Giancarlo Bigi

Second Order Optimality Conditions for Differentiable Multiobjective Problems

A second order optimality condition for multiobjective optimization with a set constraint is developed; this condition is expressed as the impossibility of nonhomogeneous linear systems. When the constraint is given in terms of inequalities and equalities, it can be turned into a John type multipliers rule, using a nonhomogeneous Motzkin Theorem of the Alternative. Adding weak second order regularity assumptions, Karush, Kuhn-Tucker type conditions are therefore deduced.

A new solution method for equilibrium problems

A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required.

Regularity conditions in vector optimization

By exploiting very recent results concerning linear separation between a set and a convex cone, the image space approach allows us to achieve new regularity conditions for vector minimum problems. In particular, we obtain very general Mangasarian–Fromovitz type conditions.

Existence results for strong vector equilibrium problems and their applications

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.

K-epiderivatives for set-valued functions and optimization

Exploiting different tangent cones, many derivatives for set-valued functions have been introduced and considered to study optimality. The main goal of the paper is to address a general concept of K-epiderivative and to employ it to develop a quite general scheme for necesary optimality conditions in set-valued problems.

On sufficient second order optimality conditions in multiobjective optimization

A second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimization

Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints

This paper deals with equilibrium problems with nonlinear constraints. Exploiting a gap function which relies on a polyhedral approximation of the feasible region, we propose two descent methods. They are both based on the minimization of a suitable exact penalty function, but they use different rules for updating the penalization parameter and they rely on different types of line search. The convergence of both algorithms is proved under standard assumptions.

Twelve monotonicity conditions arising from algorithms for equilibrium problems

In the last years many solution methods for equilibrium problems (EPs) have been developed. Several different monotonicity conditions have been exploited to prove convergence. The paper investigates all the relationships between them in the framework of the so-called abstract EP. The analysis is further detailed for variational inequalities and linear EPs, which include also Nash EPs with quadratic payoffs.

Outer approximation algorithms for canonical DC problems

The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. The algorithms rely on a polar reformulation of the problem and exploit an approximated oracle in order to check global optimality. Consequently, approximate optimality conditions are introduced and bounds on the quality of the approximate global optimal solution are obtained. A thorough analysis of properties which guarantee convergence is carried out; two families of conditions are introduced which lead to design six implementable algorithms, whose convergence can be proved within a unified framework.

Uniqueness of KKT multipliersin multiobjective optimization

Abstract The uniqueness of the KKT multipliers of a nonlinear program has been studied ina well-known paper by Kyparisis. In the first part of this note, we show that the characterization obtained in that paper does not provide a satisfactory result for the multiobjective case. Thus, we introduce a new regularity condition, which involves also the objective functions, and we show that it is necessary and sufficient in order to have unique KKT multipliers. Moreover, we use this condition to refine a second-order necessary optimality condition, which has been obtained in a recent paper.