
results for the control of a heating process confirm our
theoretical findings.},
summary = {A superlinearly convergent affine-scaling interior-point
Newton method for infinite-dimensional problems with
pointwise bounds in $L^p$-space is analysed. The problem
formulation is motivated by optimal control problems with
$L^p$-controls and pointwise control constraints.
Adaptations are made to the proposal by \citebb{ColeLi96b}
for the infinite-dimensional setting. The main building
block is a Newton-like iteration for an affine-scaling
formulation of the KKT-condition. Using a pointwise
projection, superlinear convergence under a weak strict
complementarity condition and convergence with $Q-$rate
$>1$ under a slightly stronger condition if a smoothing
step is available are established. It is shown how the
algorithm can be embedded in the class of globally
convergent trust-region interior-point methods of
\citebb{UlbrUlbrHein99}. Numerical results for
the control of a heating process confirm the theoretical
findings.}}
@misc{UlbrUlbr99,
author = {S. Ulbrich and M. Ulbrich},
title = {Nonmonotone Trust Region Methods for Nonlinear Equality
Constrained Optimization Without a Penalty Function},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999,
abstract = {We propose and analyze a class of nonmonotone trust region
methods for nonlinear equality constrained optimization
problems. The algorithmic framework yields a global
convergence without using a merit function like the
augmented Lagrangian and allows nonmonotonicity
independently for both the constraint violation and the
objective function value. Similar to the augmented
Lagrangian-based algorithm by \citebb{DennElAlMaci97},
each step is composed of a quasi-normal and a tangential
step. Both steps are required to satisfy a fraction of
Cauchy decrease condition for their respective trust
region subproblems. Our mechanism for accepting steps
combines nonmonotone decrease conditions on the constraint
violation and/or the objective function, which leads to a
flexibility and acceptance behaviour compared to
filter-based methods. Preliminary numerical results for
the {\sf CUTE} test set confirm that our approach is very
promising. The proposed class of algorithms can be extended
in a natural way to multilevel trust region algorithms.
Hereby, the constraints are grouped in blocks and for each
block a normal step is computed that acts tangential to
the previous blocks. The generalization of our analysis
to this class of algorithms is making progress and we plan
to present convergence results also for this extended
framework. The extension of our concept to general NLP
is under development.},
summary = {A class of non-monotone trust-region methods for nonlinear
equality constrained optimization problems is proposed,
where each step is composed of a quasi-normal and a
tangential step. Both steps are required to satisfy a
fraction of Cauchy decrease condition for their respective
trust-region subproblems and the mechanism for accepting
them combines non-monotone decrease conditions on the
constraint violation and/or the objective function.
Preliminary numerical results show considerable promise.}}
@article{UlbrUlbrHein99,
author = {M. Ulbrich and S. Ulbrich and M. Heinkenschloss},
title = {Global convergence of trust-region interior-point
algorithms for infinite dimensional nonconvex minimization
subject to pointwise bounds},
journal = SICON,
volume = 37, number = 3, pages = {731--764}, year = 1999,
abstract = {A class of interior-point trust-region algorithms for
infinite-dimensional nonlinear optimization subject to
pointwise bounds in $L^p$-Banach spaces, $2 \leq p \leq
\infty$, is formulated and analyzed. The problem
formulation is motivated by optimal control problems, with
$L^p$-controled, and pointwise control constraints. The
interior-point trust-region algorithms are generalizations
of those recently introduced by \citebb{ColeLi96b} for
finite dimensional problems. Many of the generalizations
derived in this paper are also important in the finite
dimensional context. They lead to a better understanding of
the method and to considerable improvements in their
performance. All first- and second-order global convergence
results known for trust-region methods in the
finite-dimensional setting are extended to the
infinite-dimensional framework of this paper.},
summary = {Interior-point trust-region algorithms for
infinite-dimensional nonlinear optimization subject to
pointwise bounds in $L^p$-Banach spaces, $2 \leq p \leq
\infty$, are analyzed. The problem formulation is motivated
by optimal control problems, with $L^p$-controled, and
pointwise control constraints. The interior-point
trust-region algorithms are generalizations of those
introduced by \citebb{ColeLi96b} for finite-dimensional
problems. Many of the generalizations lead to a
better understanding of the methods and to considerable
improvements in their performance. All first- and
second-order global convergence results known
in the finite-dimensional setting are
extended to the infinite-dimensional framework.}}
@article{UlbrUlbrVice00,
author = {M. Ulbrich and S. Ulbrich and L. Vicente},
title = {A globally convergent primal-dual interior point filter
method for nonconvex nonlinear programming},
institution = COIMBRA, address = COIMBRA-ADDRESS,
number = {TR00-11}, year = 2000,
abstract = {In this paper, the filter technique of \citebb{FletLeyf97}
is used to globalize the primal-dual interior-point
algorithm for nonlinear programming, avoiding the use of
merit functions and the updating of penalty parameters.
The new algorithm decomposes the primal-dual step obtained
from the perturbed first-order necessary conditions into a
normal and a tangential step, whose sizes are controlled by
a trust-region parameter. Each entry in the filter is a
pair of coordinates: one resulting from feasibility and
centrality, and associated with the normal step; the other
resulting from optimality (complementarity and duality) and
related with the tangential step. Global convergence to
first-order critical points is proved for the new
primal-dual interior-point filter algorithm.},
summary = {The filter technique of \citebb{FletLeyf97} is used to
globalize the primal-dual interior-point algorithm for
nonlinear programming. The new algorithm decomposes the
primal-dual step obtained from the perturbed first-order
necessary conditions into a normal and a tangential step,
whose sizes are controlled by a trust-region parameter.
Each entry in the filter is a pair of coordinates: one
resulting from feasibility and centrality, and associated
with the normal step; the other resulting from optimality
(complementarity and duality) and related with the
tangential step. Global convergence to first-order
critical points is proved for the new primal-dual
interior-point filter algorithm.}}
@techreport{UrbaTitsLawr98,
author = {T. Urban and A. L. Tits and C. L. Lawrence},
title = {A primal-dual interior-point method for nonconvex
optimization with multiple logarithmic barrier parameters
and with strong convergence properties},
institution = {Electrical Engineering and the Institute for Systems
Research, University of Maryland},
address = {College Park, USA},
number = {TR 98-27}, year = 1998,
abstract = {It is observed that an algorithm proposed in the 1980s for
the solution of nonconvex constrained optimization problems
is in fact a primal-dual logarithmic barrier interior-point
method closely related to methods under current
investigation in the research community. Its main
distinguishing features are judicious selection and update
of the multiple barrier parameters (one per constraint),
use of the objective function as merit function, and a
careful bending of the search direction. As a pay-off,
global convergence and fast local convergence ensue. The
purpose of the present note is to describe the algorithm in
the interior-point framework and language and to provide a
preliminary numerical evaluation. The latter shows that the
method compares well with algorithms recently proposed by
other research groups.},
summary = {An algorithm proposed in 1988 by Panier, Tits and
Herskovits for the solution of non-convex constrained
optimization problems is shown to be a primal-dual
logarithmic barrier interior-point method. Its
distinguishing features are multiple barrier parameters,
the use of the objective function as a merit function, and
the bending of the search direction. Global convergence and
fast local convergence ensue. The algorithm is described in
the interior-point framework and preliminary numerical
results are discussed.}}
%%% V %%%
@article{VandBoyd96,
author = {L. Vandenberghe and S. Boyd},
title = {Semidefinite programming},
journal = SIREV,
volume = 38, number = 1, pages = {49--95}, year = 1996}
@article{VandWhin69,
author = {Van de Panne, C. and A. Whinston},
title = {The symmetric formulation of the simplex method for
quadratic programming},
journal = {Econometrica},
volume = 37, number = {}, pages = {507--527}, year = 1969}
@inproceedings{Vand85,
author = {J. S. Vandergraft},
title = {Efficient optimization methods for maximum likelihood
parameter estimation},
booktitle = {Proceedings of the 24th IEEE Conference on Decision and
Control},
publisher = {IEEE}, address = {New York, NY, USA},
volume = 3, pages = {1906--1909}, year = {1985},
abstract = {Recent research in numerical optimization has led to
development of efficient algorithms based on update
methods and model trust region techniques. The update
methods are a class of iterative schemes that avoid
expensive evaluations of (approximate) Hessians, yet
retain the rapid convergence properties of
Newton-like methods that require second-derivative
(Hessian) information. Model trust region techniques
avoid the costly step-length calculations that are
required by standard iterative methods based on
approximate Newton methods. The author describes the
most successful of the update methods and shows how
it, or more conventional methods such as scoring, can
be combined with a model trust region technique to
produce numerical algorithms that are ideally suited
to maximum-likelihood (ML) parameter
estimation. Specific properties of these algorithms
include a fast (superlinear) rate of convergence
together with the ability to handle parameter
constraints easily and efficiently.},
summary = {It is shown how the most successful of the
quasi-Newton methods and more conventional
methods such as scoring can be combined with
a trust-region technique to produce numerical
algorithms that are ideally suited to
maximum-likelihood parameter estimation. Specific
properties of these algorithms include a superlinear
rate of convergence and the ability to handle parameter
constraints easily and efficiently.}}
@book{VanHVand91,
author = {Van Huffel, S. and J. Vandewalle},
title = {The Total Least-Squares Problem: Computational Aspects and
Analysis},
publisher = SIAM, address = SIAM-ADDRESS,
number = 9, series = {Frontiers in Applied Nathematics}, year = 1991}
@techreport{Vand94,
author = {R. J. Vanderbei},
title = {{LOQO}: an interior point code for quadratic programming},
institution = {Program in Statistics and Operations,Research},
address = {Princeton University, New Jersey, USA},
type = {Technical Report}, number = {SOR 94-15}, year = 1994}
@techreport{VandShan97,
author = {R. J. Vanderbei and D. F. Shanno},
title = {An interior point algorithm for nonconvex nonlinear
programming},
institution = {Program in Statistics and Operations Research},
address = {Princeton University, New Jersey, USA},
type = {Technical Report}, number = {SOR 97-21}, year = 1997}
@article{Vard85,
author = {A. Vardi},
title = {A trust region algorithm for equality constrained
minimization: convergence properties and implementation},
journal = SINUM,
volume = 22, number = 3, pages = {575--591}, year = 1985,
abstract = {In unconstrained minimization, trust region algorithms use
directions that are a combination of the quasi-Newton
direction and the steepest descent direction, depending on
the fit between the quadratic approximation of the function
and the function itself. Algorithms for nonlinear
constrained optimization problems usually determine a
quasi-Newton direction and use a line search technique to
determine the step. Since trust region strategies have
proved to be successful in unconstrained minimization, we
develop a new trust region strategy for equality
constrained minimization. This algorithm is analyzed and
global as well as local superlinear convergence theorems
are proved for various versions. We demonstrate how to
implement this algorithm in a numerically stable way. A
computer program based on this algorithm has performed very
satisfactorily on test problems; numerical results are
provided.},
summary = {A trust-region strategy for equality constrained
minimization is developed. This algorithm is analyzed and
