Study On Some Algorithms For Structural Variational Inequality And Convex Optimization Problems

Variational inequality and Convex Programming problems play very important roles in studying a wide class of problems originating from mathematics, management science and engineering science, and they have very closely relationship, namely, the first order optimality conditions of a convex programming problem can be characterized by a variational inequality. With the increase of the cross study between subjects, these two kinds of problems are widely used to characterize more problems in new areas, such as image processing, statistical learning and so on. Hence, it is very necessary and very important to devise some stable and fast numerical methods for solving these problems. In the last decades, some series of numerical algorithms are studied and designed to solve variational inequality and convex programming problems, such as projection operator algorithm, augmented Lagrangian method, interior point method, proximal point algorithm, operator splitting method and so on. These numerical algorithms are widely applicable in economic equilibrium, image processing, statistical learning, matrix optimization and other fields.Recently, as the development of information science, the researches of the model with special structures and properties have become a hot spot in many fields, such as information transmission and data processing. These problems have the characteristics of large scale, separation structure and linear constraints. This thesis is based on these features to design the effective algorithms.In this dissertation, we mainly investigate and design the efficient numerical algorithm for variational inequality and convex separate optimization problems. The dissertation is divided into seven chapters and organized as follows:In Chapter 1, we first describe the research status of projection algorithms for solving variational inequality problems. Then, we introduce the development and present researches on the topics of proximal point algorithms and operator splitting methods for inclusions and convex optimization problem. Finally, we briefly describe the motivation and the main research work of this dissertation.In Chapter 2, we describe some basic symbol, definitions, concepts, properties, some resolvent operator of set-valued monotone operators and convex functions, which will be used in algorithm analysis. Lastly, we introduce the quality standards to evaluate algorithm.In Chapter 3, we investigate the new parallel method for solving a class of structure variational inequalities problem. The parallel algorithm is devised by the projecting method and the new inexact criteria. Based on the properties of projecting method, we prove global convergence and the convergence rate. Finally, numerical experiments show new algorithm with inaccurate criterion is effective and stable structure for solving variational inequalities.In Chapter 4, we research the convergence rate of operator splitting method for solving variational inequalities with a special structure. Based on the monotone property of mappings in variational inequalities, we establish the convergence rate of operator splitting method.In Chapter 5, we study a parallel algorithm for solving linear constrained convex separate optimization problem. Firstly, based on the issue of separation structure and augmented Lagrangian method, we devise a new parallel method. Then, we prove the global convergence of the algorithm and establish the convergence rate in an ergodic sense and in a nonergodic sense. Finally, numerical experiments show that the parallel algorithm is effective and stable for solving the convex optimization problem with linear constraints and separation.In Chapter 6, we investigate an implementable Douglas-Rachoford operator splitting method for linear constraint separating convex optimization problem. In view of the classic Peaceman-Rachoford and Douglas-Rachoford operator splitting method in solving certain convex optimization problem, only one problem has no closed form solution. Then, we design an implementable Douglas-Rachoford operator splitting method to sovle all the subproblem. Then, based on the convexity of functions, we established the global convergence and the convergence rate in a nonergodic sense. The algorithms make use of the separation of structural problems and each subproblem has closed form solution in the iteration. Finally, numerical experiments show that the algorithm is effective and stable for solving separation convex optimization problem.In Chapter 7, we briefly summary the main results of this thesis and some problems which are remained and thought over in future are put forward.