Bienvenido Jiménez

A Set-Valued Ekeland's Variational Principle in Vector Optimization

This paper deals with Ekelands variational principle for vector optimization problems. By using a set-valued metric, a set-valued perturbed map, and a cone-boundedness concept based on scalarization, we introduce an original approach to extending the well-known scalar Ekelands principle to vector-valued maps. As a consequence of this approach, we obtain an Ekelands variational principle that does not depend on any approximate efficiency notion. This result is related to other Ekelands principles proved in the literature, and the finite-dimensional case is developed via an

On Approximate Solutions in Vector Optimization Problems Via Scalarization

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Strict Efficiency in Set-Valued Optimization

-efficiency notion that we introduced in [Math. Methods Oper. Res., 64 (2006), pp. 165-185; SIAM J. Optim., 17 (2006), pp. 688-710].

On Approximate Efficiency in Multiobjective Programming

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems.

Improvement sets and vector optimization

In this paper, we develop the notion of a

Optimality conditions via scalarization for a new [epsilon]-efficiency concept in vector optimization problems

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Strict Minimality Conditions in Nondifferentiable Multiobjective Programming

-minimizer (or strict efficiency) for a set-valued map. Its properties and relations with other similar notions are studied. In special circumstances, under suitable conditions, we prove that a point is a

Scalarization in set optimization with solid and nonsolid ordering cones

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On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

-minimizer of a vector function if and only if it is a

FIRST AND SECOND ORDER SUFFICIENT CONDITIONS FOR STRICT MINIMALITY IN MULTIOBJECTIVE PROGRAMMING

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