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Convergence Study on the Proximal Alternating Direction Method with Larger Step Size

Bingsheng He(hebma***at***nju.edu.cn)
Feng Ma(mafengnju***at***gmail.com)

Abstract: The alternating direction method of multipliers (ADMM) is a popular method for the separable convex programming with linear constraints, and the proximal ADMM is its important variant. Previous studies show that the relaxation factor $\gamma\in (0, \frac{1+\sqrt{5}}{2})$ by Fortin and Glowinski for the ADMM is also valid for the proximal ADMM. In this paper, we further demonstrate that the feasible region of $\gamma$ depends on the proximal term added on the second subproblem, and can be enlarged when the proximal factor is positive. We derive the exact relationship between the relaxation factor $\gamma$ and the proximal factor. Finally, we prove the global convergence and derive a worst-case $O(1/t)$ convergence rate in the ergodic sense for this generalized scheme.

Keywords: Alternating Direction Method of Multipliers, Convex Programming, Proximal Regularization, Contraction, Convergence Analysis.

Category 1: Convex and Nonsmooth Optimization

Citation:

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Entry Submitted: 02/12/2017
Entry Accepted: 02/12/2017
Entry Last Modified: 02/12/2017

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