Generalized symmetric ADMM for separable convex optimization

• Published in: Computational optimization and applications Vol. 70; no. 1; pp. 129 - 170
• Main Authors: Bai, Jianchao, Li, Jicheng, Xu, Fengmin, Zhang, Hongchao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2018; Springer Nature B.V
• Subjects: Convergence; Convex analysis; Convex and Discrete Geometry; Convexity; Data management; Economic models; Integers; Lagrange multiplier; Management Science; Mathematics; Mathematics and Statistics; Numerical methods; Operations Research; Operations Research/Decision Theory; Optimization; Statistics; Variables; Separable convex programming; Alternating direction method of multipliers; Global convergence; 68W40; Parameter convergence domain; Statistical learning; Multiple blocks; 90C06; 65E05; 65C60; Complexity
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-017-9971-0

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 ≥ 1 and q ≥ 1 are two integers. The two grouped variables are updated in a Gauss–Seidel scheme, while the variables within each group are updated in a 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 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.