An iterative working-set method for large-scale non-convex 
                          quadratic programming

                     Nicholas I. M. Gould and Philippe L. Toint

Abstract.We consider a working-set method for solving 
large-scale quadratic programming problems for which there is no requirement 
that the objective function be convex. The methods are iterative 
at two levels, one level relating to the selection of the current working
set, and the second due to the method used to solve the equality-constrained
problem for this working set. A preconditioned conjugate gradient method is
used for this inner iteration, with the preconditioner chosen especially
to ensure feasibility of the iterates. The preconditioner is updated 
at the conclusion of each outer iteration to ensure that this feasibility
requirement persists. The well-known equivalence between the
conjugate-gradient and Lanczos methods is exploited when  finding directions
of negative curvature. Details of an  implementation---the Fortran 90 package
QPA in the forthcoming GALAHAD library---are given.