Numerical analysis of leaving-face parameters in bound-constrained quadratic minimization

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 targe-scale bound-constrained quadratic problems. To fill in some gaps that have appeared in previous work, we propose 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 have shown to be crucial, particularly when dual degenerate problems are solved.151456

Derivative-free methods for nonlinear programming with general lower-level constraints

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martinez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivative-free subalgorithms for solving lower-level constrained subproblems. Numerical experiments are presented.3011952Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [PRONEX - CNPq/FAPERJ E-26/171.510/2006 - APQ1]FAPESP [2006/53768-0, 2004/15635-2

Discrete Newton's method with local variations for solving large-scale nonlinear systems

A globally convergent discrete Newton method is proposed for solving large-scale nonlinear systems of equations. Advantage is taken from discretization steps so that the residual norm can be reduced while the Jacobian is approximated, besides the reduction at Newtonian iterations. The Curtis-Powell-Reid (CPR) scheme for discretization is used for dealing with sparse Jacobians. Global convergence is proved and numerical experiments are presented.524176341744

Augmented Lagrangian algorithms based on the spectral projected gradient method for solving nonlinear programming problems

The spectral projected gradient method (SPG) is an algorithm for large-scale bound-constrained optimization introduced recently by Birgin, Martinez, and Raydan. It is based on the Raydan unconstrained generalization of the Barzilai-Borwein method for quadratics. The SPG algorithm turned out to be surprisingly effective for solving many large-scale minimization problems with box constraints. Therefore, it is natural to test its perfomance for solving the subproblems that appear in nonlinear programming methods based on augmented Lagrangians. In this work, augmented Lagrangian methods which use SPG as the underlying convex-constraint solver are introduced (ALSPG) and the methods are tested in two sets of problems. First, a meaningful subset of large-scale nonlinearly constrained problems of the CUTE collection is solved and compared with the perfomance of LANCELOT. Second, a family of location problems in the minimax formulation is solved against the package FFSQP.123349751