Sensitivity Analysis in Parametrized Convex Vector Optimization

Abstract

This paper provides some results concerning sensitivity analysis in parametrized Ž convex vector optimization. We consider three types of perturbation maps i.e., . perturbation map, proper perturbation map, and weak perturbation map accord-Ž ing to three kinds of solution concepts i.e., minimality, proper minimality, and . weak minimality with respect to a fixed ordering cone for a vector optimization problem. As for general vector optimization, authors have already established the behavior of the above three types of perturbation maps by using the concept of contingent derivatives for set-valued maps in finite dimensional Euclidean spaces. In this paper we concentrate on convex vector optimization and provide quantitative properties of the perturbation maps under some convexity assumptions. Namely, we investigate the relationships between the contingent derivatives of the perturbation maps and those of the feasible set map in the objective space.

Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem

“…Recently, multi-objective parametric mathematical programming problems have been studied by several mathematicians. We refer the reader to [11,22,23,33,34,35,37,38] for papers which have a close connection to the present work. Tanino's papers [37,38] are among the first results in this direction.…”

Global and local optimality conditions in set-valued optimization problems

“…The study of multiobjective optimization problems involving set-valued maps (as well as those involving vector-valued functions) has developed in the recent years some generalizations of convexity concepts. Among these generalizations, a widely used class of concepts is the so-called convexlikeness which contains in the case of set-valued maps at least four items used in the papers [11,14,18,20,21,22]. The main purpose of this section is to complete this class and to study the connections between these concepts; also we prove here an alternative Gordan-Farkas-type theorem which generalizes some results in [22] and this gives us the possibility to extend in a more general setting some necessary optimality conditions for weak minimum points from [14,21,22].…”

Clarke coderivatives of efficient point multifunctions in parametric vector optimization

“…For example, [2,3] has studied the behavior of F via the concept of contingent derivative introduced by Aubin [4]. Various sensitivity analysis results in this direction can be found in [5][6][7][8]. Bednarczuk and Song [9] (also see [10]) introduced the notion of generalized contingent epiderivative for a multifunction and used it to study the sensitivity of a family of parametric optimization problems with multifunctions.…”