global as well as local superlinear convergence theorems
are proved. It is demonstrated how to implement this
algorithm in a numerically stable way.}}
@article{Vard92,
author = {A. Vardi},
title = {New Minimax Algorithms},
journal = JOTA,
volume = 75, number = 3, pages = {613--634}, year = 1992,
abstract = {The purpose of this paper is to suggest a new, efficient
algorithm for the nonlinear minimax problem. The problem
is transformed into an equivalent inequality-constraint
minimization problem. The algorithm has these features: an
active-set strategy with three types of constraints; the
use of slack variables to handle inequality constraints;
and a trust-region strategy taking advantage of the
structure of the problem. Following Tapia, this problem is
solved by an active set strategy which uses three types of
active constraints (called here nonactive, semiactive and
active). Active constraints are treated as equality
constraints and are assigned slack variables. This strategy
helps to prevent zigzagging. Numerical results are
provided.},
summary = {A algorithm for nonlinear min-max is described that
reformulates the min-max function as a set of inequality
constraints and uses an active-set and a trust-region
method that exploits the structure of the
problem. Numerical results are presented.}}
@book{Vava92,
author = {S. A. Vavasis},
title = {Nonlinear Optimization: Complexity Issues},
publisher = OUP, address = OUP-ADDRESS,
series = {International Series of Monographs on Computer Science},
year = 1992}
@article{Vava92b,
author = {S. A. Vavasis},
title = {Approximation algorithms for indefinite quadratic
programming},
journal = MP,
volume = 57, number = 2, pages = {279--311}, year = 1992}
@techreport{VavaZipp90,
author = {S. A. Vavasis and R. Zippel},
title = {Proving polynomial-time for sphere-constrained quadratic
programming},
institution = CS-CORNELL, address = CORNELL-ADDRESS,
number = {TR 90-1182}, year = 1990,
abstract = {Recently \citebb{Ye89} and Karmarkar have proposed similar
algorithms for minimizing a nonconvex quadratic function on
a sphere. These algorithms are based on trust-region work
going back to \citebb{Leve44} and \citebb{Marq63}. Although
both authors state that their algorithm is polynomial time,
neither makes estimates necessary to prove that conclusion
in a formal sense. In this report we derive estimates for
the convergence of the algorithm. Our estimates are based
on bounds for separation of roots of polynomials. These
bounds prove that the underlying decision problem is
polynomial time in the Turing machine sense.},
summary = {\citebb{Ye89} and Karmarkar have proposed similar
algorithms for minimizing a non-convex quadratic function on
a sphere. Estimates are derived for
the convergence of the algorithm, based on bounds for
separation of roots of polynomials. These bounds prove that
the underlying decision problem is polynomial time in the
Turing machine sense.}}
@phdthesis{Vice96,
author = {L. N. Vicente},
title = {Trust-region interior-point algorithms for a class of
nonlinear programming problems},
school = CAAM, address = RICE-ADDRESS,
year = 1995,
note = {Report TR96-05},
abstract = {This thesis introduces and analyzes a family of
trust-region interior-point (TRIP) reduced sequential
quadratic programming (SQP) algorithms for the solution of
minimization problems with nonlinear equality constraints
and simple bounds on some of the variables. These nonlinear
programming problems appear in applications in control,
design, parameter identification, and inversion. In
particular they often arise in the discretization of
optimal control problems. The TRIP reduced SQP algorithms
treat states and controls as independent variables. They
are designed to take advantage of the structure of the
problem. In particular they do not rely on matrix
factorizations of the linearized constraints, but use
solutions of the linearized state and adjoint equations.
These algorithms result from a successful combination of a
reduced SQP algorithm, a trust-region globalization, and a
primal-dual affine scaling interior-point method. The TRIP
reduced SQP algorithms have very strong theoretical
properties. It is shown in this thesis that they converge
globally to points satisfying first and second order
necessary optimality conditions, and in a neighborhood of a
local minimizer the rate of convergence is quadratic. Our
algorithms and convergence results reduce to those of
\citebb{ColeLi96b} for box-constrained optimization. An
inexact analysis is presented to provide a practical way of
controlling residuals of linear systems and directional
derivatives. Complementing this theory, numerical
experiments for two nonlinear optimal control problems are
included showing the robustness and effectiveness of these
algorithms. Another topic of this dissertation is a
specialized analysis of these algorithms for
equality-constrained optimization problems. The important
feature of the way this family of algorithms specializes
for these problems is that they do not require the
computation of normal components for the step and an
orthogonal basis for the null space of the Jacobian of the
equality constraints. An extension of \citebb{MoreSore83}'s
result for unconstrained optimization is presented, showing
global convergence for these algorithms to a point
satisfying the second-order necessary optimality conditions},
summary = {A family of trust-region interior-point (TRIP) reduced
sequential quadratic programming (SQP) algorithms, for the
solution of minimization problems with nonlinear equality
constraints and simple bounds, is introduced and analyzed.
These problems appear in control, design,
parameter identification, and inversion. In particular they
often arise in the discretization of optimal control
problems. The TRIP reduced SQP algorithms treat states and
controls as independent variables: they do not rely on
matrix factorizations of the linearized constraints, but
use solutions of the linearized state and adjoint
equations. These algorithms result from a combination of a
reduced SQP algorithm, a trust-region globalization, and a
primal-dual affine-scaling interior-point method. They
converge globally and quadratically to points satisfying
first and second order necessary optimality conditions. The
algorithms and convergence results reduce to those of
\citebb{ColeLi96b} for box-constrained optimization. An
inexact analysis is presented to provide a practical way of
controlling residuals of linear systems and directional
derivatives. Numerical experiments for two nonlinear
optimal control problems illustrate the robustness and
effectiveness of these algorithms. A specialized analysis
for equality-constrained problems shows that these
algorithms do not require the computation of normal
components for the step and an orthogonal basis for the
null space of the Jacobian of the equality constraints.}}
@article{Vice96b,
author = {L. N. Vicente},
title = {A comparison between line searches and trust regions for
nonlinear optimization},
journal = {Investiga\c{c}\~{a}o Operacional},
volume = 16, number = 2, pages = {173--179}, year = 1996,
abstract = {Line searches and trust regions are two techniques to
globalize nonlinear optimization algorithms. We claim that
the trust-region technique has built-in an appropriate
regularization of ill-conditioned second-order
approximation. The question we ask and then answer in this
short paper supports this claim. We force the trust-region
technique to act like a line search and we accomplish this
by always choosing the step along the quasi-Newton
direction. We obtain global convergence to a stationary
point as long as the condition number of the second-order
approximation is uniformly bounded, a condition that is
required in line searches but not in trust regions.},
summary = {It is claimed that the trust-region technique has built-in
an appropriate regularization of ill-conditioned
second-order approximation which is lacking in linesearch
methods. The trust-region technique is forced to act like a
linesearch by always choosing the step along the
quasi-Newton direction. Global convergence to a critical
point is obtained so long as the condition number of the
second-order approximation is uniformly bounded, a
condition required in linesearches but not in trust
regions.}}
@article{Voge90,
author = {C. R. Vogel},
title = {A constrained least squares regularization method for
nonlinear ill-posed problems},
journal = SICON,
volume = 28, number = 1, pages = {34--49}, year = 1990,
abstract = {This paper deals with a method for solving ill-posed,
nonlinear Hilbert space operator equations
$F(x)=y$. Regularization is obtained by solving a
constrained least squares regularization problem min
$\|F(x)-y\|_2$ subject to $J(x) \leq beta_2$.
$\beta$ serves as a regularization parameter, and
$J(x)$ is a quadratic penalty functional. To robustly
and efficiently solve this regularization problem,
the author applies a trust region method. At each
iteration, the quadratic penalty constraint is
retained, a Gauss-Newton approximation to the
objective functional is taken, and he adds a
quadratic trust region constraint. The resulting
quadratic subproblem is then reformulated as a
nonlinear complementarity problem and solved using
Newton's method. He then applies methods to find
approximate solutions to a severely ill-posed
nonlinear first kind integral equation arising in
geophysics. The method of generalized cross
validation is used to pick the regularization
parameter when random error is present in the
discrete data.},
summary = {A trust-region method is applied for regularizing
ill-posed, nonlinear Hilbert space operator equations.
The subproblem is reformulated as a
nonlinear complementarity problem and solved using
Newton's method. The method of generalized cross
validation is used to pick the regularization
parameter when random error is present in the
discrete data. The method is applied to find
approximate solutions to a severely ill-posed
nonlinear first kind integral equation arising in
geophysics.}}
%%% W %%%
@book{Wats80,
author = {G. A. Watson},
title = {Approximation Theory and Numerical Methods},
publisher = WILEY, address = WILEY-ADDRESS,
year = 1980}
@article{WatsBillMorg87,
author = {L. T. Watson and S. C. Billups and A. P. Morgan},
title = {{HOMPACK}: a suite of codes for globally convergent
homotopy algorithms},
journal = TOMS,
volume = 13, number = 3, pages = {281--310}, year = 1987}
@article{WatsKamReas85,
author = {L. T. Watson and M. P. Kamat and M. H. Reaser},
title = {A robust hybrid algorithm for computing multiple
equilibrium solutions},
journal = {Engineering Computations},
volume = 2, pages = {30--34}, year = 1985,
abstract = {A hybrid method is described that seeks to combine the
efficiency of a quasi-Newton method capable of locating
stable and unstable equilibrium configurations with a
robust homotopy method that is capable of tracking
equilibrium paths with turning points while exploiting
symmetry and sparsity of the Jacobian matrices. Numerical
results are presented for a shallow arch problem.},
summary = {A hybrid method is described that combines the efficiency
of a quasi-Newton method capable of locating stable and
unstable equilibrium configurations with a robust homotopy
method that is capable of tracking equilibrium paths with
turning points while exploiting symmetry and sparsity of
the Jacobian matrices. The quasi-Newton method uses a
double dogleg trust-region strategy. Numerical results are
presented for a shallow-arch problem.}}
@article{WeihCalzPana87,
author = {C. Weihs and G. Calzolari and L. Panattoni},
title = {The behavior of trust-region methods in {FIML}-estimation},
journal = {Computing},
volume = 38, number = 2, pages = {89--100}, year = 1987,
abstract = {This paper presents a Monte-Carlo study of the practical
reliability of numerical algorithms for FIML- estimation in
nonlinear econometric models. The performance of different
techniques of Hessian approximation in trust-region
algorithms is compared regarding their ``robustness''
against ``bad'' starting points and their ``global'' and
``local'' convergence speed, i.e. the gain in the objective
function caused by individual iteration steps far off from
and near to the optimum. Concerning robustness and global
convergence speed, the crude GLS-type Hessian
approximations performed best, efficiently exploiting the
special structure of the likelihood function. But,
concerning local speed, general purpose techniques were
strongly superior. So, some appropriate mixture of these
two types of approximations turned out to be the only
techniques to be recommended.},
summary = {The reliability of numerical algorithms for FIML-estimation
in nonlinear econometric models is explored by a
Monte-Carlo study. Techniques of Hessian approximation in
trust-region algorithms are compared regarding their
robustness and their global and local convergence speed.
