On H(?)lder Continuity And Other Properties In Vector Optimization

In this thesis, we investigated the H?lder continuity of perturbed set-valued solution mapping, the lower and upper, Lipshitz/H?lder continuity of approximate set-valued solution mapping for (vector) equilibrium problems or (vector) quasiequilibrium problems, vector minimax problems, vector saddle point problems and multiobjective Lagrangian duality problems in the lexicographic order, respectively. This thesis is divided into eight chapters. It is organized as follows:In Chapter 1, we describe the development and current researches on the topic of vector optimization, including the semicontinuity, Lipshitz/H?lder continuity of solution mappings to the vector optimization and other problems, and the vector optimization in lexicographic order. We also give the motivation and the main research work.In Chapter 2, some definitions and their properties, which are frequently used, are shown.In Chapter 3, we introduce the concept of weak pseudomontone for set-valued mapping. Based on this assumption, we discuss the H?lder continuity of perturbed solution mapping for generalized vector quasiequilibrium problems with set-valued mapping. As an application of our main results, we establish the error estimates of the distance between perturbed solution mapping and exact solution mapping for vector variation inequalities and vector equilibrium problems, respectively.In Chapter 4,we obtain some results on the error estimates of perturbed solutions to parametric vector quasiequilibrium problems in metric spaces. Under some special cases, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point.In Chapter 5, we introduce the concept of strongly monotone for vector-valued mapping. Then, the various properties of this mapping are discussed in this chapter. Based on the assumptions of strongly monotone, we investigate the H?lder continuity of the perturbed solution mapping, which is set-valued one, for a weak vector equilibrium problem. Since the strongly monotone mapping for vector-valued mapping does not imply the singleton of the solution mapping of the weak vector equilibrium problem, our main results in this chapter are a correction of the corresponding ones of Anh and Khanh (2008c).In Chapter 6, by virtue of the set inclusion relation, we investigate the lower semicontinuity and continuity of the approximate solution mapping for equilibrium problems. Based on these main results, a scalarization representation of the solution mapping and a property involving the union of a family of lower semicontinuous set-valued mappings, we also discuss the lower semicontinuity and continuity of the approximate solution mapping for vector equilibrium problems. Futhermore, we investigate the Lipshitz/H?lder continuity of the approximate solution mapping for equilibrium problems, which strengthenes the results of continuity of the approximate solution mapping for vector equilibrium problems. Appling our main results to variational inequalties and optimization problems, we establish the Lipshitz/H?lder continuity of their approximate solution mapping, respectively.In Chapter 7, we first study the properties of the efficient points in lexicographic order, and introduce a new vector-valued mapping in lexicographic order. Based on these concepts and their properties, we investigate the vector minimax and saddle point problems. The relations between the minimax theorem and the saddle point theorem are also discussed in this chapter. Then, based on the efficient point and vector-valued Lagrangian function, we study the multiobjective programming and Lagrangian duality problems, and establish the weak (strong) duality theorems and Lagrangian multiplier rules. We furthermore discuss the relations between the efficient point of multiobjecive programming problems and saddle point problems in lexicographic order.In Chapter 8, we summarize the results of this thesis and make some discussions.