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Generalized Symmetric ADMM for Separable Convex Optimization

Jianchao Bai (bjc1987***at***163.com)
Jicheng Li (jcli***at***mail.xjtu.edu.cn)
Fengmin Xu (fengminxu***at***mail.xjtu.edu.cn)
Hongchao Zhang (hozhang***at***math.lsu.edu)

Abstract: The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of $p$ block variables while the other has $q$ block variables, where $p \ge 1$ and $q \ge 1$ are two integers. The two grouped variables are updated in a {\it Gauss-Seidel} scheme, while the variables within each group are updated in a {\it Jacobi} scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that GS-ADMM is globally convergent with a worst-case $\C{O}(1/t)$ ergodic convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible and attractive on choosing and using larger stepsizes of the dual variable. Besides, two special cases of GS-ADMM, which allows using zero penalty terms, are also discussed and analyzed. Compared with several state-of-the-art methods, preliminary numerical experiments on solving a sparse matrix minimization problem in the statistical learning show that our proposed method is effective and promising.

Keywords: Separable convex programming, Multiple blocks, Parameter convergence domain, Alternating direction method of multipliers, Global convergence, Complexity, Statistical learning

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Global Optimization (Theory )

Citation: Accepted by Comput. Optim. Appl.

Download: [PDF]

Entry Submitted: 10/31/2016
Entry Accepted: 10/31/2016
Entry Last Modified: 11/13/2017

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