New second-order radial epiderivatives and applications to optimality conditions

Abstract

In this paper, we introduce the second-order weakly composed radial epiderivative of set-valued maps, discuss its relationship to the second-order weakly composed contingent epiderivative, and obtain some of its properties. Then we establish the necessary optimality conditions and sufficient optimality conditions of Benson proper efficient solutions of constrained set-valued optimization problems by means of the second-order epiderivative. Some of our results improve and imply the corresponding ones in recent literature.

New Properties of Higher-Order Radial Sets and Higher-Order Radial Derivatives and Applications to Optimality Conditions

“…Wang et al [14] proposed the higher-order weak radial epiderivative of a set-valued map, and obtained the optimal-ity conditions for non-convex set-valued optimization problems under the weakly efficiency. Zhang and Wang [15] introduced the second-order weakly composed radial epiderivative of set-valued maps, and obtained the necessary optimality conditions of Benson proper efficient solutions for the constrained set-valued optimization problems without the assumptions of generalized cone-convexity. Peng et al [16] provided the higher-order weak lower inner Studniarski epiderivative for set-valued maps, and obtained KarushCKuhnCTucker necessary optimality conditions for Benson proper efficient solutions of the constrained set-valued optimization problems.…”

“…Motivated by the work reported in [10][11][12][13][14][15][16], several new properties are obtained for higher-order upper radial sets and higher-order upper radial derivatives introduced in [13],…”