Concerning robustness and global convergence speed, the
crude GLS-type Hessian approximations performed best,
efficiently exploiting the special structure of the
likelihood function. Concerning local speed, general
purpose techniques were strongly superior. Some appropriate
mixture of these two types of approximations is
recommended.}}
@book{Wilk63,
author = {J. H. Wilkinson},
title = {Rounding Errors in Algebraic Processes},
publisher = {Her Majesty's Stationery Office}, address = {London},
year = 1963}
@book{Wilk65,
author = {J. H. Wilkinson},
title = {The Algebraic Eigenvalue Problem},
publisher = OUP, address = OUP-ADDRESS,
year = 1965}
@inproceedings{Wilk68,
author = {J. H. Wilkinson},
title = {A priori error analysis of algebraic processes},
booktitle = {Proceedings of the International Congress of Mathematicians},
editor = {I. G. Petrovsky},
publisher = {Mir Publishers}, address = {Moscow, USSR},
pages = {629--640}, year = 1968}
@article{Will64,
author = {J. W. J. Williams},
title = {Algorithm 232, {H}eapsort},
journal = {Communications of the ACM},
volume = 7, pages = {347--348}, year = 1964}
@techreport{Will90,
author = {K. A. Williamson},
title = {A Robust Trust Region Algorithm for Nonlinear Programming},
institution = CAAM, address = RICE-ADDRESS,
number = {TR90-22}, year = 1990,
abstract = {This work develops and tests a trust region algorithm for
the nonlinear equality constrained optimization problem.
Our goal is to develop a robust algorithm that can handle
lack of second-order sufficiency away from the solution in
a natural way. \citebb{CeliDennTapi85} give a trust region
algorithm for this problem, but in certain situations their
trust region subproblem is too difficult to solve. The
algorithm given here is based on the restriction of the
trust region subproblem given by \citebb{CeliDennTapi85} to
a relevant two dimensional subspace. This restriction
greatly facilitates the solution of the subproblem. The
trust region subproblem that is the focus of this work
requires the minimization of a possibly non-convex
quadratic subject to two quadratic constraints in two
dimensions. The solution of this problem requires the
determination of all the global solutions, and the
non-global solution, if it exists, to the standard
unconstrained trust region subproblem. Algorithms for
approximating a single global solution to the unconstrained
trust region subproblem have been well-established.
Analytical expressions for all of the solutions will be
derived for a number of special cases, and necessary and
sufficient conditions are given for the existence of a
non-global solution for the general case of the
two-dimensional unconstrained trust region subproblem.
Finally, numerical results are presented for a preliminary
implementation of the nonlinear programming algorithm, and
these results verify that it is indeed robust.},
summary = {A variant of the trust-region algorithm by
\citebb{CeliDennTapi85} for general nonlinear programming
is developed that uses a two-dimensional subproblem. This
subproblem consists of globally minimizing a possibly
non-convex quadratic subject to two quadratic constraints
in two dimensions. A detailed study of this subproblem is
supplied for a number of special cases. Preliminary
numerical experiments illustrate the robustness of the
resulting algorithm.}}
@phdthesis{Wils63,
author = {R. B. Wilson},
title = {A simplicial algorithm for concave programming},
school = {Harvard University}, address = {Massachusetts, USA},
year = 1963}
@phdthesis{Winf69,
author = {D. Winfield},
title = {Function and functional optimization by interpolation in
data tables},
school = {Harvard University}, address = {Cambridge, USA},
year = 1969}
@article{Winf73,
author = {D. Winfield},
title = {Function Minimization by Interpolation in a Data Table},
journal = JIMA,
volume = 12, pages = {339--347}, year = 1973,
abstract = {A method is described for unconstrained function
minimization using function values and no derivatives. A
quadratic model of the function is formed by interpolation
to points in a table of function values. The quadratic
model (not necessarily positive definite) is minimized over
a constraining region of validity to locate the next trial
point. The points of interpolation are chosen from a data
table containing function values at an initial grid and at
subsequent trial points. The method is efficient in its use
of function evaluations, but expensive in computation
required to choose new trial points.},
summary = {A method is described for unconstrained function
minimization using function values and no derivatives. A
quadratic model of the function is formed by interpolation
to points in a table of function values. The quadratic
model (not necessarily positive definite) is minimized over
a constraining region of validity to locate the next trial
point. The points of interpolation are chosen from a data
table containing function values at an initial grid and at
subsequent trial points. The method is efficient in its use
of function evaluations, but expensive in computation
required to choose new trial points.}}
@book{Wlok87,
author = {J. Wloka},
title = {Partial Differential Equations},
publisher = CUP, address = CUP-ADDRESS,
year = 1987}
@article{Wolf59,
author = {P. Wolfe},
title = {The {S}implex method for quadratic programming},
journal = {Econometrica},
volume = 27, pages = {382--398}, year = 1959}
@article{Wome82,
author = {R. S. Womersley},
title = {Optimality conditions for piecewise smooth functions},
journal = MPS,
volume = 17, pages = {13-27}, year = 1982}
@book{WonnWonn90,
author = {T. H. Wonnacott and R. J. Wonnacott},
title = {Introductory Statistics},
publisher = WILEY, address = WILEY-ADDRESS,
edition = {fifth}, year = 1990}
@phdthesis{Wrig76,
author = {M. H. Wright},
title = {Numerical methods for nonlinearly constrained optimization},
school = STANFORD, address = STANFORD-ADDRESS,
year = 1976}
@article{Wrig92,
author = {M. H. Wright},
title = {Interior methods for constrained optimization},
journal = {Acta Numerica},
volume = 1, pages = {341--407}, year = 1992}
@article{Wrig95,
author = {M. H. Wright},
title = {Why a pure primal {N}ewton barrier step may be infeasible},
journal = SIOPT,
volume = 5, number = 1, pages = {1--12}, year = 1995}
@inproceedings{Wrig98b,
author = {M. H. Wright},
title = {The interior-point revolution in constrained optimization},
crossref = {DeLeMurlPardTora98}, pages = {359--381}}
@article{Wrig99,
author = {M. H. Wright},
title = {Ill-conditioning and computational error in interior
methods for nonlinear programming},
journal = SIOPT,
volume = 9, number = 1, pages = {84--111}, year = 1999}
@article{WrigHolt85,
author = {S. J. Wright and J. N. Holt},
title = {Algorithms for nonlinear least squares with linear
inequality constraints},
journal = SISSC,
volume = 6, number = 4, pages = {1033--1048}, year = 1985,
abstract = {Two algorithms for solving nonlinear least squares
problems with general linear inequality constraints
are described. At each step, the problem is reduced
to an unconstrained linear least squares problem in
the subspace defined by the active constraints, which
is solved using the Levenberg-Marquardt method. The
desirability of leaving an active constraint is
evaluated at each step, using a different technique
for each of the two algorithms. Each step is
constrained to be within a circular region of trust
about the current approximate minimiser, whose radius
is updated according to the quality of the step after
each iteration. Comparisons of the relative
performance of the two algorithms on small problems
and on a larger exponential data-fitting problem are
presented.},
summary = {Two algorithms for solving nonlinear least squares
problems with general linear inequality constraints
are described. At each step, the problem is reduced
to an unconstrained linear least squares problem in
the subspace defined by the active constraints, which
is solved using the Levenberg-Morrison-Marquardt method. The
desirability of leaving an active constraint is
evaluated at each step, using a different technique
for each of the two algorithms. Comparisons of the relative
performance of the two algorithms on small problems
and on a larger exponential data-fitting problem are
presented.}}
@article{Wrig87,
author = {S. J. Wright},
title = {Local properties of inexact methods for minimizing
nonsmooth composite functions},
journal = MP,
volume = 37, number = 2, pages = {232--252}, year = 1987}
@article{Wrig89,
author = {S. J. Wright},
title = {Convergence of {SQP}-like methods for constrained
optimization},
journal = SICON,
volume = 27, number = 1, pages = {13--26}, year = 1989}
@article{Wrig89b,
author = {S. J. Wright},
title = {An inexact algorithm for composite nondifferentiable
optimization},
journal = MP,
volume = 44, number = 2, pages = {221--234}, year = 1989,
abstract = {We describe an inexact version of \citebb{Flet87}'s $QL$
algorithm with second-order corrections for
minimizing composite nonsmooth functions. The method
is shown to retain the global and local convergence
properties of the original version, if the parameters
are chosen appropriately. It is shown how the inexact
method can be implemented for the case in which the
function to be minimized is an exact penalty function
arising from the standard nonlinear programming
problem. The method can also be applied to the
problems of nonlinear $\ell_1$- and $\ell_{\infty}$-
approximation.},
summary = {An inexact version of \citebb{Flet87}'s $QL$
algorithm with second-order corrections for
minimizing composite non-smooth functions, which
retains the global and local convergence properties
of the original version, is given. It is shown how
the inexact method can be implemented for exact
penalty functions arising from nonlinear programming
problems, as well as problems of nonlinear $\ell_1$-
and $\ell_{\infty}$- approximation.}}
@article{Wrig90,
author = {S. J. Wright},
title = {Convergence of an inexact algorithm for composite nonsmooth
optimization},
journal = IMAJNA,
volume = 10, number = 3, pages = {299--321}, year = 1990,
abstract = {This paper describes an inexact version of \citebb{Flet82}'s
second-order correction algorithm for minimizing
composite nondifferentiable functions, and adds a
test which allows global convergence to be
demonstrated without the assumption that a global
minimum of the model function is found at each
iteration. Implementable criteria for accepting
inexact solutions of the subproblem, while retaining
local convergence properties, are also given.},
summary = {An inexact version of \citebb{Flet82}'s
second-order correction algorithm for minimizing
composite non-differentiable functions is described. A
test is suggested, which allows global convergence to be
proved without the assumption that a global
minimum of the model function is found at each
iteration. Implementable criteria for accepting
inexact solutions of the subproblem, while retaining
local convergence properties, are also given.}}
@book{Wrig97,
author = {S. J. Wright},
title = {Primal-Dual Interior-Point Methods},
publisher = SIAM, address = SIAM-ADDRESS,
year = 1997}
@techreport{Wrig98,
author = {S. J. Wright},
title = {Effects of finite-precision arithmetic on interior-point
methods for nonlinear programming},
institution = ANL, address = ANL-ADDRESS,
number = {MCS-P705-0198}, year = 1998}
@article{Wrig99b,
author = {S. J. Wright},
title = {Superlinear convergence of a stabilized {SQP} method to a
degenerate solution},
journal = COAP,
volume = 11, number = 3, pages = {253--275}, year = 1999}
@misc{WrigOrba99,
author = {S. J. Wright and D. Orban},
title = {Properties of the Log-Barrier Function for Degenerate
Nonlinear Programs},
howpublished = {Presentation at the First Workshop on Nonlinear
Optimization ``Interior-Point and Filter Methods'',
Coimbra, Portugal},
year = 1999}
@article{Wome85,
author = {R. S. Womersley},
title = {Local properties of algorithms for minimizing nonsmooth
composite functions},
journal = MP,
volume = 32, number = 1, pages = {69--89}, year = 1985}
%%% X %%%
@article{XiaoZhou92,
author = {Y. Xiao and F. Zhou},
title = {Nonmonotone Trust Region Methods with Curvilinear Path in
Unconstrained Optimization},
journal = {Computing},
volume = 48, number = {3--4}, pages = {303--317}, year = 1992,
abstract = {A general nonmonotone trust region method with curvilinear
path for unconstrained optimization problem is presented.
Although this method allows the sequence of objective
function values to be nonmonotone, convergence properties
similar to those of the usual trust region methods with
curvilinear path are proved under certain conditions.
Some numerical results are reported which show the
superiority of the nonmonotone trust region method
with respect to the numbers of gradient evaluations and
function evaluations.},
summary = {A non-monotone trust-region algorithm is proposed for
unconstrained optimization, whose convergence properties
are similar to those of the monotone versions. Numerical
experiments show the potential benefits of the approach.}}
@techreport{XiaoChu95,
author = {Y. Xiao and E. K. W. Chu},
title = {Nonmonotone Trust Region Methods},
institution = {Monash University}, address = {Clayton, Australia},
number = {95/17}, year = 1995,
abstract = {In this report, we study non-monotone techniques in trust
region methods for unconstrained minimization problems. Two
non-monotone trust region methods are developed and
assessed based on extensive numerical experiments with {\sf
CUTE}. Strategies for automatic selection and adjustment of
parameters are discussed, which enable switching between
non-monotone and monotone algorithms at different stages of
calculation according to the intermediate information
obtained. Numerical results show that these strategies
improve the efficiency and act as safeguards against the
possible inferior behavior of non-monotone algorithms. In
addition, the global convergence of the algorithms are
proved, and further modifications and possible improvements
are discussed.},
summary = {Two non-monotone trust-region methods are developed and
assessed based on extensive numerical experiments with {\sf
CUTE}. Strategies for automatic adjustment of parameters
are discussed, which enable switching between non-monotone
and monotone algorithms at different stages of calculation
according to the intermediate information obtained.
Numerical results show that these strategies improve the
efficiency of non-monotone algorithms. Global convergence
of the algorithms is proved, and further modifications
discussed.}}
@phdthesis{Xiao96,
author = {Y. Xiao},
title = {Non-monotone algorithms in optimization and their
applications},
school = {Monash University}, address = {Clayton, Australia},
year = 1996}
@article{XuZhan95,
author = {C. Xu and J. Zhang},
title = {An active set method for general $\ell_1$ linear problem
subject to box constraints},
journal = {Optimization},
volume = 34, number = 1, pages = {67--80}, year = 1995,
abstract = {An active set algorithm is presented for the solution
of general $\ell_1$ linear problem with simple bound
constraints on variables. These problems appear as
subproblems when trust region type linear
approximation methods are used to minimize an
unconstrained nonsmooth composite function. The
method finds an optimal solution among dead points of
the problem and eventually terminates at an optimal
solution in a finite number of steps.},
summary = {An active set algorithm is presented for the solution
of general $\ell_1$ linear problem with simple bound
constraints on its variables.}}
@article{XuZhan99,
author = {C. Xu and J. Zhang},
title = {A Scaled Optimal Path Trust Region Algorithm},
journal = JOTA,
volume = 102, number = 1, year = 1999,
abstract = {Optimal path trust algorithm intends to determine a
trajectory along which the solution to a trust region
subproblem at a given point with any trust region radius is
located. Although its idea is attractive, the existing
optimal path method seems impractical because it requires,
in addition to a factorization, the calculation of full
eigensystem of the working matrix. We propose a scaled
optimal path trust region algorithm which finds a solution
of the subproblem in full dimensional space by just one
\citebb{BuncParl71} factorization for working matrix at
each iteration and by using the resulting unit lower
triangular factor to scale the variables in the problem. A
scaled optimal path can then be formed easily. The
algorithm has good convergence properties under commonly
used conditions. Computational results are presented to
show that this algorithm is robust and effective.},
summary = {A scaled optimal path trust-region algorithm is proposed,
which finds a solution of the subproblem in
full-dimensional space by just one \citebb{BuncParl71}
factorization and by using the resulting
unit lower triangular factor to scale the variables.
The resulting algorithm has good convergence
properties. Computational results show that this algorithm
is robust and effective.}}
%%% Y %%%
@article{YabeYama97,
author = {H. Yabe and H. Yamashita},
title = {Q-superlinear Convergence of Primal-Dual Interior Point
Quasi-{N}ewton Methods for Constrained Optimization},
journal = {Journal of the Operations Research Society of Japan},
volume = 40, number = 3, pages = {415--436}, year = 1997}
@article{YamaFukuIbar89,
author = {E. Yamakawa and M. Fukushima and T. Ibaraki},
title = {An efficient trust region algorithm for minimizing
nondifferentiable composite functions},
journal = SISSC,
volume = 10, number = 3, pages = {562--580}, year = 1989,
abstract = {This paper presents a trust region method for solving the
following problem. Minimize $\phi(x)=f(x)+c(x(x))$ over
$x \in \Re^n$, where $f$ and $c$ are smooth functions and
$h$ is a polyhedral convex function. Problems of this form
include various important applications such as min-max
optimization, Chebychev approximation and minimization of
exact penalty functions in nonlinear programming. The
algorithm is an adaptation of a recently proposed
successive quadratic programming method for nonlinear
programming and makes use of second-order approximations
to both $f$ and $c$ in order to avoid the Maratos effect.
It is proved under appropriate assumptions that the
algorithm is globally and quadratically convergent to a
solution of the problem. Some numerical results
exhibiting the effectiveness of the algorithm are also
reported.},
summary = {A trust-region method for solving the problem of
minimizing $\phi(x)=f(x)+h(c(x))$ over
$x \in \Re^n$, where $f$ and $c$ are smooth functions and
$h$ is a polyhedral convex function. The
algorithm is an adaptation of the SQP method by
\citebb{Flet82b} and makes use of second-order approximations
to both $f$ and $c$ in order to avoid the Maratos effect.
Global and quadratic convergence is proved.
Numerical results illustrate the effectiveness of
the algorithm.}}}
@article{Yama82,
author = {H. Yamashita},
title = {A globally convergent constrained quasi-{N}ewton method
with an augmented {L}agrangian type penalty-function},
journal = MP,
volume = 23, number = 1, pages = {75--86}, year = 1982}
@techreport{YamaYabe96,
author = {H. Yamashita and H. Yabe},
title = {Nonmonotone {SQP} methods with global and superlinear
convergence properties},
institution = {Mathematical Systems, Inc.},
address = {Sinjuku-ku, Tokyo, Japan},
year = 1996}
@article{YamaYabe96b,
author = {H. Yamashita and H. Yabe},
title = {Superlinear and quadratic convergence of some primal-dual
interior point methods for constrained optimization},
journal = MPA,
volume = 75, number = 3, pages = {377--397}, year = 1996}
@techreport{YamaYabeTana97,
author = {H. Yamashita and H. Yabe and T. Tanabe},
title = {A Globally and superlineraly convergent primal-dual point
trust region method for large scale constrained
optimization},
institution = {Mathematical Systems, Inc.},
address = {Sinjuku-ku, Tokyo, Japan},
year = 1997,
abstract = {This paper proposes a primal-dual interior point method for
solving large scale nonlinearly constrained optimization
problems. To solve large scale problems, we use a trust
region method that uses second derivatives of functions for
minimizing the barrier-penalty function instead of the
usual line search strategies. By carefully controlling
parameters in the algorithm, superlinear convergence of the
iteration is also proved. A nonmonotone strategy is adopted
to avoid Maratos effect as in the nonmonotone SQP method of
\citebb{YamaYabe96}. The method is implemented and tested
with a variety of problems given by Hock and Schittkowski's
book and by {\sf CUTE}. The results of our numerical
experiment show that the given method is efficient for
solving large scale nonlinearly constrained optimization
problems.},
summary = {A primal-dual interior-point method is proposed, where the
barrier function is minimized by a trust-region method that
uses second derivatives. Superlinear convergence is proved.
A non-monotone strategy is adopted to avoid the Maratos
effect as in \citebb{YamaYabe96}. Results are reported for
a variety of problems given by Hock and Schittkowski and by
{\sf CUTE}. }}
@article{YamaFuku95,
author = {N. Yamashita and M. Fukushima},
title = {On Stationary Pointss of the Implicit {L}agrangian for
Nonlinear Complementarity Problems},
journal = JOTA,
volume = 84, number = 3, pages = {653--663}, year = 1995}
@techreport{YangLiZhou98,
author = {Y. Yang and D. Li and S. Zhou},
title = {A trust region method for a semismooth reformulation to
variational inequality problems},
institution = {Department of Applied Mathematics, Hunan University},
address = {Changsha, China},
number = {May, 15}, year = 1998,
abstract = {We consider the variational inequality problem (denoted
by $VI(X,F)$): find an $x^* \in X$ such that
$F(x^*)^T(x-x^*)\geq 0, \forall x \in X$, where
$F:\Re^n \rightarrow \Re^n$ is continuously differentiable
and the set $X$ has the following form
$X:= \{x \in \Re^n \mid g(x) \geq 0, h(x) = 0 \}$ where
$g: \Re^n \rightarrow \Re^m$ and
$h: \Re^n \rightarrow \Re^l$ are twice continuously
differentiable. The KKT system of the above $VI(X,F)$ is
\[F(x)-g'(x)^Ty + h'(x)^Tz = 0,\]
\[ h(x) = 0,\ms g(x) \geq 0,\ms y \geq 0,\ms y^Tg(x) = 0.\]
In this paper, we present a well-defined trust region
method for solving this system based on its semismooth
reformulation. The proposed method solves subproblems
inexactly. We show that the proposed method converges
globally without monotone assumption on the function $F$.
Moreover, the rate of convergence is
Q-superlinear/Q-quadratic even if strict complementarity
does not hold at the solution.},
summary = {A trust region method is proposed for solving general
nonlinearly constrained variational inequality problems.
It is based on the semi-smooth reformulation of the
first-order optimality conditions, allows for inexact
solution of subproblems and is globally convergent even in
the non-monotone case. Its rate of convergence is
Q-superlinear or Q-quadratic even without strict
complementarity.}}
@article{YangToll91,
author = {E. K. Yang and J. W. Tolle},
title = {A class of methods for solving large, convex quadratic
programs subject to box constraints},
journal = MP,
volume = 51, number = 2, pages = {223--228}, year = 1991}
@article{YangZhanYou96,
author = {B. Yang and K. Zhang and Z. You},
title = {A successive quadratic programming method that uses new
corrections for search directions},
journal = JCM,
volume = 71, number = 1, pages = {15--31}, year = 1996}
@inproceedings{Ye89,
author = {Y. Ye},
title = {An extension of {K}amarkar's algorithm and the trust region
method for quadratic-programming},
booktitle = {Progress in Mathematical Programming},
editor = {N. Megiddo},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
pages = {49--63}, year = 1989,
abstract = {An extension of \citebbs{Karm84} algorithm and the trust
region method is developed for solving quadratic
programming problems. This extension is based on the affine
scaling technique, followed by optimization over a trust
ellipsoidal region. It creates a sequence of interior
feasible points that converge to the optimal feasible
solution. The initial computational results reported here
suggest the potential usefulness of this algorithm in
practice.},
summary = {An extension of \citebbs{Karm84} algorithm and the
trust-region method is developed for solving quadratic
programming problems. It is based on the affine scaling
technique, followed by optimization over a trust
ellipsoidal region, and creates a sequence of interior
feasible points that converge to the optimal solution.
