Improving the global convergence of Inexact Restoration methods for constrained optimization problems

Inexact restoration (IR) methods are an important family of numerical methods for solving constrained optimization problems with applications to electronic structures and bilevel programming among others areas. In these methods, the minimization is divided in two phases: decreasing infeasibility (feasibility phase) and improving optimality (optimality phase). The feasibility phase does not require the generated points … Read more

A globally trust-region LP-Newton method for nonsmooth functions under the Hölder metric subregularity

We describe and analyse a globally convergent algorithm to find a possible nonisolated zero of a piecewise smooth mapping over a polyhedral set, such formulation includes Karush-Kuhn-Tucker (KKT) systems, variational inequalities problems, and generalized Nash equilibrium problems. Our algorithm is based on a modification of the fast locally convergent Linear Programming (LP)-Newton method with a … Read more

On the best achievable quality of limit points of augmented Lagrangian schemes

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper we propose a somewhat different approach, in the sense that the algorithm itself is … Read more

On the convergence of augmented Lagrangian strategies for nonlinear programming

Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping … Read more

Constraint Qualifications for Karush-Kuhn-Tucker Conditions in Constrained Multiobjective Optimization

The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is … Read more

On optimality conditions for nonlinear conic programming

Sequential optimality conditions have played a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions have been described in conic contexts, where many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and … Read more

On Constraint Qualifications for Second-Order Optimality Conditions Depending on a Single Lagrange Multiplier.

Second-order optimality conditions play an important role in continuous optimization. In this paper, we present and discuss new constraint qualifications to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single … Read more

New Constraint Qualifications with Second-Order Properties in Nonlinear Optimization

In this paper we present and discuss new constraint qualifications to ensure the validity of well known second-order properties in nonlinear optimization. Here, we discuss conditions related to the so-called basic second-order condition, where a new notion of polar pairing is introduced in order to replace the polar operation, useful in the first-order case. We … Read more

Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set … Read more

Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points

Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the … Read more