Existence And Iterative Algorithm Of Solutions For Bilevel Generalized Mixed Equilibrium Problems

Bilevel optimization problems such as bilevel programming,bilevel variational inequality and bilevel equilibrium problem have been widely used in many fields.The bilevel optimization problem mainly studies the existence of solutions and iterative algorithms.In this paper,we investigate the existence and uniqueness of solutions and the strong convergence of the iterative sequences generated by the algorithms of the new bilevel generalized mixed equilibrium problem in Banach space.we investigate an existence theorem of the solution for the new bilevel generalized mixed equilibrium problems in Hausdorff Topological vector space.we investigate the strong convergence of the iterative sequences generated by the algorithms for bilevel mixed equilibrium problems in a real Hilbert space.The structure of this dissertation is as follows:In the first chapter,we mainly introduce the research background,basic definitions and concepts of this dissertation,and briefly explain the main contents of this dissertation.In the second chapter,a new class of bilevel generalized mixed equilibrium problems is introduced and studied in Banach spaces.First,an auxiliary generalized mixed equilibrium problem to compute the approximate solutions of the bilevel generalized mixed equilibrium problems is introduced.By using a minimax inequality,the existence and uniqueness of solutions of the auxiliary generalized mixed equilibrium problem is proved under quite mild conditions.By using auxiliary principle technique,new iterative algorithm to compute the approximate solutions of the bilevel generalized mixed equilibrium problems are suggested and analyzed.The strong convergence of the iterative sequences generated by the algorithms are proved under quite mild assumptions.The properties of solution set for a class of bilevel generalized mixed equilibrium problems is studied.In the third chapter,a new bilevel generalized mixed equilibrium problem is introduced and studied in topological vector spaces.By using a minimax inequality,the existence of solutions and the behavior of solution set for the BNGMEP are studied under quite mild conditions.These results are new and generalize some recent results in this field.In the fourth chapter,we propose a new algorithm for solving a bilevel mixed equilibrium problem in a real Hilbert space.The subgradient method proposed in this paper requires only to calculate,at each iteration,four subgradients of convex functions and one projection onto a convex set.We prove a strong convergence theorem for the proposed algorithm.In the last chapter,we make a brief summary of the research and introduce the future research work.