A practical second-order optimality condition for cardinality-constrained problems with application to an augmented Lagrangian method

This paper addresses the mathematical programs with cardinality constraints (MPCaC). We first define two new tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC. We then propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. All the concepts proposed … Read more

Global convergence of a derivative-free inexact restoration filter algorithm for nonlinear programming

In this work we present an algorithm for solving constrained optimization problems that does not make explicit use of the objective function derivatives. The algorithm mixes an inexact restoration framework with filter techniques, where the forbidden regions can be given by the flat or slanting filter rule. Each iteration is decomposed in two independent phases: … Read more

A globally convergent trust-region algorithm for unconstrained derivative-free optimization

In this work we explicit a derivative-free trust-region algorithm for unconstrained optimization based on the paper (Computational Optimization and Applications 53: 527-555, 2012) proposed by Powell. The objective function is approximated by quadratic models obtained by polynomial interpolation. The number of points of the interpolation set is fixed. In each iteration only one interpolation point … Read more

Theoretical aspects of adopting exact penalty elements within sequential methods for nonlinear programming

In the context of sequential methods for solving general nonlinear programming problems, it is usual to work with augmented subproblems instead of the original ones, tackled by the $\ell_1$-penalty function together with the shortcut usage of a convenient penalty parameter. This paper addresses the theoretical reasoning behind handling the original subproblems by such an augmentation … Read more

Global convergence of trust-region algorithms for constrained minimization without derivatives

In this work we propose a trust-region algorithm for the problem of minimizing a function within a convex closed domain. We assume that the objective function is differentiable but no derivatives are available. The algorithm has a very simple structure and allows a great deal of freedom in the choice of the models. Under reasonable … Read more