A global hybrid derivative-free method for high-dimensional systems of nonlinear equations

Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) under the projects PTDC/MAT-APL/28400/2017 and UID/MAT/00297/2019. Marcia A. Gomes-Ruggiero: Partially supported by PRONEX-Optimization; CNPq (Grants 306220/2009-1 and 309733/2013-8); FAPESP (Grants 2013/05475-7 and 2013/07375-0) and CAPES.This work concerns the numerical solution of high-dimensional systems of nonlinear equations, when derivatives are not available for use, but assuming that all functions defining the problem are continuously differentiable. A hybrid approach is taken, based on a derivative-free iterative method, organized in two phases. The first phase is defined by derivative-free versions of a fixed-point method that employs spectral parameters to define the steplength along the residual direction. The second phase consists on a matrix-free inexact Newton method that employs the Generalized Minimal Residual algorithm to solve the linear system that computes the search direction. This second phase will only take place if the first one fails to find a better point after a predefined number of reductions in the step size. In all stages, the criterion to accept a new point considers a nonmonotone decrease condition upon a merit function. Convergence results are established and the numerical performance is assessed through experiments in a set of problems collected from the literature. Both the theoretical and the experimental analysis support the feasibility of the proposed hybrid strategy.authorsversionpublishe

Solving the dual subproblem of the Method of Moving Asymptotes using a trust-region scheme

An alternative strategy to solve the subproblems of the Method of Moving Asymptotes (MMA) is presented, based on a trust-region scheme applied to the dual of the MMA subproblem. At each iteration, the objective function of the dual problem is approximated by a regularized spectral model. A globally convergent modification to the MMA is also suggested, in which the conservative condition is relaxed by means of a summable controlled forcing sequence. Another modification to the MMA previously proposed by the authors [Optim. Methods Softw., 25 (2010), pp. 883-893] is recalled to be used in the numerical tests. This modification is based on the spectral parameter for updating the MMA models, so as to improve their quality. The performed numerical experiments confirm the efficiency of the indicated modifications, especially when jointly combined.151170Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq