Approximate norm descent  methods for constrained nonlinear systems

          by B. Morini, M. Porcelli and Ph. L. Toint
                    Report NAXYS-05-2016

Abstract.
We address the solution of convex-constrained nonlinear systems of equations
where the Jacobian matrix is unavailable or its computation/storage is
burdensome.  In order to efficiently solve such problems, we propose a new
class of algorithms which are ``derivative-free'' both in the computation of
the search direction and in the selection of the steplength.  Search
directions comprise the residuals and Quasi-Newton directions while the
steplength is determined by using a new linesearch strategy based on a
nonmonotone approximate norm descent property of the merit function. We
provide a theoretical analysis of the proposed algorithm and we discuss
several conditions ensuring convergence to a solution of the constrained
nonlinear system.  Finally, we illustrate its numerical behaviour also in
comparison with existing approaches.

Keywords: nonlinear systems of equations, bound constraints,
numerical algorithms, convergence theory.