Volume 13, Issue 1
A Fast Symmetric Alternating Direction Method of Multipliers

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 200-219.

Published online: 2019-12

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• Abstract

In recent years, alternating direction method of multipliers (ADMM) and its variants are popular for the extensive use in image processing and statistical learning. A variant of ADMM: symmetric ADMM, which updates the Lagrange multiplier twice in one iteration, is always faster whenever it converges. In this paper, combined with Nesterov's accelerating strategy, an accelerated symmetric ADMM is proposed. We prove its $\mathcal{O}(\frac{1}{k^2})$ convergence rate under strongly convex condition. For the general situation, an accelerated method with a restart rule is proposed. Some preliminary numerical experiments show the efficiency of our algorithms.

• Keywords

Nesterov's accelerating strategy, alternating direction method of multipliers, symmetric ADMM, separable linear constrained optimization.

90C25, 90C30, 49M29, 65B99

luogangnk@gmail.com (Gang Luo)

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@Article{NMTMA-13-200, author = {Luo , Gang and Yang , Qingzhi}, title = {A Fast Symmetric Alternating Direction Method of Multipliers}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {200--219}, abstract = {

In recent years, alternating direction method of multipliers (ADMM) and its variants are popular for the extensive use in image processing and statistical learning. A variant of ADMM: symmetric ADMM, which updates the Lagrange multiplier twice in one iteration, is always faster whenever it converges. In this paper, combined with Nesterov's accelerating strategy, an accelerated symmetric ADMM is proposed. We prove its $\mathcal{O}(\frac{1}{k^2})$ convergence rate under strongly convex condition. For the general situation, an accelerated method with a restart rule is proposed. Some preliminary numerical experiments show the efficiency of our algorithms.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0108}, url = {http://global-sci.org/intro/article_detail/nmtma/13437.html} }
TY - JOUR T1 - A Fast Symmetric Alternating Direction Method of Multipliers AU - Luo , Gang AU - Yang , Qingzhi JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 200 EP - 219 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2018-0108 UR - https://global-sci.org/intro/article_detail/nmtma/13437.html KW - Nesterov's accelerating strategy, alternating direction method of multipliers, symmetric ADMM, separable linear constrained optimization. AB -

In recent years, alternating direction method of multipliers (ADMM) and its variants are popular for the extensive use in image processing and statistical learning. A variant of ADMM: symmetric ADMM, which updates the Lagrange multiplier twice in one iteration, is always faster whenever it converges. In this paper, combined with Nesterov's accelerating strategy, an accelerated symmetric ADMM is proposed. We prove its $\mathcal{O}(\frac{1}{k^2})$ convergence rate under strongly convex condition. For the general situation, an accelerated method with a restart rule is proposed. Some preliminary numerical experiments show the efficiency of our algorithms.

Gang Luo & Qingzhi Yang. (2019). A Fast Symmetric Alternating Direction Method of Multipliers. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 200-219. doi:10.4208/nmtma.OA-2018-0108
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