Improved Stochastic Alternating Direction Method Of Multipliers

Recently, due to the rapid development of information technology and the arrival of the era of big data, we often encounter large-scale problems in solving the optimization problems. Therefore, it is increasingly important to be able to find an effective method to solve such problems. Alternating direction method of multipliers is a simple but powerful algorithm for solving convex optimization problems with separable structures. Its main idea is to decompose the complex large-scale optimization problem into several sub-problems, the solution of the original problem is obtained by alternately solving a series of sub-problems. Because of its generality and effectiveness, alternating direction method of multipliers has attracted the attention of a large number of scholars. At present, there have been a lot of generalized and improved algorithms for this method, where the stochastic alternating direction method of multipliers has recently become a hot research topic in the field of optimization.In this paper, we propose a new improved stochastic alternating direction method of multipliers, and give the convergence analysis of the algorithm. At the same time, the numerical experiment results also show the efficiency of the algorithm.This paper is organized as follows.At first, the first chapter mainly introduces the theory and research backgrounds of the optimization problems. Then, aiming at the large-scale convex optimization problem model with linear constraints, we briefly introduce the application background and their merit and shortcoming of several stochastic alternating direction method of multipliers recently proposed, finally the main content and chapters arrangement are given.The second chapter proposes a new improved stochastic alternating direction method of multipliers.the iteration format and the corresponding algorithm of the proposed method are established.In the third chapter, for the new algorithm proposed by the second chapter, we show the convergence rate of the new algorithm in theory by using two different iterative aver-aging way.For generalized lasso model, the fourth chapter carries out numerical experiments on the new algorithm and several stochastic algorithms recently proposed.and then the feasibility and efficiency of the new proposed algorithm are verified by comparing the numerical results.In the last chapter, some results are summarized. And we give several issues that could be considered next.