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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.

In various fields of applied mathematics and engineering, many problems can be equivalently formulated as a separable convex optimization problem with two blocks; that is, given two closed convex functions

In this paper, we concentrate on the linearly constrained convex programming problem with three blocks:

In order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM with three blocks which involves an additional computation of optimal step size. Specifically, with the triple

where

The remaining parts of this paper are organized as follows. In Section

Throughout this paper, we assume

The next lemma introduced in [

Let

In this section, we first investigate the contractive property of the distance between the sequence generated by ADMM with three blocks and the solution set under the condition that

Let

Since

With the above preparation, we are ready to prove the convergence of the ADMM with three blocks for solving (

Let

By the inequality (

the authors proved the convergence of the ADMM under the conditions that

For the ADMM with two blocks, Ye and Yuan [

Let the sequence

By direct computations to

Based on the above inequality, we are able to prove the following convergence result of the relaxed ADMM with three blocks. Since the proof is in line with that of Theorem

Let

In this paper, we take a step to investigate the ADMM for separable convex programming problems with three blocks. Based on the contractive analysis of the distance between the sequence and the solution set, we establish theoretical results to guarantee the global convergence of ADMM with three blocks under weaker conditions than those employed in [

The first author is supported by the Natural Science Foundation of Jiangsu Province and the National Natural Science Foundation of China under Project no. 71271112. The second author is supported by university natural science research fund of jiangsu province under grant no. 13KJD110002.