Vector Equilibrium And Efficiency

Vector equilibrium and efficiency are extensive problems involved in many subjectsand practical fields such as optimization, investment decision, economy model, optimalcontrol and engineering technology, and contain a lot of specific mathematics theoryproblems such as extreme value, variation, saddle point, vector implicit omplementarityand variational inequality, etc. Many results have been applied to economy equilibrium,game, economy management and so on. In this paper, the problems of vector equilibriumand efficiency are researched with the tools of analysis and topology in the case oftopological vector space. For detail, it can be stated as follows:1. A survey for vector equilibrium, efficiency and variational inequality ispresented. With a few of examples in finite dimensional case, we show the vectorequilibrium problems containing many typical mathematical problems such asoptimization, variational inequality, saddle point and vector implicitcomplementarity.2. The vector equilibrium problems for bi-variable mapping from a topologicalvector space to another are researched. The existence theorems of solutions for weakvector equilibrium problems are shown. The vector equilibrium problems of conepseudo-monotone, cone quasi-convex, cone-strictly quasi-convex mappings withhemi-continuity are discussed. The relationships of solutions among them are obtained.With Ky Fan's fixed-point theorem, the existences of solutions are proved. The existenceand relationships of solutions of vector equilibrium and vector implicit complementproblems for quasi - monotonic and hemi -continuous mapping are shown.3. The existence theorem of generalized weak efficient solutions with respect tovariable cone for a linear Gateaux differentiable mapping is proved with set valuedmapping fixed-point theorem and the relation between a vector optimization and avariational inequality problem. The existence of weak efficient solutions formulti-objective convex vector optimization is characterized. Existence and equivalenceof solutions for a non-convex vector optimization and a variational inequality arepresented. By level function,ε-subdifferential,normal cone and dual function,globaloptimal solutions for a convex function in locally convex topological vector space anda few of equivalent forms on it are obtained. Maximum points are characterized.4.The optimality conditions of Kuhn-Tucker and Fritz John's saddle points of vetoroptimization for generalized subconvex-like mapping are presented. Lagrange dualmapping and relationships among optimality conditions of saddle points, efficient andweak efficient solutions for vector and scalar optimization are established. ByLagrange dual and sealarization of vector optimization, characterizations of solutionsfor which objective mapping satisfies weak convexity conditions are shown. The stronglyand weak dual theorems for generalized subconvex-like mapping are obtained. With thehelp of contingent, normal and feasible direction cones for a set at a point, thecharacterizations of Pareto, weak and proper efficient points for convex set arepresented. In non-convex case, locally efficiency is discussed. Several kinds oflocally efficient points are characterized with different cones.5. Gordan-Farkas type theorems for several generalized cone subconvex-like arepresented. Some applications to vector optimization problems in infinite dimensionalspace are shown. The strong Lagrange duality and saddle point condition are obtained.It generalized some results of vector function to that of mapping in topological vectorspace6. The perpendicular points and semi-continuous conceptions are introduced, andthey are applied to existence of solution on generalized vector variationalinequalities and dual problems for semi-strictly quasi-monotone set-valued mappings.Generalized variational inequality induced by a convex function is discussed. Existenceof solution is obtained with Gap function properties and dual problems. Unique...