César Gutiérrez

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

\varepsilon

On Approximate Efficiency in Multiobjective Programming

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

Improvement sets and vector optimization

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.

Multiplier Rules and Saddle-Point Theorems for Helbig's Approximate Solutions in Convex Pareto Problems

AbstractThis paper is focused on approximate (

Scalarization in set optimization with solid and nonsolid ordering cones

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A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

-efficient) solutions of multiobjective mathematical programs. We introduce a new