Tangent Derivatives Of Set-Valued Maps And Applications To Group Decision Making Problems

A kind of second-order tangent derivatives is introduced,with which a second-order necessary optimality condition is established for set-valued optimization with variable ordering structure in the sense of local weakly nondominated points.Under special circumstances,a first-order necessary optimality condition is obtained.The relationship to second-order contingent tangent derivatives for the sum of two set-valued maps is given under some constraint qualification indued by modified Dubovitskij-Miljutin tangent cones.Further more,a necessary optimality condition is obtained where the objective and constraining functions are considered separately with respect to second-order contingent tangent derivatives.The relation between optimal equilibrium solution and weakly effective solution of group decision is discussed.Under the assumption of generalized cone-subconvexlikeness,scalarization theorem of optimal equilbrium solution is established.By a new kind of second-order tangent derivatives,necessary optimality conditions are established for an optimal equilbrium solution of group decision making problems.