Study On Vector Optimization Theory And Related Problems

Vector optimization theory is one of the main research fields of optimization theory and applications. The study of this topic involves many disciplines, such as:convex analysis, nonlinear analysis, nonsmooth analysis, partial ordering theory, and so on. The theory and methods for the vector optimization are widely used in the areas of economic analysis, financial investment, engineering design, environmental protection, military decision making, etc. Thus, the research for this topic has important theo-retical value and practical significance. In this thesis, we mainly study the theory of vector optimization in four aspects:the generalized convexity of vector-valued maps; the approximate solutions of vector optimization problems; the optimality conditions and duality of vector optimization problems and vector variational inequalities, etc.The main research contents are as follows:In Chapter 1, we give brief introduction to the research significance and develop-ment of vector optimization, then we outline the contents studied in this thesis. And we recall some basic concepts and results.In Chapter 2, the concept ofε-cone subpreinvex functions is introduced in Banach space. The relationships with cone convex, cone subconvex, cone preinvex functions are explored. Some properties of cone subpreinvex functions are given. Under the condition of cone subpreinvexity, the optimality conditions and duality results of approxmate efficient solutions for vector optimization problems are obtained.In Chapter 3, a new class of generalized vector-valued arcwise connected functions, termedε-subarcwise connected functions, is introduced. The properties ofε-subarcwise connected function are derived. The optimality conditions and duality of approxmate efficient solutions for vector optimization problems withε-subarcwise connectivity are studied.In Chapter 4, we study the Lagrange multiplier rule for approximate efficient solutions and approximate quasi-efficient solutions of vector optimization problems in Asplund space. Firstly, in the case that the interior of ordering cone is nonempty, we establish the relationships between the vector optimization with set constraint and the scalar optimization problems, by using the separating function as scalar function. And we apply limiting subdifferential and Frechet subdifferential respectively to obtain the Lagrange multiplier type necessary optimality conditions of approximate efficient solutions and approximate quasi-efficient solutions. Secondly, in the case that the interior of ordering cone is empty, we establish the relationships between the vector optimization with cone constraint and the scalar optimization problems, by using the oriented distance function as scalar function. And in this case, we obtain the Lagrange multiplier type necessary optimality conditions of approximate efficient solutions and approximate quasi-efficient solutions. Finally, we apply the partial results presented in this chapter to study the vector variational inequalities. We establish the relationships between the vector optimization problems and vector variational inequalities and give the properties of the solutions for vector variational inequalities.In Chapter 5, we study the approximate quasi-efficient solutions of vector op-timization problems. We obtain the necessary and sufficient optimality conditions of approximate quasi-efficient solutions in terms of Gateaux differentiability at some point. We formulate an approximate Mond-Weir type dual model and establish the duality results.In Chapter 6, we study the solvability for vector variational inequalities and the relationships between the vector variational-like inequalities and the vector optimiza-tion problems. Firstly, we introduce the concept of relaxedεe-pseudomonotone map-ping (weaker than pseudomonotone mapping), under the assumption of relaxedεe-pseudomonotone mapping, some existence results of solutions for vector variational inequalities and weak vector variational inequalities are established. At the same time, we present the solvability results for two classes of approximate vector varia-tional inequalities. Secondly, by using the ideas of penalty function, we establish the relationships between the vector variational-like inequalities and a class of a vector optimization problems.In Chapter 7, the concept of nondifferentiable pseudoinvex functions defined by upper Dini directional derivatives are introduced. Some properties of this class of pseudoinvex functions are given. Several new characterizations of the solution set of nondifferentiable pseudoinvex optimization problems are obtained. By establishing the relationships between the variational-like inequalities and the nondifferentiable pseu-doinvex optimization problems, we give the characterizations of the solution sets of the variational-like inequalities.