The Optimality Conditons, Saddle Points And Duality For The Nearly Subconvexlike Vector Optimization Of Set-Maps In Ordered Linear Space

In this thesis, some topics in vector optimization of set-valued maps are discussed. The concept of nearly subconvexlike set-valued map is introduced and some important properties of the new concept are obtained in the linear space. Under the assumption, a theorem of the Gordan-Farkas type alternative is built up, by using the separation theorem of convex sets in a real linear space. In order linear space, the mathematics models for vector optimization problems of nearly subconvexlike set-valued maps are established. By the theorem of alternative, the optimality sufficient condition and necessary condition and the theorems of scalarization for the vector optimization of nearly subconvexlike set-valued maps are established in the sense of the weakly efficient solution. After a Lagrange map is brought in vector optimization of set-valued maps, the concept of the saddle points is defined. The theorem of existence of Lagrange multipliers is proved, and the relations between saddle points and weakly efficient solutions are discussed in order linear space. Based on these, Lagrange duality results of vector optimization of set-valued maps are given, including weak duality theorem, strong duality theorem, converse duality theorem.