Research On Optimality Conditions And Algorithms Of Vector Equilibrium Problems

In this paper,we use the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping to establish the optimality conditions of two types vector quasi-equilibrium problems with variable ordering structures.The paper also uses projection iterative methods to give an iterative algorithm for finding the common solution of the strong vector equilibrium problem and the fixed point problem of set-valued mapping,and obtains convergence results.These results not only develop and generalize some related research results in recent literature,but also enrich the theory and algorithm of vector equilibrium itself,thus,it provides an indispensable theoretical basis for the practical application of vector equilibrium in economy,production and life.The full text is divided into four chapters,which are as follows:In the first chapter,we mainly introduce the historical background and development process and present situation of the vector equilibrium problem,and also briefly describes the main research content and the paper structure of this paper.In the second half of this chapter,we give some concepts,lemmas and known conclusions to be used in this paper.In the second chapter,we use the cosmically upper continuity condition instead of upper semicontinuity condition for cone-valued mapping to discuss the vector quasi-equilibrium problems with variable ordering structures.Under conditions of cosmically upper continuity for cone-valued mapping and suitable continuity and cone-convexity conditions for equilibrium mapping,some existence theorems of optimal solution are obtained for weak/Stampacchia vector quasi-equilibrium problems with variable ordering structures by using fixed point method and maximal element method.These main results obtained in this paper generalize and develop some recent ones in the literature.In the third chapter,It mainly considers the iterative algorithm for finding common solutions of strong vector equilibrium problems and fixed point problems of multivalued mappings.Firstly,the Minty vector equilibrium problem related to the strong vector equilibrium problem is introduced and the relationship between the Minty vector equilibrium problem and the strong equilibrium problem is discussed.Then,by applying the Minty vector equilibrium problem,projection iterative methods are proposed and the feasibility of the algorithm is analyzed.Further,some convergence results are established in Hilbert spaces.In the last chapter,we summarize the results and conclusions obtained in this paper and puts forward a new outlook.