The Subgradient Extragradient Method For Solving The Monotone Bilevel Equilibrium Problem With Constraints

Bilevel optimization problems such as bilevel variational inequalities,bilevel equilibrium problems and bilevel programming problems have been widely applied in many fields,and bilevel optimization problems also have attracted the attention of scholars.This thesis mainly investigate several classes of bilevel equilibrium problems.We propose several algorithms for solving these problems and establish their strong convergence theorems under suitable conditions.The results presented in this thesis extend and improve some corresponding results in the literature.This thesis consists of seven chapters.In Chapter 1,we introduce the purpose of several problems which including fixed point problems,equilibrium problems and variational inequalities,then we get research background and present situation of monotone bilevel equilibrium problems and state the main work of this thesis.In Chapter 2,we present some basic concepts and theories which will be used in the main results.In Chapter 3,we study the monotone bilevel equilibrium problem with constraints of the general system of variational inequalities and common fixed points problems.In this chapter,via a subgradient extragradient implicit rule,we introduce and analyze an iterative algorithms for solving the monotone bilevel equilibrium problems.Some strong convergence results for the proposed algorithms are established under the mild assumptions,and they are also applied for finding a common solution of the general system of variational inequality,a variational inequality problem,and a fixed point problem.Numerical example illustrates the feasibility of theoretical result.In Chapter 4,we continues to study the quasi-monotone bilevel equilibrium problem with constraints of the general system of variational inequalities and common fixed point problems.Based on the conclusion of Chapter 3,via a new subgradient extragradient implicit rule,we construct an Mann-type implicit iterative algorithms for solving the monotone bilevel equilibrium problems by using demiclosedness principle of nonexpansive mapping and asymptotically nonexpansive mapping.Some strong convergence results for the proposed algorithms are established under the mild assumptions,and they are also applied for finding a common solution of the general system of variational inequality,a variational inequality problem,and a fixed point problem.Numerical example illustrates the feasibility of theoretical result.In Chapter 5,we study a class of monotone bilevel equilibrium problem.we expand the problems studied in chapters 3 and propose the mixed monotone bilevel equilibrium problem with constraints of the general system of variational inequalities and common fixed point problems.In order to improve the accuracy of the algorithm,via a implicit midpont rule,we introduce and analyze an implicit midpont iterative algorithm for solving the monotone bilevel equilibrium problems.Some strong convergence results for the proposed algorithms are established under the suitable assumptions,and they are also applied for finding a common solution of the general system of variational inequality,a variational inequality problem,and and a fixed point problem.Numerical example illustrates the feasibility of theoretical result.In Chapter 6,we continue to study the mixed monotone bilevel equilibrium problem above.Based on the conclusion of Chapter 5,in order to improve the efficiency of the algorithm,via a inertial method,we construct an iterative algorithm for solving the monotone bilevel equilibrium problem.Some strong convergence results for the proposed algorithms are established under the suitable assumptions.Numerical example illustrates the feasibility of theoretical result.In Chapter 7,we summarize the work of this thesis and the prospects of the future research.