Vector Quasi-equilibrium Problem With Set Constraints: Linear Separation, Saddle Points,optimality Conditions And Error Bounds

Vector Equilibrium problems, as the meaningful extension of variational inequalities and complementarity problems, have been widely used in many fields,such as economic and financial, engineering, transportation, optimization control. The optimal condition, Lagrangian saddle point theory and linear separation are the main courses of vector quasi(relatively weak-)equilibrium problems(for short, VQEP,qrw-VQEP).The method of image space analysis has a strong advantage on the study of the equilibrium problem and optimization problem. Especially, in recent years, there has been an increasing interest in investigating vector optimization problem by using the method of image space analysis. However, this method is seldom used in vector quasi equilibrium problems at present, especially when in an infinite dimensional image.Therefore, the main purpose of this dissertation is to study linear separation,optimality conditions, saddle point theorems, gap functions and solution set of error bounds of vector quasi-equilibrium problem with set constraints with infinite dimensional image by utilizing the image analysis.The main results of this dissertation related to vector quasi equilibrium problems with set constraints are as follows:(1) Chapter 3 characterizes the linear separation for VQEP(res.,qrw-VQEP) and saddle points theorem of generalized Lagrangian functions by using the quasi interior of a regularization of the image.(2) In Chapter 4, we characterize the optimality conditions of VQEP(res.,qrw-VQEP). Under the generalized Slater condition, we obtain Lagrangian type sufficient optimality conditions for qrw-VQEP. In addition, under additional assumptions, we obtain Lagrangian type sufficient optimality conditions of VQEP.(3)In Chapter 5, we shall give gap functions for VQEP(res.,qrw-VQEP) and under strong monotonicity conditions, we prove that the solution set of VQEP(res.,qrw-VQEP) admits an error bound with respect to the gap function.