Linear Scalarization Characterizations Of Solutions To Generalized Vector Equilibrium Problems And The Application Analysis

Vector equilibrium problems,also known as generalized Ky Fan inequalities,are important mathematical models and powerful tools in the field of vector optimization,which contain vector variational inequalities,vector optimization problems,vector complementarity problems,vector saddle point problems and Nash equilibrium problems.The stability analysis is an important topic in the research of vector optimization and equilibrium problems,which includes the error bound analysis,the continuity of perturbed solution mappings and so on.The linear scalarization characterizations of solutions are effective tools to study error bounds and the continuity of solution mappings.Therefore,this thesis mainly establishes the linear scalarization characterizations of solutions to generalized vector equilibrium problems via improvement sets,and then applies them to study error bounds and the continuity of solution mappings.The results obtained improve and generalize the corresponding conclusions in related literatures.Firstly,under the weaker setting of generalized convexity than that of the related literatures,the linear scalarization characterizations of solutions for unified weak set-valued vector equilibrium problems via improvement sets and Benson proper efficient solutions of unified set-valued vector equilibrium problems via improvement sets are established,respectively.Then,based on the linear scalarization characterizations of solutions,the continuity of solution mappings to parametric unified weak set-valued vector equilibrium problems and the continuity of Benson proper efficient solution mappings for parametric unified set-valued vector equilibrium problems are studied by means of monotonicity assumptions and the scalarization methods.Finally,combined with the linear scalarization characterization properties of solutions,(regularized)gap functions for unified weak vector equilibrium problems through improvement sets are constructed,and then error bounds of the problems using(regularized)gap functions are established under the assumptions of generalized monotonicity.