Second-Order M-Composed Tangent Derivative and Its Applications

Abstract

A new kind of second-order tangent derivative, second-order [Formula: see text]-composed tangent derivative, for a set-valued function is introduced with help of a modified Dubovitskij–Miljutin cone. By using the concept, several generalized convex set-valued functions are introduced. When both the objective function and constrained function are second-order [Formula: see text]-composed derivable, under the assumption of nearly cone-subconvexlikeness, by applying a separation theorem for convex sets, Fritz John and Kuhn–Tucker second-order necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn–Tucker second-order sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem. A unified second-order necessary and sufficient optimality condition is derived in terms of second-order [Formula: see text]-composed tangent derivatives.