New second-order radial epiderivatives and nonconvex set-valued optimization problems

Abstract

In this paper, we introduce the generalized second-order composed radial epiderivative of set-valued maps and establish a few relations between the epiderivative and the generalized second-order composed contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed radial epiderivative, we obtain the necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions of unconstrained set-valued optimization problems without convexity conditions. Furthermore, the necessary optimality conditions and sufficient optimality conditions are obtained for Henig efficient solutions of constrained set-valued optimization problems. Some of obtained results improve or imply the corresponding ones in recent literature.