Characterizations Of Generalized E-Benson Proper Efficient Solutions In Vector Optimization

Study on characterizations of properly efficient solutions of vector optimization problems is an important research aspect in the field of the theory and methods of vector optimization problems.Scalarization methods are important methods to study the characterizations of solution for vector optimization problems,mainly including linear scalarization method based on generalized convexity and corresponding alternative theorem,and nonlinear scalarization method based on nonlinear scalarization function and corresponding non-convex separation theorem.This paper mainly focuses on some characterizations of the generalized E-Benson properly efficient solutions of the vector optimization problems by means of improvement sets,algebraic interior,algebraic closure and quasi-interior,including linear scalarization characterizations of the generalized E-Benson properly efficient solutions in the interior of algebra,Lagrangian multiplier theorem,saddle point theorem,duality theorem,and linear scalarization theorem for generalized E-Benson properly efficient solutions under quasi-interior conditions.Chapter 1 mainly introduces the research background and advancements of the vector optimization problems,and some concepts and basic tools needed for the research work.Chapter 2 mainly focuses on the concept of generalized E-Benson properly efficient solutions and scalarization properties in the algebraic internal sense.By means of algebraic internality and algebraic closure the generalized E-Benson properly efficient solutions are proposed in the real linear space.And under the assumption of the near E-subconverity,linear scalarization theorem,Lagrangian multiplier theorem,,saddle point theorem and duality theorem for E-Benson's real efficient solutions of set-valued vector optimization problem is established.Chapter 3 mainly focuses on quasi-interior properties of the improved set and linear scalarization properties of the generalized E-Benson properly efficient solutions in the sense of quasi-interior.Firstly,the quasi-interior properties of the improved set are obtained in the real nondegenerate segregating locally convex topological vector space by means of the recovery cone and quasi-interior,and then by means of separation theorem in the sense of quasi-interior linear scalarization theorem of the generalized E-Benson properly efficient solutions of set-valued vector optimization problems is established under the assumption of adjacent E-submergence convexity.