The Optimality Conditions And Duality Of Set-Valued Maps

In this thesis,some topics in vector optimization theory with set-valued maps are discussed.The concept of generalized subconvexlike set-valued map is defined,and some important properties of the new concept are discussed in linear space.Under the assuption of generalized subconvexlikeness,a Gordan-Farkas type alternative theorem is proved. The concept of -generalized convex set-valued map is defined in linear topological space. Under the assuption of -generalized convexity,relative interior is introduced,and a Farkas-Minkowski type alternative theorem is proved. Under the assuption of generalized subconvexlikeness ,the optimality conditions of set-valued optimization problems in linear space are established by using obtained Gordan-Farkas type alternative theorem.Under the assuption of near subconvexlikeness and -generalized convexity, the optimality conditions of set-valued optimization problems in linear topological space are established by using alternative theorem of near subconvexlikeness and obtained Farkas-Minkowski type alternative theorem.The concepts of super efficient solution and -super efficient solution are defined in normed space,and the optimality conditions of set-valued optimization problems are established under the assuption of semi-preinvexity. In normed space,the theorem of existence of Lagrangian multipliers is proved,and -super saddle point is defined,and the relations between -super saddle point and existence of -super efficient solution are discuss.Based on these, Lagrange -super dual results of set-valued optimization problem are given,including weak duality,strong duality,converse duality etc.