Some Characterizations For Approximate Benson Efficient Solutions In General Linear Spaces

The vector equilibrium problem has a wide application prospect in many fields,such as social economic system,engineering technology.It unifies and broadens variational inequality,economic equilibrium problem,vector optimization,vector complementation and so on.Good results have been obtained in the study of vector equalization in topological linear space.In this thesis,the existence of an approximate Benson efficient solution for constrained set-valued equilibrium problems is studied in a general linear space.The main contents are as follows:Firstly,by using the generalized cone-subconvex-like hypothesis and the separation theorem of convex sets,we discuss the Kuhn-Tucker type and Lagrange type optimality conditions for the approximate Benson efficient solutions of set-valued equalization problems with constraints in general linear spaces.Secondly,the Lagrange multiplier theorem in general linear space is derived,and the concept of saddle point for Lagrange mapping is proposed,and the relationship between the effective solution and the saddle point is given,and the relationship between the effective solution and the duality problem of proper form is discussed.Finally,the research work on approximate Benson efficient solutions for constrained set-valued equilibrium problems in general linear spaces is summarized,and the future research problems are prospected.