Optimality criteria in set-valued optimization

Abstract

The main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.2000 Mathematics subject classification: primary 49J53, 90C30.

Weak Clarke epiderivative in set-valued optimization

“…The following notion of Clarke epiderivative for set-valued maps was introduced by Lalitha, Dutta and Govil [10] where the epiderivative is a single valued map.…”

“…It can be seen that in case of contingent derivative, necessary and sufficient optimality conditions do not coincide under the standard assumptions [5]. Therefore, while characterizing optimality conditions, it is useful to consider derivatives involving epigraph of set-valued maps rather than their graph [8,10,13].…”

“…Since contingent cones are not necessarily convex, Sach and Craven [13] introduced a derivative by relating Clarke tangent cone to the epigraph where the derivative is a setvalued map. Later on Lalitha, Dutta and Govil [10] introduced the notion of Clarke epiderivative of a set-valued map in terms of Clarke tangent cone where epiderivative is a single valued map.…”

About contingent epiderivatives

“…On the one hand, using different kinds of minimal elements, see [11,13]; on the other hand, considering different types of tangent cones, see [3,6,12,15]. However, all definitions have the same mathematical structure.…”

Proper Clarke Epiderivative in Set-Valued Optimization

“…Chen [4] introduced the notion of tangent epiderivative (also termed Clarke epiderivative in [15]) of a set-valued map in terms of Clarke tangent cone where epiderivative is a single valued map. Lalitha, Dutta and Govil [15] obtained optimality conditions using this notion Clarke epiderivative for weak mnimizers. Recently Lalitha and Arora [16] introduced the notion of weak Clarke epiderivative in terms of weak minimizers of Clarke tangent cone.…”