The Research Of The ADMM Algorithm For Solving The Optimization Problem

The alternating direction method of multipliers?ADMM?, is a simple and effective method to solve the convex programming problem with separable structure, and especially to large-scale problems. The algorithm makes full use of the separability of the objective function, and decomposes the original problem into several minimization problems more easily getting local solution alternately. In recent years, ADMM algorithms are widely applied to various optimization problems, such as the least squares problem, matrix problem completely,l₁-norm optimization problem, convex composite optimization, etc.In this paper, we study a majorized ADMM algorithm with indefinite proximal terms, the differences between this algorithm and general ADMM algorithm: firstly, the definition of the augmented Lagrangian function is different; secondly, the adjacent mode of the linear operators S and T are different. This article uses the majorized ADMM algorithm with indefinite proximal terms solving linear constrainted 2-block convex composite optimization problems, and analyzes the convergence and complexity of the algorithm.Moreover, the computational bebefit of using indefinite proximal terms within the ADMM framework instead of the positive semidefinite proximal terms is demonstrated by the numerical experimentation. And in this process, the author finds the values of parameter ? has a certain impact on the performance of the majorized ADMM algorithm with indefinite proximal terms,the fact that the different value of parameter t in the interval [1.0,1.618] has less effect on the algorithm for solving small or medium scale optimization problems is demonstrated through the numerical experimentation. But compared with other values in this interval, when the ? equals 1.618, iterative error of the algorithm is smaller and the optimal objective function value is faster to reach.In the first chapter, this article mainly expounds the research background and the current situation. In the second chapter, the paper introduces a majorized ADMM withindefinite proximal terms, and its global convergence and worst-case o?1/k? complexity are given certificates. In the third chapter, the majorized ADMM with indefinite proximal terms is better than the ADMM with positive semifinite terms demonstrated by the numerical experiments, and through adjusting the different value of parameter ?, the numerical experiments show the influence of parameter ? to the majorized ADMM algorithm with indefinite proximal terms.