Saddle Point Of Cone-constrained Vector Equilibrium Problem And Related Problems

The vector equilibrium problem is a hot issue in the field of operations research and nonlinear analysis.It is widely used in many fields such as technology,mathematical economy and social economic system.In this dissertation,the existence condition of the saddle point of the cone-constrained vector equilibrium problem is considered in the Hausdorff locally convex topological vector space,and the saddle point and related problems of a special kind of cone-constrained vector equilibrium problem are considered.The dissertation consists of the following five chapters:In chapter 1,the research background and present situation of this dissertation are briefly introduced,and the related knowledge of cone-constrained vector equalization problem is also introduced.In chapter 2,some basic notations and notions are given,as well as the lemmas and propositions needed in the later proofs.In chapter 3,it is proved that some equivalent forms about the existence of saddle points in the cone-constrained vector equilibrium problem.First we consider that the Lagrange multiplier belongs to the quasi-interior of the polar cone of the ordering set,that is,in the case of θ~*(?)P~*\{0},the equivalent form of the saddle point is established;in addition,under the Slater-type constraint qualifications,other equivalent forms of the saddle point are also obtained.In chapter 4,we consider the saddle-point optimality conditions for a weak efficient solution of the cone-constrained vector equilibrium problem.Under the assumption of θ~*(?)P~*\{0},it is described that the existence of a saddle point is a sufficient optimality condition for a weak efficient solution of the vector equilibrium problem;and under the assumption of generalized subconvexlike or preconvexlike,it is obtained that the existence of saddle points is the necessary optimality condition for a weak effificient solution of the vector equilibrium problem.In chapter 5,we consider a special kind of cone-constrained vector equilibrium problem,and under the assumption of convexity of the objective function h,we obtained some equivalent forms about the existence of saddle points for the penalty function.