Second-Order Optimality Conditions For Strongly Efficient Elements Of Set-valued Optimization

The strong efficiency of set-valued optimization in real normed spaces was considered.By applying the second-order contingent derivatives, the properties of basic functional and strong efficient element, the second-order necessary optimality condition of unconstrained set-valued optimization was established for the objective function being nearly cone-subconvexlike, and sufficient condition was presented under cone-convexity hypothesis.With the properties of a generalized second-order contingent set and a generalized second-order cone-directed contingent derivative for set-valued maps, the generalized second-order necessary and sufficient conditions were obtained for a set-valued optimization problem, whose constraint condition was determined by a fixed set or a set-valued map.