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Abstract and Applied Analysis
Volume 2013, Article ID 183961, 7 pages
Research Article

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

1International Center of Management Science and Engineering, School of Management and Engineering, Nanjing University, Nanjing 210093, China
2School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China
3Department of Mathematics, Nanjing University, Nanjing 210093, China

Received 4 July 2013; Accepted 5 September 2013

Academic Editor: Xu Minghua

Copyright © 2013 Caihua Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.