Studies On Well-posedness And Stability Of Set Optimization Problem With Variable Ordering Structures

In this paper,we focuses on the well posedness and stability of set optimization problems in the sense of variable order relations of sets.Under appropriate conditions,the sufficient and necessary conditions of LP Well-posedness and the sufficient conditions and noncompact characterizations of generalized LP Well-posedness are given for set optimization problems in the sense of two different partial order structures.Further,the stability theorems of Painlev ′e-Kuratowski upper(lower)convergence of approximate solution sets and approximate weak solution sets are ginen for set optimization problem under different variable set order relations.The main results obtained in this paper extend and develop some corresponding ones existing in the relevant research results in the sense of fixed order relations,and enrich the theoretical research of set optimization.The research methods and processing skills used also provide reference for the research of other aspects of set optimization in this paper.This paper is divided into four chapters,the specific contents are as follows:In chapter 1,this chapter gives an overview of the historical background of set optimization problem,and then analyzes the research status of set optimization problem,and then puts forward the specific content and research significance of this paper.In the last part of this chapter,some definitions and known conclusions are given,which are necessary for the follow-up research.In chapter 2,we mainly discusses the Levitin-Polyak Well-posedness of set optimization problems with variable partial order structure.Firstly,based on the variable partial order structure,two different variable order relations of sets are given,and the optimal solution,LP-approximate solution sequence and the concepts of LP Wellposedness and generalized LP Well-posedness of set optimization problems in the sense of these two different variable order relations are given.Then,the analytical properties of the approximate solution sets are discussed.Furthermore,using these properties,and under the appropriate assumptions of cone continuity and cone convexity,the sufficient and necessary conditions of LP Well-posedness and the sufficient conditions and noncompact characterizations of generalized LP Well-posedness are given for set optimization problems in the sense of two different partial order structures.In chapter 3,the stability of set optimization problem based on variable partial order structure is considered.According to the variable partial order structure,a variety of variable order relations of sets are introduced,including upper(lower)order relation,weak upper(lower)order relation,ε-upper(lower)order relation and ε-weak upper(lower)order relation,and the corresponding concepts of optimal solution and optimal solution sets are given.Then,the relationship between these optimal solution sets is discussed,and the equivalence characterization of approximate solution is established.Finally,under the infinite upper continuity of cone-valued mapping,the semi-continuity and cone convexity conditions of target set-valued mapping,the stability theorems of Painlev ′e-Kuratowski upper(lower)convergence of approximate solution sets and approximate weak solution sets are proved for set optimization problem under different variable set order relations.Moreover,a concrete example is given to illustrate the effectiveness of the main results.In chapter 4,we briefly summarizes the main research results,and put forward the future research ideas and ideas.