The Research On Super Efficient Solutions And Strictly Efficient Solutions For Set-Valued Optimization Problems

In Hausdorff locally convex spaces,for the set-valued vector optimization problem(SOP),the concept of the super efficient generalized gradient of set-valued map is introduced by means of contingent epiderivative.Under the condition of connectedness,its existence is proved and the equivalent characterization of this super efficient generalized gradient is established by the separation theorem of convex sets and contingent epiderivative.In Hausdorff locally convex spaces,for group multiobjective decision making problems,the joint super efficient solutions of set-valued maps are introduced by means of super efficient numbers of alternatives.The optimality necessary and sufficient conditions for the joint super efficient solutions are obtained in the sense of generalized gradient,and the Kuhn-Tucker and Lagrange optimality conditions are established.In Hausdorff locally convex spaces,for the set-valued vector optimization problem(SOP),the concept of super saddle point for a set-valued map is introduced by means of Lagrange set-valued map.Two scalarization lemmas are proved,and the theorem of super saddle points and the equivalent characterization of super saddle points are obtained by the separation theorem of convex sets.We introduce a new notion of convexity for set-valued maps with quasi interior, called qc-cone-convexlikeness,and compare this convexity notion with ic-cone-convexlikeness and near cone-subconvexlikeness.In spaces where intD=(?)and qiD≠(?) are satisfied,this convexity notion can be used as the main tool to establish an alternative theorem.Using this theorem,we obtain necessary conditions of strictly efficient solutions for a class of set-valued vector optimization problems with both the objective ordering cone and the constraint ordering cone having empty interior.