The Optimality Conditions And Applications Of Vector Equilibrium Problems With Constraints

Vector optimization、vector variational inequality are widely used in the areas of the economic analysis, financial management, ecological protection, systems engineering, etc. Vector equilibrium problem is a natural extension of vector optimization, vector variational inequality. Vector equilibrium problems also include vector Nash equilibrium problems and vector complementarity problems. Therefore, the study of vector equi-librium problem will promote the investigation of these related issues, The ideas and methods on vector optimization and vector variational inequality could also contribute to the development of vector equilibrium theory.Optimality conditions plays a prominent part in the study of the algorithm of vec-tor equilibrium problem and establishing the duality theory. While most of the vector equilibrium problems have constraints, it has very important theoretical and practical significance to study the optimality conditions of vector equilibrium problem with con-straint.In this paper, we discuss necessary conditions of weakly efficient solutions. Henig efficient solutions for nonconvex vector equilibrium problems with constraint from two aspects of nondifferentiable and differentiable. Furthermore, by the assumption of invex-ity and cone-convexity, we get sufficient conditions for these two kinds solutions. Some new results are obtained. This paper is divided into four chapters:In chapter one, we briefly introduce the research background and motivation of this dissertation, and sone difficulty may appear. Some definitions and lemmas are also introduced in this chapter. In chapter two, we provided the necessary conditions of weakly efficient solutions、Henig efficient solutions for vector equilibrium problems with constraints under the conditions of strongly compactly Lipschitz and stricrly differentiable. Moreover, by the assump-tion of invexity and convexity, we give sufficient conditions for those solutions. Futher some necessary and sufficient conditions of corresponding solutions for vector equilibrium problems with constraint. As applications, we give necessary and sufficient conditions for corresponding solutions to vector optimization problems. In chapter three, there is no sig-nificance of discussing weakly efficient solutions when the inter of ordering cone xis empty, so we should discuss efficient solutions. Using the function of△s, we obtain the neces-sary conditions of efficient solutions. And further assume that objective and constraint functions to meet certain conditions (for example, invex functions and cone-convex func-tions), we can get sufficient conditions for corresponding solutions to vector equilibrium problems with constraint. In chapter four, we discuss second order optimality conditions for vector optimization problems with constraint. Since the first derivative equal to zero cannot be determined whether there are extremes, we must use two order derivative to judge. We study vector optimization problem with constraint in real Hausdorff locally convex space in this chapter. Sufficient conditions for weakly efficient solutions, Henig efficient solutions, and supper efficient solutions are obtained by the second order invex function. Then the relations between saddle points and these kinds of solutions are also given, and the duality of vector optimization problem are provided.