Computational results suggest its potential usefulness.}}
@article{YeTse89,
author = {Y. Ye and E. Tse},
title = {An extension of {K}armarkar's projective algorithm for
convex quadratic programming},
journal = MP,
volume = 44, number = 2, pages = {157--179}, year = 1989}
@article{Ye92,
author = {Y. Ye},
title = {On an affine scaling algorithm for nonconvex quadratic
programming},
journal = MP,
volume = 56, pages = {285--300}, year = 1992,
abstract = {We investigate the use of interior algorithms, especially
the affine-scaling algorithm, to solve
nonconvex---indefinite or negative definite---quadratic
programming
(QP) problems. Although the nonconvex QP with a polytope
constraint is a "hard" problem, we show that the problem
with an ellipsoidal constraint is "easy". When the "hard"
QP is solved by successively solving the "easy" QP, the
sequence of points monotonically converge to a feasible
point satisfying both the first and the second order
optimality conditions.},
summary = {The use of interior algorithms, especially the
affine-scaling algorithm, to solve non-convex---indefinite or
negative definite---quadratic programming (QP) problems is
investigated. Although the non-convex QP with a polytope
constraint is a ``hard'' problem, it is shown that the
problem with an ellipsoidal constraint is ``easy''. When the
``hard'' QP is solved by successively solving the "easy" QP,
the sequence of points monotonically converge to a feasible
point satisfying both the first and the second order
optimality conditions.}}
@techreport{Ye97,
author = {Y. Ye},
title = {Approximating quadratic programming with bound constraints},
institution = {Department of Management Sciences},
address = {The University of Iowa, USA},
type = {Working Paper}, year = 1997,
abstract = {We consider the problem of approximating the global maximum
of a quadratic program with $n$ variables subject to bound
constraints. Based on the results of Goemans and Williamson
(1996) and Nesterov (1997), we show that a $4/7$
approximate solution can be obtained in polynomial time},
summary = {The problem of approximating the global maximum of a
quadratic program with $n$ variables subject to bound
constraints is considered. Based on the results of Goemans
and Williamson (1996) and Nesterov (1997), it is shown that
a $4/7$ approximate solution can be obtained in polynomial
time.}}
@misc{YinHan98,
author = {H. Yin and J. Han},
title = {A new interior-point trust-region algorithm for nonlinear
minimization problems with simple bound constraints},
howpublished = {Presentation at the International Conference on Nonlinear
Programming and Variational Inequalities, Hong Kong},
year = 1998,
abstract = {In the paper, a new interior-point trust-region algorithm
is given for nonlinear minimization problems with simple
bound constraints. The objective function of the quadratic
subproblem model is obtained based on the Newton step for
the first-order KKT condition of the problem, and the
constraints are a sphere trust-region constraints and a
system of bound constraints. We accept or reject the trial
steps only depending on the objective function and its
approximation. The global convergence to a KKT point is
obtained for our algorithm under some mild conditions.
The second-order convergence is also hold under some
assumptions},
summary = {In the method of \citebb{ColeLi96b}, feasibility of the
trial step is obtained by backtracking from a possibly
infeasible step into the interior of the feasible region.
A variant of this method is discussed, in which the initial
trial step is not allowed to be infeasible, therefore
avoiding the need of backtracking.}}
@book{Yosi70,
author = {K. Yosida},
title = {Functional Analysis},
publisher = SPRINGER, address = SPRINGER-ADDRESS,
year = 1970}
@techreport{Yuan83,
author = {Y. Yuan},
title = {Global convergence of trust region algorithms for nonsmooth
optimization},
institution = DAMTP, address = DAMTP-ADDRESS,
number = {DAMTP/NA13}, year = 1983,
summary = {See \citebb{Yuan85a}.}}
@article{Yuan84,
author = {Y. Yuan},
title = {An example of only linear convergence of trust region
algorithms for nonsmooth optimization},
journal = IMAJNA,
volume = 4, number = 3, pages = {327--335}, year = 1984,
abstract = {Most superlinear convergence results about trust-region
algorithms for non-smooth optimization are dependent on the
inactivity of trust region restrictions. An example is
constructed to show that it is possible that at every
iteration the trust region bound is active and the rate of
convergence is only linear, though strict complementarity
and second order sufficiency conditions are satisfied.},
summary = {An example is constructed where, at every iteration of a
trust-region method for non-smooth optimization the
trust-region bound is active and the rate of convergence is
only linear, though strict complementarity and second order
sufficiency conditions hold.}}
@article{Yuan85a,
author = {Y. Yuan},
title = {Conditions for convergence of trust region algorithms for
nonsmooth optimization},
journal = MP,
volume = 31, number = 2, pages = {220--228}, year = 1985,
abstract = {This paper discusses some properties of trust-region
algorithms for nonsmooth optimization. The problem is
expressed as the minimization of a function $h(f(x))$ where
$h(\cdot)$ is convex and where $f$ is a continuously
differentiable mapping from $\Re^n$ to $\Re^n$. Bounds for
the second order derivative approximation matrices are
discussed. It is shown that the results of \citebb{Powe75}
and \citebb{Powe84} hold for nonsmooth optimization.},
summary = {Properties of trust-region algorithms for non-smooth
optimization are discussed. The problem is expressed as the
minimization of a function $h(f(x))$ where $h(\cdot)$ is
convex and where $f$ is a continuously differentiable
mapping from $\Re^n$ to $\Re^n$. It is shown that the
results of \citebb{Powe75} and \citebb{Powe84} hold for
non-smooth optimization.}}
@article{Yuan85b,
author = {Y. Yuan},
title = {On the superlinear convergence of a trust region algorithm
for nonsmooth optimization},
journal = MP,
volume = 31, number = 3, pages = {269--285}, year = 1985,
abstract = {It is proved that the second-order correction trust-region
algorithm of \citebb{Flet82} ensures superlinear
convergence if some mild conditions are satisfied.},
summary = {It is proved that the second-order correction trust-region
algorithm of \citebb{Flet82} ensures superlinear
convergence if some mild conditions are satisfied.}}
@article{Yuan90,
author = {Y. Yuan},
title = {On a Subproblem of Trust Region Algorithms for Constrained
Optimization},
journal = MP,
volume = 47, number = 1, pages = {53--63}, year = 1990,
abstract = {We study a subproblem that arises in some trust region
algorithms for equality constrained optimization. It is the
minimization of a general quadratic function with two
special quadratic constraints. Properties of such
subproblems are given. It is proved that the Hessian of the
Lagrangian has at most one negative eigenvalue, and an
example is presented to show that the Hessian may have a
negative eigenvalue when one constraint is inactive at the
solution.},
summary = {A subproblem that arises in some trust-region algorithms
for equality constrained optimization is studied. It is the
minimization of a general quadratic function with two
special quadratic constraints. Properties of such
subproblems are given. It is proved that the Hessian of the
Lagrangian has at most one negative eigenvalue, and an
example is presented to show that the Hessian may have a
negative eigenvalue when one constraint is inactive at the
solution.}}
@article{Yuan91,
author = {Y. Yuan},
title = {A dual algorithm for minimizing a quadratic function with
two quadratic constraints},
journal = {Journal of Computational Mathematics},
volume = 9, number = 4, pages = {348--359}, year = 1991,
abstract = {In this paper, we present a dual algorithm for minimizing
a convex quadratic function with two quadratic constraints.
Such a minimization problem is a subproblem that appears in
some trust region algorithms for general nonlinear
programming. Some theoretical properties of the dual
problem are given. Global convergence of the algorithm
is proved and a local superlinear result is presented.
Numerical examples are also provided.},
summary = {A dual globally and superlinearly convergent algorithm is
proposed for minimizing a convex quadratic objective
subject to two quadratic constraints. Numerical examples
are provided.}}
@inproceedings{Yuan93,
author = {Y. Yuan},
title = {A new trust-region algorithm for nonlinear optimization},
booktitle = {Proceedings of the First International Colloquium on
Numerical Analysis},
editor = {D. Bainov and V. Covachev},
publisher = {VSP}, address = {Zeist, The Netherlands},
pages = {141--152}, year = 1993,
abstract = {Trust region algorithms are a class of numerical algorithms
for optimization. In this paper we present a new trust
region algorithm for general nonlinear constrained
optimization problems. The algorithm is bases on the
$\ell_{\infty}$ exact penalty function. Under very mild
conditions, global convergence results for the algorithm
are given.},
summary = {Trust-region algorithms are a class of numerical algorithms
for optimization. A trust-region algorithm for general
nonlinear constrained optimization problems is presented.
The algorithm is bases on the $\ell_{\infty}$ exact penalty
function. Under very mild conditions, global convergence
results for the algorithm are given.}}
@inproceedings{Yuan94,
author = {Y. Yuan},
title = {Trust region algorithms for nonlinear programming},
booktitle = {Contemporary Mathematics},
editor = {Z. C. Shi},
publisher = AMS, address = AMS-ADDRESS,
volume = 163, pages = {205--225}, year = 1994,
abstract = {Nonlinear programming, or nonlinear optimization, is
to minimize or maximize a nonlinear function, possibly
subject to finitely many algebraic equations and
inequalities. Trust region algorithms are a class of
numerical algorithms for optimization. In this paper we
review some main results of trust region algorithms for
nonlinear optimization.},
summary = {A review of the many results in the domain of trust-region
methods for nonlinear optimization and the solution of
nonlinear systems of algebraic equations is presented.}}
@article{Yuan94b,
author = {Y. Yuan},
title = {On the convergence of trust region algorithms},
journal = {Mathematica Numerica Sinica},
volume = 16, number = 3, pages = {333-346}, year = 1994,
abstract = {Trust region algorithms for nonlinear optimization
and their convergence properties are
discussed. Convergence results and techniques for
convergence analysis are studied. An $A ( \delta , \eta)$
descent trial step is defined, and is used to
obtain a unified proof for global convergence of
trust region algorithms.},
summary = {Trust region algorithms for nonlinear optimization
and their convergence properties are discussed. An
$A(\delta ,\eta)$ descent trial step is defined, and
is used to obtain a unified proof for global convergence
of such algorithms.},
note = {(in chinese)}}
@inproceedings{Yuan94c,
author = {Y. Yuan},
title = {Trust region algorithms for constrained optimization},
booktitle = {Proceedings of Conference on Scientific and Engineering
Computing for Young Chinese Scientists},
editor = {J.Z. Cui and Z.C. Shi and D.L. Wang},
publisher = {National Defence Industry Press}, address = {Beijing, China},
pages = {105--110}, year = 1994,
abstract = {In this paper, we review the trust region algorithms for
nonlinear optimization and the fundamental ideas of trust
region algorithms are discussed. Model algorithms for
unconstrained optimization, constrained optimization, and
nonsmooth optimization are given. Main techniques for
global convergence and local superlinear convergence are
analyzed.},
summary = {A survey of trust-region methods for nonlinear optimization
is given, with emphasis on global and locallys superlinear
convergence properties.}}
@inproceedings{Yuan94d,
author = {Y. Yuan},
title = {Nonlinear Programming: trust region algorithms},
editor = {S. T. Xiao and F. Wu},
booktitle = {Proceedings of Chinese SIAM annual meeting},
publisher = {Tsinghua University Press}, address= {Beijing, China},
pages = {83--97}, year = 1994,
abstract = {We review the main techniques used in trust region
algorithms for nonlinear constrained optimization},
summary = {A brief survey of trust-region methods for constrained
nonlinear optimization is presented.}}
@article{Yuan95,
author = {Y. Yuan},
title = {On the convergence of a new trust region algorithm},
journal = NUMMATH,
volume = 70, number = 4, pages = {515--539}, year = 1995,
abstract = {In this paper we present a new trust region algorithm for
general nonlinear constrained optimization problems. The
algorithm is based on the $L_{\infty}$ exact penalty
function. Under very mild conditions, global convergence
results for the algorithm are given. Local convergence
properties are also studied. It is shown that the penalty
parameter generated by the algorithm will be eventually not
less than the $l_1$ norm of the Lagrange multipliers at the
accumulation point. It is proved that the method is
equivalent to the sequential quadratic programming method
for all large $k$, hence superlinearly convergent results
of the SQP method can be applied. Numerical results are
also reported.},
summary = {A trust-region algorithm is presented for general nonlinear
constrained problems, based on the $L_{\infty}$ exact
penalty function. Under very mild conditions, global
and local convergence results for the algorithm are given.
