scholarly journals On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Caihua Chen ◽  
Yuan Shen ◽  
Yanfei You
Keyword(s):  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Objective Functions ◽  
Step Size ◽  
Mild Conditions ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
Optimal Step Size ◽  
Linearly Constrained ◽  
Kkt Points

We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.

A Linearized Alternating Direction Method of Multipliers with Substitution Procedure

2015 ◽  
Vol 32 (03) ◽  
pp. 1550011 ◽  
Author(s):  
Miantao Chao ◽  
Caozong Cheng ◽  
Haibin Zhang
Keyword(s):  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Method Of Multipliers ◽  
Worst Case ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
Contractive Type ◽  
Linearly Constrained ◽  
Contraction Type ◽  
Substitution Procedure

We consider the linearly constrained separable convex programming problem whose objective function is separable into m individual convex functions with non-overlapping variables. The alternating direction method of multipliers (ADMM) has been well studied in the literature for the special case m = 2, but the direct extension of ADMM for the general case m ≥ 2 is not necessarily convergent. In this paper, we propose a new linearized ADMM-based contraction type algorithms for the general case m ≥ 2. For the proposed algorithm, we prove its convergence via the analytic framework of contractive type methods and we derive a worst-case O(1/t) convergence rate in ergodic sense. Finally, numerical results are reported to demonstrate the effectiveness of the proposed algorithm.

scholarly journals SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure

2018 ◽  
Vol 12 (1) ◽  
pp. 224-243 ◽  
Author(s):  
Abdellah Bnouhachem ◽  
Themistocles Rassias
Keyword(s):  
Variational Inequality ◽  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Descent Direction ◽  
Computational Results ◽  
Step Size ◽  
Sqp Method ◽  
Separable Structure ◽  
Alternating Direction ◽  
Optimal Step Size

In this paper, we suggest and analyze a new alternating direction scheme for the separable constrained convex programming problem. The theme of this paper is twofold. First, we consider the square-quadratic proximal (SQP) method. Next, by combining the alternating direction method with SQP method, we propose a descent SQP alternating direction method by using the same descent direction as in [6] with a new step size ?k. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1/t) convergence rate for the SQP alternating direction method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

scholarly journals Convergence Analysis of Alternating Direction Method of Multipliers for a Class of Separable Convex Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zehui Jia ◽  
Ke Guo ◽  
Xingju Cai
Keyword(s):  
Convergence Analysis ◽  
Convex Programming ◽  
Alternating Direction Method ◽  
Linear Functions ◽  
Method Of Multipliers ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Alternating Direction ◽  
Separable Convex Programming ◽  
The Individual

The purpose of this paper is extending the convergence analysis of Han and Yuan (2012) for alternating direction method of multipliers (ADMM) from the strongly convex to a more general case. Under the assumption that the individual functions are composites of strongly convex functions and linear functions, we prove that the classical ADMM for separable convex programming with two blocks can be extended to the case with more than three blocks. The problems, although still very special, arise naturally from some important applications, for example, route-based traffic assignment problems.

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