The convergence rate of the proximal alternating direction method of multipliers with indefinite proximal regularization

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• 作者：Min Sun ; Jing Liu
• 关键词：proximal alternating direction method of multipliers ; two ; block separable convex minimization problem ; compressive sensing
• 刊名：Journal of Inequalities and Applications
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1440KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X
• 卷排序：2017

文摘

The proximal alternating direction method of multipliers (P-ADMM) is an efficient first-order method for solving the separable convex minimization problems. Recently, He et al. have further studied the P-ADMM and relaxed the proximal regularization matrix of its second subproblem to be indefinite. This is especially significant in practical applications since the indefinite proximal matrix can result in a larger step size for the corresponding subproblem and thus can often accelerate the overall convergence speed of the P-ADMM. In this paper, without the assumptions that the feasible set of the studied problem is bounded or the objective function’s component \(\theta_{i}(\cdot)\) of the studied problem is strongly convex, we prove the worst-case \(\mathcal{O}(1/t)\) convergence rate in an ergodic sense of the P-ADMM with a general Glowinski relaxation factor \(\gamma\in(0,\frac{1+\sqrt{5}}{2})\), which is a supplement of the previously known results in this area. Furthermore, some numerical results on compressive sensing are reported to illustrate the effectiveness of the P-ADMM with indefinite proximal regularization.