It is shown that
the penalty parameter generated by the algorithm will be
eventually not less than the $l_1$ norm of the Lagrange
multipliers at the accumulation point. It is proved that
the method is equivalent to the sequential quadratic
programming method for all large $k$, hence superlinearly
convergent results for the SQP method can be applied.
Numerical results are reported.}}
@article{Yuan96,
author = {Y. Yuan},
title = {A short note on the {D}uff-{N}ocedal-{R}eid algorithm},
journal = {SEA Bull. Math.},
volume = 20, number = 3, pages = {137--144}, year = 1996,
abstract = {In this short note, an example is given to show that the
algorithm of \citebb{DuffNoceReid87} for nonlinear
equations may converge to a non-optimal solution. It is
also shown that a slightly modification can ensure the
global convergence of the algorithm.},
summary = {An example is given to show that the algorithm of
\citebb{DuffNoceReid87} for nonlinear equations may
converge to a non-optimal solution. It is also shown that a
slightly modification can ensure the global convergence of
the algorithm.}}
@misc{Yuan97,
author = {Y. Yuan},
title = {Some properties of a trust region subproblem},
howpublished = {Presentation at the XVIth International Symposium on
Mathematical Programming, Lausanne},
year = 1997,
summary = {It is shown that the model reduction obtained by applying
the Steihaug-Toint algorithm on a convex quadratic model
in dimension two is at least half of that obtained by the
exact minimizer of the model within the trust region.}}
@article{Yuan98a,
author = {Y. Yuan},
title = {Trust region algorithms for nonlinear equations},
journal = {Information},
volume = 1, pages = {7--20}, year = 1998,
abstract = {In this paper, we consider the problem of solving nonlinear
equations $F(x)=0$, where $F(x)$ from $\Re^n$ to $\Re^m$ is
continuously differentiable. We study a class of general
trust region algorithms for solving nonlinear equations by
minimizing a given norm $\|F(x)\|$. The trust region
algorithm for nonlinear equations can be viewed as an
extension of the Levenberg-Marquardt algorithm for
nonlinear least squares. Global convergence of trust region
algorithms for nonlinear equations are studied and local
convergence analyses are also given.},
summary = {The problem of solving nonlinear equations $F(x)=0$, where
$F(x)$ from $\Re^n$ to $\Re^m$ is continuously
differentiable, is considered. We study a class of general
trust-region algorithms for solving nonlinear equations by
minimizing a given norm $\|F(x)\|$. The trust-region
algorithm for nonlinear equations can be viewed as an
extension of the Levenberg-Morrison-Marquardt algorithm for
nonlinear least-squares. Global convergence of trust-region
algorithms for nonlinear equations are studied and local
convergence analyses are also given.}}
@inproceedings{Yuan98b,
author = {Y. Yuan},
title = {An example of non-convergence of trust region algorithms},
crossref = {Yuan98}, pages = {205--218},
abstract = {It is well known that trust region algorithms have very
nice convergence properties. Trust region algorithms can be
classified into two kinds: one requires sufficient
reduction in objective function value (merit function
value, in the case of constrained optimization), the other
only needs reduction in objective function value. In
general, it can be shown that the algorithms that require
sufficient reductions have strong convergence result,
namely all accumulation points are stationary points. The
algorithms that do not require sufficient reductions have
the nice properties of accepting any better iterates, but
the convergence result is weak, only one accumulation point
is a stationary point. In this paper, we construct an
example to show that it can happen that for a class of
trust region algorithms that do not require sufficient
reductions the whole sequence need not to converge. In our
example, only one accumulation point is a stationary point
while all other accumulation points are non-stationary
points.},
summary = {An example is constructed which shows that it can happen
that for a class of trust-region algorithms that do not
require sufficient reductions the whole sequence need not
to converge. In the example, only one accumulation point is
a stationary point while all other accumulation points are
non-stationary.}}
@misc{Yuan98c,
author = {Y. Yuan},
title = {Optimality conditions for the {C}elis-{D}ennis-{T}apia
subproblems},
howpublished = {Presentation at the Optimization 98 Conference, Coimbra},
year = 1998,
abstract = {We give necessary and sufficient optimality conditions
which can be easily verified for local solutions of
Celis-Dennis-Tapia subproblem, which is a subproblem in
trust region algorithms for nonlinear constrained
optimization. If the CDT suproblem has no global solution
with the Hessian of Lagrangian positive semi-definite, the
Hessian at any local solution has at least negative
eigenvalue. Some other characters of local solutions are
also given. We also discuss the gap between necessary
conditions and sufficient conditions.},
summary = {Easily verifiable necessary and sufficient optimality
conditions are given for local solutions of the
Celis-Dennis-Tapia (CDT) subproblem, which is a subproblem
in trust-region algorithms for nonlinear constrained
optimization. If the CDT suproblem has no global solution
at with the Hessian of the Lagrangian is positive
semi-definite, the Hessian at any local solution has at
least one negative eigenvalue. Some other characteristics
of local solutions are also given. The gap between
necessary and sufficient conditions is also discussed.}}
@inproceedings{Yuan98d,
author = {Y. Yuan},
title = {Matrix computation problems in trust region algorithms
for optimization},
booktitle = {Proceedings of the 5th CSIAM annual meeting},
editor = {Q. C. Zeng and T. Q. Li and Z. S. Xue and Q. S. Cheng},
publisher = {Tsinghua University Press},
address = {Beijing, China},
pages = {54--64}, year = 1998,
abstract = {Trust region algorithms are a class of recently developed
algorithms for solving optimization problems. The
subproblems appeared in trust region algorithms are
usually minimizing a quadratic function subject to one
or two quadratic constraints. In this paper we review
some of the widely used trust region subproblems and some
matrix computation problems related to these trust region
subproblems.},
summary = {The linear algebra aspects of the solution methods for
various trust-region subproblems are reviewed.}}
@techreport{Yuan99,
author = {Y. Yuan},
title = {On the Truncated Conjugate-Gradient Method},
institution = ICMSEC, address = ICMSEC-ADDRESS,
number = {ICM-99-003}, year = 1999,
abstract = {In this paper, we consider the truncated conjugate-gradient
method for minimizing a convex quadratic function subject
to a ball trust region constraint. It is showed that the
reduction in the objective function by the solution
obtained by the truncated CG method is at least half of the
reduction by the global minimizer in the trust region.},
summary = {It is shown that the model reduction obtained by applying
the Steihaug-Toint algorithm on a convex quadratic model is
at least half of that obtained by the exact minimizer of
the model within the trust region.}}
@techreport{Yuan99b,
author = {Y. Yuan},
title = {A review of trust region algorithms for optimization},
institution = ICMSEC, address = ICMSEC-ADDRESS,
number = {ICM-99-038}, year = 1999,
abstract = {Iterative methods for optimization can be classified
into two categories: line search methods and trust region
methods. In this paper we give a review on trust region
algorithms for nonlinear optimization. Trust region methods
are robust, and can be applied ti ill-conditioned problems.
A model trust region algorithm is presented to demonstrate
the trust region approaches. Various trust region
subproblems and their properties are presented.
Convergence properties of trust region algorithms are given.
Techniques such as backtracking, non-monotone and
second-order correction are also briefly discussed.},
summary = {A survey of trust-region methods is presented, with special
emphasis on the possible subproblem solution techniques.
