The New Description Of Super Efficient Solutions And Henig Efficient Solutions

A type of nonderivative super subgradient is introduced for a set-valued map. Under some condition its existence theorem is proved. As an appication, under the assumption of near-subconvexlikeness, the sufficient and necessary optimality conditions are established for set-valued optimization problem to obtain super efficient elements.With the help of the properties of generalized second-order composed contingent epiderivative, the necessary optimality conditions are established for set-valued optimization problem to obtain Henig efficient element. The relationship between quasi-invex convex functions which satisfy the nature of control and generalized contingent epiderivative is given. By making use of this properties and the properties of the expansion of the cone, we get the sufficient optimality conditions.