The Optimality Conditions For The Solution Of The Vector Equilibrium Problems Via Improvement Set

Vector equilibrium problems is a hot issues of the oprational research and nonlin-ear analysis,which also has been widely used in engineering,mathematical economic and social economic system,etc.In the study of vector equilibrium problems,there are various forms of the solution,for example,efficient solutions,weakly efficient solutions,approximate efficient solutions,approximate weakly efficient solutions,etc.How to put forward a unified concept of vector equilibrium problems and study their properties under the unified framework is very important.In this paper,firstly,we define the E-efficient solutions,the E-weakly efficient solutions,the E-Henig properly efficient solutions of the vector equilibrium problems.Secondly,some properties and the optimality conditions for the E-efficient solutions,the E-weakly efficient solutions,the E-Henig properly efficient solutions of the vector equilibrium problems are studied.The main contents are presented as follows:In the first chapter,we briefly introduce the current development of vector equilib-rium problems and some concepts and lemmas related to this paper.In the second chapter,first of all,we introduce the concept of E-efficient solutions,E-weakly efficient solutions,E-Henig properly efficient solutions via improvement set for the vector equilibrium problems,and obtain their properties.Then the nonlinear scalar characterization of the E-efficient solution and E-weakly efficient solutions are given.In the third chapter,by improvement set,we define the concept of ic-E-convexlikeness mapping and obtain a new alternative theorem.Under the condition of the objective func-tion be generalized convex function,the optimality conditions for E-efficient solutions,the E-weakly efficient solutions,the E-Henig properly efficient solutions of the vector equilibrium problems are established by using the separation theorem of convex set and the alternative theorem.