Convergence properties for the unconstrained case are
reviewed and techniques such as backtracking, non-monotone
and second-order correction are also briefly discussed.}}
%%% Z %%%
@inproceedings{Zara71,
author = {E. H. Zarantonello},
title = {Projections on convex sets in {H}ilbert space and spectral
theory},
booktitle = {Contributions to Nonlinear Functional Analysis},
editor = {E. H. Zarantonello},
publisher = AP, address = AP-ADDRESS,
pages = {237--424}, year = 1971}
@article{Zhan89,
author = {J. Zhang},
title = {Superlinear convergence of a trust region-type successive
linear-programming method},
journal = JOTA,
volume = 61, number = 2, pages = {295--310}, year = 1989,
abstract = {The convergence rate of the SLP method suggested by
\citebb{ZhanKimLasd85} is discussed for composite
nondifferentiable optimization problems. A superlinear rate
is assured under a growth condition, and it is further
strengthened to a quadratic rate if the inside function is
twice differentiable. Several sufficient conditions are
given which make the growth condition true. The conditions
can be relaxed considerably in practical use.},
summary = {The convergence rate of the SLP method suggested by
\citebb{ZhanKimLasd85} is discussed for composite
non-differentiable optimization problems. A superlinear rate
is assured under a growth condition, and it is further
strengthened to a quadratic rate if the inside function is
twice differentiable. Several sufficient conditions are
given which make the growth condition true. The conditions
can be relaxed considerably in practical use.}}
@article{ZhanZhu90,
author = {J. Zhang and D. Zhu},
title = {Projected quasi-{N}ewton algorithm with trust-region for
constrained optimization},
journal = JOTA,
volume = 67, pages = {369--393}, year = 1990,
abstract = {\citebb{NoceOver85} proposed a two-sided projected Hessian
updating technique for equality constrained optimization
problems. Although local two-step Q-superlinear rate was
proved, its global convergence is not assured. In this
paper, we suggest a trust-region-type, two-sided, projected
quasi-Newton method, which preserves the local two-step
superlinear convergence of the original algorithm and also
ensures global convergence. The subproblem that we propose
is as simple as the one often used when solving
unconstrained optimization problems by trust-region
strategies and therefore is easy to implement.},
summary = {A trust-region-type, two-sided, projected quasi-Newton method
is proposed, which preserves the local two-step superlinear
convergence of the original algorithm of \citebb{NoceOver85}
and also ensures global convergence. The proposed subproblem
is as simple as the one used when solving unconstrained
problems by trust-region strategies.}}
@article{ZhanZhu94,
author = {J. Zhang and D. Zhu},
title = {A projective quasi-{N}ewton method for nonlinear
optimization},
journal = JCAM,
volume = 53, number = 3, pages = {291-307}, year = 1994,
abstract = {A trust region method for nonlinear optimization
problems with equality constraints is proposed in
this paper. This method incorporates quadratic
subproblems in which orthogonal projective matrices
of the Jacobian of constraint functions are used to
replace QR decompositions. As QR decomposition does
not ensure continuity, but projective matrix does,
convergence behaviour of the new method can be
discussed under more reasonable assumptions. The
method maintains a two-step feature: one movement in
the range space of the Jacobian, whereas the other
one in the null space. It is proved that all
accumulation points of iterates are KKT
(Karush-Kuhn-Tucker) points and the method has a
one-step superlinear convergence rate.},
summary = {A trust region method for nonlinear optimization
problems with equality constraints is proposed
that incorporates quadratic subproblems in which orthogonal
projective matrices of the Jacobian of constraint functions
are used to replace QR decompositions. As QR decomposition
does not ensure continuity, but projective matrix does,
convergence behaviour of the new method is studied by
exploiting the continuity of these matrices. A
one-step superlinear convergence rate is also proved.}}
@article{ZhanZhu99,
author = {J. Zhang and D. Zhu},
title = {A nonmonotonic trust region method for constrained
optimization problems},
journal = {Journal of the Australian Mathematical Society (Series B)},
volume = 40, number = 4, pages = {542--567}, year = 1999,
abstract = {In this paper we propose an easy-to-implement algorithm
for solving general nonlinear optimization problems
with nonlinear equality constraints. A nonmonotonic
trust region strategy is suggested which does not
require the merit function to reduce its value in
every iteration. In order to deal with large
problems, a reduced Hessian is used to replace a full
Hessian matrix. To avoid solving quadratic trust
region subproblems exactly which usually takes
substantial computation, we only require an
approximate solution which requires less
computation. The calculation of correction steps,
necessary from a theoretical view point to overcome
the Maratos effect but which often brings in negative
results in practice, is avoided in most cases by
setting a criterion to judge its necessity. Global
convergence and. a local superlinear rate are then
proved. This algorithm has a good performance. },
summary = {A nonmonotonic, reduced Hessian trust-region method is
used to solve equality constrained nonlinear
optimization problems. Approximate solutions to the
trust-region subproblems are allowed. Although theory
dictates that second-order correction steps be used
to overcome the Maratos effect, a suitable scheme is
developed to ensure that they are only used when
absolutely necessary. Global convergence at a local
superlinear rate is established, and the resulting
algorithm performs well in practice.}}
@article{ZhanZhuFan93,
author = {J. Zhang and D. Zhu and Y. Fan},
title = {A practical trust region method for equality constrained
optimization problems},
journal = OMS,
volume = 2, number = 1, pages = {45--68}, year = 1993,
abstract = {In this paper we propose an easy-to-implement algorithm for
solving general nonlinear optimization problems with
nonlinear equality constraints. In order to deal with large
scale problems, a reduced Hessian is used to replace the
full Hessian matrix. To avoid solving quadratic trust
region subproblems exactly, which usually takes most
computing time, we only require an approximate solution
with less computation. The calculation of correction steps,
that is necessary from the theoretical point of view to
overcome the \citebb{Mara78} effect but often brings in
negative results in practice, is avoided in most cases by
setting a criterion to judge its necessity. Global
convergence and a local superlinear rate are proved.
Numerical results are reported to show that this algorithm
has good performance.},
summary = {An easy-to-implement algorithm for solving general
nonlinear optimization problems with nonlinear equality
constraints is proposed, that uses a reduced Hessian matrix.
The quadratic trust-region subproblems are solved
approximately. The calculation of correction steps, that are
necessary from the theoretical point of view to overcome
the Maratos effect but prove costly in practice, is
avoided in most cases by a suitable test. Global
and superlinear convergence are proved. Numerical results
are reported.}}
@article{ZhanKimLasd85,
author = {J. Zhang and N. H. Kim and L. S. Lasdon},
title = {An improved successive linear programming algorithm},
journal = {Management Science},
volume = 31, pages = {1312--1331}, year = 1985}
@article{ZhanXu99,
author = {J. Zhang and C. Xu},
title = {A Class of Indefinite Dogleg Path Methods for
Unconstrained Optimization},
journal = SIOPT,
volume = 9, number = 3, pages = {646--676}, year = 1999,
abstract = {In this paper we propose a convenient curvilinear search
method to solve the trust region subproblems arising from
unconstrained optimization problems. The curvilinear paths
we set forth are dogleg paths, generated mainly by
employing Bunch-Parlett factorizations for general
symmetric matrices that may be indefinite. This method is
easy to implement and globally convergent. It is proved
that the method satisfies the first- and second-order
stationary point convergence properties and that the
convergence rate is quadratic under commonly used
conditions on functions. Numerical experiments are
conducted to compare this method with some existing
methods.},
summary = {A variant of the dogleg method for approximately solving
the trust-region subproblem is proposed. This variant is
adequate for the case where the subproblem is non-convex.
In this case, it defines a family of path that use a
direction of negative curvature obtained from the
Bunch-Parlett factorization of the Hessian. Global
convergence to unconstrained second-order stationary points
is proved for the resulting trust-region method, as well as
quadratic convergence of the associated version of Newton's
method. Numerical results are shown.}}
@article{ZhanXu99b,
author = {J. Zhang and C. Xu},
title = {Trust region dogleg path algorithms for unconstrained
minimization},
journal = AOR,
volume = 87, pages = {407--418}, year = 1999,
abstract = {In this paper, we propose a class of convenient curvilinear
search algorithms to solve trust region problems arising
from unconstrained optimization. The curvilinear paths we
set are dogleg paths, generated mainly by employing the
Bunch-Parlett factorization for general symmetric matrices
which may be indefinite. These algorithms are easy to use
and globally convergent. It is proved that these
algorithms satisfy the first- and second-order stationary
point convergence properties and that the rate of
convergence is quadratic under commonly used assumptions},
summary = {Trust-region algorithms using curvilinear search
to approximately solve the possibly nonconvex subproblem
are proved to be globally convergent to first- and
second-order critical points. The rate of convergence is
quadratic under typical assumptions.}}
@article{ZhanXu99c,
author = {J. Zhang and C. Xu},
title = {A projected indefinite dogleg-path method for equality
constrained optimization},
journal = BIT,
volume = 39, number = 3, pages = {555-578}, year = 1999,
abstract = {In this paper, we propose a 2-step trust-region indefinite
dogleg path method for the solution of nonlinear equality
constrained optimization problems. The method is a globally
convergent Fontecilla method and an indefinite dogleg path
method is proposed to get approximate solutions of quadratic
programming subproblems even if the Hessian of the model
is indefinite. The dogleg path lie in the null space of
the Jacobian matrix of the constraints. An $\ell_1$ exact
penalty function method is used in the method to determine
if a trial point is accepted. The global convergence and
the local two-step superlinear convergence rate is proved.
Some numerical results are presented.},
summary = {A two-step trust-region algorithm is presented for the
solution of optimization problems with nonlinear equality
constraints. The method is based on that of
\citebb{Font90}, but uses an indefinite dogleg strategy in
the nullspace of the constraints Jacobian as a way to obtain
an approximate solution of the quadratic programming
subproblem. Global and locally two-step superlinear
convergence are proved and some numeruical experiements
shown.}}
@inproceedings{ZhanXuDu98,
author = {J. Zhang and C. Xu and L. Du},
title = {A more efficient variation of an indefinite dogleg path
method},
booktitle = {Operations Research and its Applications.
Third International Symposium, ISORA'98.},
publisher = {World Publishing Corp}, address = {Beijing, China},
pages = {428--434}, year = 1998,
abstract = {The authors previously suggested a class of indefinite
dogleg path methods that can be used to find
approximate solutions of a trust region model in
which the working matrix is not necessary positive
definite. In this paper we propose a scaled version
of the indefinite dogleg path method. The main
advantage of the revision is that the scaled
subproblem has a 1 by 1 or 2 by 2 block diagonal matrix as
its working matrix so that the solution of the Newton
equation when the working matrix is positive definite
and the calculation of a direction of negative
curvature when the matrix is indefinite become much
easier. Also, it can save a substantial number of
matrix factorizations.},
summary = {A scaled version of the author's previous indefinite
dogleg path method is considered. The main
advantage is that the scaled
subproblem has a 1 by 1 or 2 by 2 block diagonal Hessian,
so that the solution of the Newton
equations and directions of negative
curvature are simple to obtain.}}
@article{Zhan92,
author = {Y. Zhang},
title = {Computing a {C}elis-{D}ennis-{T}apia trust-region step for
equality constrained optimization},
journal = MP,
volume = 55, number = 1, pages = {109--124}, year = 1992,
abstract = {We study an approach for minimizing a convex quadratic
function subject to two quadratic constraints. This problem
stems from computing a trust-region step for an SQP
algorithm proposed by \citebb{CeliDennTapi85} for equality
constraint optimization. Our approach is to reformulate the
problem into a univariate nonlinear equation $\phi(\mu)=0$,
where the function $\phi(\mu)$ is continuous, at least
piecewise differentiable and monotone. Well-established
methods then can be readily applied. We also consider an
extension of our approach to a class of non-convex
quadratic functions and show that our approach is
applicable to reduced Hessian SQP algorithms. Numerical
results are presented indicating that our algorithm is
reliable, robust and has the potential to be used as a
building block to construct trust-region algorithms for
small-sized problems in constrained optimization.},
summary = {An approach to minimizing a convex quadratic function
subject to two quadratic constraints is studied. This
problem stems from computing a trust-region step for an SQP
algorithm proposed by \citebb{CeliDennTapi85} for equality
constraint optimization. The approach taken is to
reformulate the problem as a univariate nonlinear equation
$\phi(\mu)=0$, where the function $\phi(\mu)$ is
continuous, at least piecewise differentiable and monotone.
Well-established methods then can be readily applied. An
extension of this approach to a class of non-convex
quadratic functions is considered, and it is shown that the
approach is applicable to reduced Hessian SQP algorithms.
Numerical results are presented.}}
@article{Zhan94,
author = {Y. Zhang},
title = {On the convergence of infeasible interior-point methods for
the horizontal linear complementarity problem},
journal = SIOPT,
volume = 4, number = 1, pages = {208--227}, year = 1994}
@inproceedings{ZhaoWang93,
author = {M. Zhao and X. Wang},
title = {Model trust region technique in parallel {N}ewton method
for training neural networks},
booktitle = {IEEE International Symposium on Circuits and Systems (ISCAS
93)},
publisher = {IEEE}, address = {New York},
volume = 4, pages = {2399--2402}, year = 1993,
abstract = {In this article, the double dogleg trust region approach of
unconstrained minimization is introduced in the parallel
Newton's (PN) algorithm proposed in Zhao (1993). The PN
algorithm uses a recursive procedure for computing both the
Hessian matrix and the Newton direction. The input weights
of each neuron in the network are updated after each
presentation of the training data with a global strategy.
