Some Property Of The Solution Set For Vector Equilibrium And Set Optimization Problems

With going deep of the research work,vector quasi-equilibrium problems(GVQEP),have been putted forward and investigated by many people in different settings.The main reason is that this model provides a unified framework for many important problems.such as vector quasi-optimization problems,vector quasivariational inequality problem,Generalized Nash equilibrium problem and vector saddle point problem,etc.Also,the research results of vector quasi-equilibrium problem are applied to various real life problems,such as semi-infinite programs,mathematical program with equilibrium control and so on.On the other hand,set optimization is also widely studied and applied to a variety of practical problems.For instance,gap functions for vector variational inequalities,fuzzy optimization,inverse problems for partial dierential equations,image processing and mathematical economics all lead to optimization problems that can be conveniently cast as set-valued optimization problems.Among them,we can see that the solvability of solutions and stability is an important topic in optimization theory and application.Therefore,this paper mainly studies the solvability of quasi-equilibrium problems and the stability of solution sets of set optimization problems.The paper is divided into four chapters.In the first chapter,we mainly introduced the historical background and research status of the quasi-vector equilibrium problem and set optimization problems.Next,the background knowledge of some concepts used in this paper and some existing conclusions are givenIn the second chapter,we mainly considered the existence of solutions for two kinds of generalized strong vector quasi-equilibrium problems with variable ordering structure.Firstly,a key local property of cosmically upper continuity for cone-valued mapping is discussed by using the concept of cosmically upper continuity rather than upper semicontinuity for cone-valued mapping.Next,under suitable conditions of cone-continuity and cone-convexity for equilibrium mapping,several existence theorems of solutions and closedness of solution sets are established for these two kinds of generalized strong vector quasi-equilibrium problems with variable ordering structure.Moreover,an example is given to illustrate the validity of our theorems.In the third chapter,we investigated the stability of the solution sets for set optimization problems via improvement sets.Firstly,we consider the relations among the solution sets for optimization problem with set optimization criterion.Furthermore,we studied the property of level mapping via important set and get some useful results.Then,by using these related conclusions,we established the upper semi-continuity,Hausdor upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets.In the fourth chapter,we summarize the results and conclusions obtained in this thesis and puts forward a new prospect.