Convergence Of The Inexact Semi-Proximal Alternating Direction Methods Of Multipliers

Semi-proximal Alternating Direction Methods of Multipliers (Semi-proximal ADMM). fo-cusing on solving convex optimization problems with separable variables, is a kind of very efficient numerical method. The convergence analysis of this method is premised on solving the exact solutions of the subproblems. However, for complex subproblems. solving exactly is a challenging task. Therefore, it is very necessary to consider solving the semi-proximal ADMM in an inexact way. In this dissertation, we propose two different inexact criteria applying to the subproblems. Furthermore, we prove the global convergence of these methods in the case of these inexact criteria.This paper is organized as follows:Chapter 2, we introduce some preliminaries about convex functions, monotone property, subdifferential, etc.. Besides, we propose the Slater Constrain Qualification, which ensures the nonempty solution set of original problem. Moreover, we describe the maximal monotone property of the subdifferential of closed proper convex functions.Chapter 3, for the first inexact algorithm, we require the distance between the exact solution and inexact solution is no more than a specific given constant. By using the error bound functions, we prove that in the case that objective; functions are continuous and differential, the transformed functions by original subproblems in the inexact algorithm are actually strong convex. In that case, the distance between the exact solution and inexact solution can be described according to the formulas including error bound functions, which means the inexact algorithm can be implemented. Finally, we prove the convergence of the method in the case of this inexact criterion.Chapter 4, for the second inexact algorithm, we add an inexact term to the optimal condi-tions in the subproblems of semi-proximal ADMM, the module of this term is actually determined by the distance between the exact solution and inexact solution. After that, we add a correct step. Based on several lemmas and propositions, we then prove the convergence of the method in the case of this inexact criterion.