Experimental results indicate that the double dogleg trust
region approach is superior to the line search technique in
the PN algorithm, and that the PN algorithm with both
global strategies exhibits better convergence performance
than the well-known backpropagation algorithm.},
summary = {The double-dogleg trust-region approach of unconstrained
minimization is introduced into the parallel Newton's (PN)
algorithm, which uses a recursive procedure for
computing both the Hessian matrix and the Newton direction.
The input weights of each neuron in the network are updated
after each presentation of the training data with a global
strategy. Experimental results indicate that the
double-dogleg trust-region approach is superior to the
linesearch technique in the PN algorithm, and that the PN
algorithm with both global strategies exhibits better
convergence performance than backpropagation.}}
@inproceedings{ZhouSi98,
author = {G. Zhou and J. Si},
title = {Subset based training and pruning of sigmoid neural
networks},
booktitle = {Proceedings of the 1998 American Control Conference,
Evanston, IL, USA},
pages = {58--62}, year = 1998,
abstract = {In the present paper we develop two algorithms,
subset based training (SBT) and subset based training
and pruning (SBTP), using the fact that the Jacobian
matrices in sigmoid network training problems are
usually rank deficient. The weight vectors are
divided into two parts during training, according to
the Jacobian rank sizes. Both SBT and SBTP are trust
region methods. Comparing to the standard
Levenberg-Marquardt (LM) method, these two algorithms
can achieve similar convergence properties as the LM
but with less memory requirements. Furthermore the
SBTP combines training and pruning of a network into
one comprehensive procedure. Some convergence
properties of the two algorithms are given to
qualitatively evaluate the performance of the
algorithms.},
summary = {Two trust-region algorithms,
subset based training (SBT) and subset based training
and pruning (SBTP), are developed using the fact that the
Jacobian matrices in sigmoid network training problems are
usually rank deficient. The weight vectors are
divided into two parts during training, according to
the Jacobian rank sizes. These two algorithms
have convergence properties similar to those of the
Levenberg-Morrison-Marquardt method but with less memory
requirements. Furthermore the SBTP combines training and
pruning of a network into one comprehensive procedure. }}
@article{ZhouSi98b,
author = {G. Zhou and J. Si},
title = {Advanced neural-network training algorithm with
reduced complexity based on Jacobian deficiency},
journal = {IEEE Transactions on Neural Networks},
volume = 9, number = 3, pages = {448--453}, year = 1998,
abstract = {We introduce an advanced supervised training method
for neural networks. It is based on Jacobian rank
deficiency and it is formulated, in some sense, in
the spirit of the Gauss-Newton algorithm. The
Levenberg-Marquardt algorithm, as a modified
Gauss-Newton, has been used successfully in solving
nonlinear least squares problems including
neural-network training. It outperforms the basic
backpropagation and its variations with variable
learning rate significantly, but with higher
computation and memory complexities within each
iteration. The mew method developed in this paper is
aiming at improving convergence properties, while
reducing the memory and computation complexities in
supervised training of neural networks. Extensive
simulation results are provided to demonstrate the
superior performance of the new algorithm over the
Levenberg-Marquardt algorithm.},
summary = {A supervised training method for neural
networks based on Jacobian rank deficiency is formulated
in the spirit of the Gauss-Newton algorithm. The new
method aims at improving convergence properties compared
to the Levenberg-Morrison-Marquardt method, while
reducing the memory and computation complexities in
supervised training of neural networks. Extensive
simulation results demonstrate the superior performance
of the new algorithm over the Levenberg-Morrison-Marquardt
algorithm.}}
@article{ZhouSi99,
author = {C. Zhou and J. Si},
title = {Subset-based training and pruning of sigmoid neural
networks},
journal = {Neural Networks},
volume = 12, number = 1, pages = {79--89}, year = 1999,
abstract = {In the present paper we develop two algorithms,
subset-based training (SBT) and subset-based training
and pruning (SBTP), using the fact that the Jacobian
matrices in sigmoid network training problems are
usually rank deficient. The weight vectors are
divided into two parts during training, according to
the Jacobian rank sizes. Both SBT and SBTP are
trust-region methods. Compared with the standard
Levenberg-Marquardt (LM) method, these two algorithms
can achieve similar convergence properties as the LM
but with fewer memory requirements. Furthermore the
SBTP combines training and pruning of a network into
one comprehensive procedure. The effectiveness of the
two algorithms is evaluated using three
examples. Comparisons are made with some existing
algorithms. Some convergence properties of the two
algorithms are given to qualitatively evaluate the
performance of the algorithms.},
summary = {Subset-based training (SBT) and subset-based training
and pruning (SBTP) trust-region algorithms are
developed to cope with the fact that the Jacobian
matrices in sigmoid network training problems are
usually rank deficient. Both methods prove to be as
effective as the Levenberg-Morrison-Marquardt
approach, but have significantly smaller memory
requirements. Additionally SBTP combines training
and pruning of a network into one comprehensive
procedure. The effectiveness of the two algorithms is
evaluated using three examples. Comparisons are made
with existing algorithms, and convergence properties
are investigated.}}
@article{ZhouXiao94,
author = {F. Zhou and Y. Xiao},
title = {A Class of nonmonotone stabilization trust region methods},
journal = {Computing},
volume = 53, number = 2, pages = {119--136}, year = 1994,
abstract = {A class of trust region methods in unconstrained
optimization is presented, by adopting a nonmonotone
stabilization strategy. Under some regularity conditions,
the convergence properties of these methods are discussed.
Extensive numerical results which are reported show that
these methods are very efficient.},
summary = {A class of trust-region methods for unconstrained
optimization is presented which use a non-monotone
stabilization strategy. Under some regularity conditions,
the convergence properties of these methods are discussed.
Extensive numerical results which are reported.}}
@article{ZhouTits93,
author = {J. Zhou and A. L. Tits},
title = {Nonmonotone line search for minimax problems},
journal = JOTA,
volume = 76, number = 3, pages = {455--476}, year = 1993}
@article{Zhu92,
author = {D. Zhu},
title = {Convergence of a projected gradient method with trust
region for nonlinear constrained optimization},
journal = {Optimization},
volume = 23, number = 3, pages = {215--235}, year = 1992,
abstract = {Describes a projected gradient algorithm with trust
region, introducing nondifferentiable merit function
for solving nonlinear constrained optimization
problems. The author shows that this method is
globally convergent even if conditions are weak. It
is also proved that, when the strict complementarity
condition holds, the proposed algorithm can be solved
by an equality constrained problem, allowing locally
rate of superlinear convergence.},
summary = {A globally convergent trust-region projected-gradient
algorithm is described that uses a non-differentiable
merit function.}}
@article{Zhu95,
author = {D. Zhu},
title = {A Nonmonotonic Trust Region Technique for nonlinear
Constrained Optimization},
journal = JCM,
volume = 13, number = 1, pages = {20--31}, year = 1995,
abstract = {In this paper, a nonmonotonic trust region method for
optimization problems with equality constraints is proposed
by introducing a nonsmooth merit function and adopting a
correction step. It is proved that all accumulation points
of the iterates generated by the proposed algorithm are
Kuhn-Tucker points and that the algorithm is
q-superlinearly convergent.},
summary = {A non-monotonic method for problems with equality
constraints is proposed by introducing a non-smooth merit
function and a correction step. It is proved that all
accumulation points of the iterates generated are
Kuhn-Tucker points and that the algorithm is
Q-superlinearly convergent.}}
@techreport{Zhu99,
author = {D. Zhu},
title = {A Family of Generalized Projected Gradient Methods with
Mixing Strategy for Convex Constrained Optimization},
institution = {Department of Mathematics, Shanghai Normal University},
address = {Shanghai, China},
number = {(na)}, year = 1999,
abstract = {A family of generalized projected gradient algorithms is
proposed for convex constrained optimization problems.
The mixed strategy in association with nonmonotone
technique is adopted in which projected gradient methods
switch to back tracking steps in trust region subproblems.
The globally convergent theoretical analysis of the proposed
algorithms are given and the local convergence rate of the
proposed algorithms are proved unders some reasonable
conditions},
summary = {A globally convergent trust-region method is proposed for
problems with convex constraints, that combines the
strategies of \citebb{ConnGoulSartToin93} with the
non-monotone technique of \citebb{DengXiaoZhou93}.}}
@article{Zhu96,
author = {C. Zhu},
title = {Asymptotic convergence analysis of some inexact proximal
point algorithms for minimization},
journal = SIOPT,
volume = 6, number = 3, pages = {626--637}, year = 1996}
@article{ZhuByrdLuNoce97,
author = {C. Zhu and R. H. Byrd and P. Lu and J. Nocedal},
title = {Algorithm 78: {L-BFGS-B}: Fortran subroutines for
large-scale bound constrained optimization},
journal = TOMS,
volume = 23, number = 4, pages = {550--560}, year = 1997}
@article{ZhuBrow87,
author = {T. Zhu and L. D. Brown},
title = {Two-dimensional velocity inversion and synthetic seismogram
computation},
journal = {Geophysics},
volume = 52, number = 1, pages = {37--50}, year = 1987,
abstract = {A traveltime inversion schemes has been developed to
estimate velocity and interface geometries of
two-dimensional media from deep reflection data. The
velocity structure is represented by finite elements,
and the inversion is formulated as an iterative,
constrained, linear least-squares problems which can
be solved by either the singular value truncation
method or the Levenberg-Marquardt method. The damping
factor of the Levenberg-Marquardt method is chosen by
the model-trust region approach. The traveltimes and
derivative matrix required to solve the least-squares
problem are computed by ray tracing. To aid seismic
interpretation, the authors also include in the
inversion scheme a fast algorithm based on asymptotic
ray theory for calculating synthetic seismograms from
the derived velocity model.},
summary = {A trust-region method is used to
estimate velocity and interface geometries of
two-dimensional media from deep-reflection data, where
the velocity structure is represented by finite elements.
The traveltimes and derivative matrix required to solve
the least-squares problem are computed by ray tracing.}}
@mastersthesis{Zupk97,
author = {M. Zupke},
title = {{T}rust-{R}egion-{V}erfahren zur {L}\"{o}sung nichtlinearer
{K}ompl\-em\-ent\-ar\-it\-\"{a}ts\-probleme},
school = HAMBURG, address = HAMBURG-ADDRESS,
year = 1997,
summary = {The nonlinear complementarity problem is reformulated as a
non-smooth system of equations by using a recently
introduced NCP-function. A trust-region-type method is then
applied to the resulting system of equations, that allows
an inexact solution of the trust-region subproblem. It is
shown that the algorithm is well-defined for a general
nonlinear complementarity problem and that it has some nice
global and local convergence properties. Numerical show the
advantage of using the non-monotone technique proposed by
\citebb{Toin96b}.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PROCEEDINGS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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publisher = SPRINGER, address = SPRINGER-ADDRESS,
number = 630, series = {Lecture Notes in Mathematics}, year = 1978}
@proceedings{Yuan98,
editor = {Y. Yuan},
title = {Advances in Nonlinear Programming},
booktitle = {Advances in Nonlinear Programming},
publisher = KLUWER, address = KLUWER-ADDRESS,
year = 1998}