Elvira Hernández

A unified vector optimization problem: complete scalarizations and applications

The aim of this article is to introduce and analyse a general vector optimization problem in a unified framework. Using a well-known nonlinear scalarizing function defined by a solid set, we present complete scalarizations of the solution set to the vector problem without any convexity assumptions. As applications of our results we obtain new optimality conditions for several classical optimization problems by characterizing their solution set.

Optimality conditions for a unified vector optimization problem with not necessarily preordering relations

This paper studies a general vector optimization problem which encompasses those related to efficiency, weak efficiency, strict efficiency, proper efficiency and approximate efficiency among others involving non necessarily preordering relations. Based on existing results about complete characterization by scalarization of the solution set obtained by the same authors, several properties of (generalized) convexity and lower semicontinuity of the composition of the scalarizing functional and the objective vector function are studied. Finally, some optimality conditions are presented through subdifferentials in the convex and nonconvex case.

Characterizing efficiency without linear structure: a unified approach

In this paper, we study a general optimization problem without linear structure under a reflexive and transitive relation on a nonempty set E, and characterize the existence of efficient points and the domination property for a subset of E through a generalization of the order-completeness condition introduced earlier. Afterwards, we study the abstract optimization problem by using generalized continuity concepts and establish various existence results. As an application, we extend and improve several existence results given in the literature for an optimization problem involving set-valued maps under vector and set criteria.

On solutions of set-valued optimization problems

We consider two criteria of a solution associated with a set-valued optimization problem, a vector criterion and a set criterion. We show how solutions of a vector type can help to find solutions of a set type and reciprocally. As an application, we obtain a sufficient condition for the existence of solutions of a set type via vector optimization.

Some equivalent problems in set optimization

Given a set-valued optimization problem (P), there is more than one way of defining the solutions associated with it. Depending on the decision makers preference, we consider the vector criterion or the set criterion. Both criteria of solution are considered together to solve problem (P) by reducing the feasible set.

Scalar multiplier rules in set-valued optimization

In this work we establish necessary and sufficient conditions for weak minimizers of a constrained set-valued optimization problem. These conditions are given by means of multiplier rules formulated in terms of contingent epiderivatives of scalar set-valued maps. We consider that the image spaces are finite dimensional and the set-valued maps are convex and stable. For these conditions, our results provide a scalar version of analogous existing results in the literature under weaker existence conditions. Moreover we prove that the multiplier rules can be computed in terms of the directional derivative of associated maps of infima.

Benson Proper Efficiency in Set-Valued Optimization on Real Linear Spaces

In this work, a notion of cone-subconvexlikeness of set-valued maps on linear spaces is given and several characterizations are obtained. An alternative theorem is also established for this kind of set-valued maps. Using the notion of vector closure introduced recently by Adan and Novo, we also provide, in this framework, an adaptation of the proper efficiency in the sense of Benson for set-valued maps. The previous results are then applied to obtain different optimality conditions for this Benson-vectorial proper efficiency by using scalarization and multiplier rules.

Lagrange duality, stability and subdifferentials in vector optimization

Abstract Lagrange duality theorems for vector and set optimization problems which are based on a consequent usage of infimum and supremum (in the sense of greatest lower and least upper bounds with respect to a partial ordering) have been recently proven. In this note, we provide an alternative proof of strong duality for such problems via suitable stability and subdifferential notions. In contrast to most of the related results in the literature, the space of dual variables is the same as in the scalar case, i.e. a dual variable is a vector rather than an operator. We point out that duality with operators is an easy consequence of duality with vectors as dual variables.

Weak and Strong Subgradients of Set-Valued Maps

We consider weak subgradients of a set-valued map and present a new notion of strong subgradient. We study their properties and compare our constructions and results with other developments. We give existence conditions of both types and establish several optimality conditions in terms of set optimization.

About Hahn-Banach extension theorems and applications to set-valued optimization

In this paper we revise several generalizations of the algebraic Hanh-Banach extension theorem for set-valued maps and give new versions in terms of set-relations and selections. We apply our results by establishing optimality conditions for a set-valued constrained optimization problem considering several types of solutions. Finally, we give necessary conditions for the existence of affine selections via affinelike maps in the framework of linear spaces.