E-global Proper Eficiency For Set-Valued Optimization Problems In Linear Spaces

The optimization theory plays a very important role in engineering technology,economic management,optimal control and other issues.When solving a real problem,decision makers often need to consider multiple indicators(even infinite or set-value).So set-valued optimization problems have received more and more attention.It is worth noting that the study of the proper efficiency for set-valued optimization problems is closely related to the topology internal of the ordered cone.But when the image space we consider is a real linear space,it does not have a topology.Therefore,how to study the concepts and optimality conditions of the proper efficiency for set-valued optimization problems in linear space has theoretical and practical significance.In this paper,we introduce the concept of global proper efficiency via improvement sets(named as -global proper efficiency)in real linear spaces.The algebraic properties of -global proper efficient point are studied.Furthermore,we obtain the linear scalar characterization,Lagrange multiplier theorem and nonlinear scalar theorem of -global proper efficient solution for set-valued optimization.The thesis is divided into four chapters,the main contents are as follows:In chapter 1,we expound the background of set-valued optimization,the significance and progress of the research of proper efficiency and optimality conditions for set-valued optimization are reviewed,some definitions and theorems related to the research are introduced.In chapter 2,the concept of global proper efficient point via improvement sets in real ordered linear spaces is introduced using the properties of algebraic interior of sets and improvement sets,the relationships between -global proper efficiency and other proper efficiency are discussed.We also explore the existence of -global proper efficient point.In chapter 3,linear scalar characterization and Lagrange multiplier theorem of -global proper efficient solutions for set-valued optimization are given.We introduce the concept of -global proper saddle point for set-valued optimization in linear spaces.The equivalent conditions of -global proper saddle points and the saddle point theorem of -global proper efficiency are obtained.In chapter 4,we generalize the nonconvex separation theorems from the topological space to the linear space by the properties of the nonlinear scalar function via improvement sets in linear spaces.The nonlinear scalar theorem of -global proper efficient solutions and -weakly efficientsolutions for set-valued optimization problems are given,in the case that the objective functions and the feasible set do not have any generalized convexity.