Second-order Optimality Conditions For Set-valued Optimization Problems

The concept of cone subarcwise connected set-valued map is introduced. Two examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then,by virtue of generalized second-order contingent epiderivative, second-order optimality necessary conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problem. When objective function is cone subarcwise connected, a second-order optimality sufficient condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problem.A new kind of second-order tangent derivative, second-order M? 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 M? composed derivative, under the assumption of nearly cone-subconvexlikeness, by applying a separation theorem for convex sets, second-order Fritz John and Kuhn-Tucker optimality necessary conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a second-order Kuhn-Tucker optimality sufficient 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 conditions are derived in terms of second-order M? composed tangent derivatives.