Vector Optimization Problem: Stability, Goodness And Sensitivity Analysis

In this thesis,we focus on stability theory of vector optimization problems based on qualitative analysis and quantitative analysis, including generic stability of weak Pareto-Nash equilibria of multi-objective game, well-posedness and sensitivity analysis for a number of vector optimization problems. It is divided into six chapters and is organized as follows:In chapter 1, we first briefly summarize the research background and significance for vector optimization problems. Moreover, we discuss the development and current researches on generic stability of the solution set, well-posedness and sensitivity analysis for vector optimization problems and others. Finally, we clarify the main research work, innovation and basic framework on the thesis.In chapter 2, we recalled some basic notions and properties in this thesis. It includes the continuity of set- valued mapping and fixed point theorems; Hausdorff distance and noncompact measure; continuity and convexity for vector-valued mapping; efficient points, weak efficient points and nonlinear scalarization function; notions of S- derivatives and Fréchet coderivatives for set- valued mapping.In chapter 3, we investigate generic stability of weak Pareto-Nash equilibria for multi-objective game models under uncertainty. Firstly, implanting the uncertain parameters into generalized game models and generalized multi-objective game models and using strong Berge equilibria to replace Nash equilibria, we can achieve the "first refinement" on solutions for these two types of game models through proof of the existence of solutions of generalized game models under uncertainty and generalized multi-objective game models under uncertainty. Secondly, based on results of the existence of these two types of game models, we give generic stability of solutions of generalized game models under uncertainty and generalized multi-objective game models under uncertainty by using the axiomatic method, namely in the sense of Baire classification the most generalized game under uncertainty and generalized multi-objective game models under uncertainty is essential, which also achieve "double refinement" on solution for these two types of game models.In chapter 4, we investigate generalized strong well-posedness and strong well-posedness(in short: GS-wp and S-wp) for a number of vector optimization problems by using bounded rationality models and investigate generic Tykhonov well-posedness on efficient solutions for a class of vector optimization problems. Firstly, based on the bounded rationality models, we define a unified notion on Levitin-Polyak and Hadamard well-posedness for a number of nonlinear problems, which is called strong well-posedness. Furthermore, a unified approach on sufficient conditions and metric characterizations of generalized well-posedness and well-posedness for nonlinear problems is established under bounded rationality models framework. Based on the unified approach and the nonlinear scalarization function, we give sufficient conditions and metric characterizations on generalized strong well-posedness and strong well-posedness for set-valued vector qusi-variational inequalities, vector qusi-equilibrium problems and symmetric vector qusi-equilibrium problems. Finally, we give generic uniqueness on Pareto efficient solution and weak Pareto efficient solution of a class of vector valued optimization problems by way of generic uniqueness of solution of nonlinear problems and two kinds of scalarization functions. Therefore, we prove that pointwise Tykhonov of the class of vector valued optimization problems based on generic property.In chapter 5, we investigate sensitivity analysis for a number of vector optimization problems by using S- derivative and Fréchet coderivatives. Firstly, we study S- derivative estimations and Fréchet subdifferential estimations on a kind of set-valued gap function by using S- derivative of the original space and Frechet derivative of the dual space. Based on above results, we give Sderivative estimation and Fréchet subdifferential estimation on perturbed mapping for the parametric vector equilibrium problem and the parametric multi-objective optimization problems because of some equivalent relationships between them. So we achieve a quantitative characterization on local robustness for the optimal value of parametric vector equilibrium problem and the parametric multi-objective optimization problems under some uncertain parameters disturbances. Finally, on the basis of S- derivative estimation of parametric vector equilibrium problem and the parametric multi-objective optimization problems, we derive Sderivative estimations of efficient solutions of parametric vector equilibrium problem and the parametric multi-objective optimization problems by using S- derivative estimations of a kind of parametric variational system. Therefore, we obtain a quantitative characterization on local robustness for solution set of parametric vector equilibrium problem and the parametric multi-objective optimization problems under some uncertain parameters disturbances.In chapter 6, we briefly summarize the results of this thesis.