Generalized symmetric ADMM for separable convex optimization

Jianchao Bai (), Jicheng Li (), Fengmin Xu () and Hongchao Zhang ()
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Jianchao Bai: Xi’an Jiaotong University
Jicheng Li: Xi’an Jiaotong University
Fengmin Xu: Xi’an Jiaotong University
Hongchao Zhang: Louisiana State University

Computational Optimization and Applications, 2018, vol. 70, issue 1, No 5, 129-170

Abstract: 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 $$p \ge 1$$ p ≥ 1 and $$q \ge 1$$ 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 $${\mathcal {O}}(1/t)$$ 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; 65C60; 65E05; 68W40; 90C06 (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10589-017-9971-0

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