Proximal Alternating Direction Methods For Convex Optimization

In mathematical programming, We always meet many large-scale problems, The main methods of solving theose problems include Conjugate Gradient Method, BFGS Method, Decomposition Method and so on. Complicated optimization problem is decomposed into several sub-problems in decomposition method. The main problems and sub-problems are non-smooth or the optimal solution does not meet the KKT conditions in some decomposition method. Therefore,these two conditions are enormous influence on the constructed algorithm. In order to avoid the emergence of these problems, scholars scholars put forward alternating direction decomposition method is applied to the augmented Lagrangian function relaxation method, and the universality and effectiveness of alternate decomposition method, attracted the attention of a large number of scholars, now has a variety of forms, including approximate alternating direction method of alternating direction method. Discuss various points to the method of convergence and alternately and inferiority and the practicability of the algorithm has become hot topics in the study of the current optimization problem.In this paper, the main work is divided into the following two aspects. Is a convex programming problem with separable structure, puts forward a new approximate alternating direction method, and proves the convergence of new methods, and finally through the numerical experiment and existing methods are compared.2it is to expand the linear constraints of original model to the problem of convex function constraint, using the approximate alternating direction method and three kinds of linearization approximation alternating direction method for solving.The paper is organized as follows. In Chapter1, we firstly give a briefly introduces the development of the method of alternating direction decomposition process, and gives the classic model of alternating direction method, and simple introduced the normal form of the variational inequality; In Chapter2, this paper puts forward a new linear approximation, and under the condition of the assumptions of convergence, and through numerical experiments compared with the existing methods; In Chapter3, the constraints from linear to approximate alternating direction method is proposed function, and three different kinds of approximate linear algorithm of alternating direction method; Finally,some conclusions are drawn in the last Chapter.