Optimality Conditions For Vector Equilibrium Problems With Constraints

Vector equilibrium problems is a kind of general model, which contains vector opti-mization problems, variational inequality problems as special cases. Vector equilibrium model can not completely instead of special purpose of these models. But, the common properties of them can be unified handled rather than study each model. On the other hand, the methods and ideas of vector optimization problems and variational inequality problems can also promote the development of vector equilibrium problems. Therefore, vector equilibrium problems becomes the hot issues of optimization problems. The s-tudy of optimality conditions can provide theoretical basis for the algorithm of vector equilibrium problems, moreover, it also will promote the investigation of dual theory and Stability theory of vector equilibrium problems. Thus, it is necessary to investigate the optimality conditions of vector equilibrium problems.In this paper, we mainly discuss the optimality conditions for weakly efficient solu-tions, efficient solutions and approximate weakly efficient solutions of vector equilibrium problems with constraints in the linear spaces and the topological spaces. The contents are as follow:In chapter one, we briefly introduce the background and status, some definitions and lemmas are also introduced in this chapter. In chapter two, we firstly discuss the necessary and sufficient conditions for weakly efficient solutions of vector equilibrium problems with constraints under the assumption of the mapping is near-subconvexlikeness in the linear spaces. As we know, it is no significant for weak efficient solution when the algebraic inte-rior of the cone is empty. Therefore, we study the necessary and sufficient conditions for efficient solutions of vector equilibrium problems with constraints under the assumption of the mapping is ic-cone-convexlikeness, moreover, we discuss the optimality condition for efficient solution of the vector equilibrium problems in locally convex spaces. Finally, we give the necessary and sufficient conditions for weakly efficient solutions to variational inequality problems as applications. As we know, there is only algebraic structure in the linear space. Thus, many properties of the functions (such as, continuity and differentia-bility) we can not use. Hence, in chapter three, we firstly discuss the optimality conditions for weakly efficient solutions of vector equilibrium problems with constraints in the topo-logical spaces by using the differentiability of mapping. Secondly, we define a new class of generalized convex functions, termed cone-subsemipreinvex functions. Moreover, we study their properties and give the example to illustrate that the cone-subsemipreinvex functions is extension for cone-semipreinvex functions. Finally, we obtain the sufficient conditions for approximate weakly efficient solutions of vector equilibrium problems with constraints under the mapping is cone-subsemipreinvex.