广义交替方向乘子法：新的理论见解与应用
Generalized Alternating Direction Method of Multipliers: New Theoretical Insights and Applications.
作者信息
Fang Ethan X, He Bingsheng, Liu Han, Yuan Xiaoming
机构信息
Department of Operations Research and Financial Engineering, Princeton University.
International Centre of Management Science and Engineering, and Department of Mathematics, Nanjing University, Nanjing, 210093, China. This author was supported by the NSFC Grant 91130007 and the MOEC fund 20110091110004.
出版信息
Math Program Comput. 2015 Jun;7(2):149-187. doi: 10.1007/s12532-015-0078-2. Epub 2015 Feb 6.
Recently, the alternating direction method of multipliers (ADMM) has received intensive attention from a broad spectrum of areas. The generalized ADMM (GADMM) proposed by Eckstein and Bertsekas is an efficient and simple acceleration scheme of ADMM. In this paper, we take a deeper look at the linearized version of GADMM where one of its subproblems is approximated by a linearization strategy. This linearized version is particularly efficient for a number of applications arising from different areas. Theoretically, we show the worst-case 𝒪(1/) convergence rate measured by the iteration complexity ( represents the iteration counter) in both the ergodic and a nonergodic senses for the linearized version of GADMM. Numerically, we demonstrate the efficiency of this linearized version of GADMM by some rather new and core applications in statistical learning. Code packages in Matlab for these applications are also developed.
最近，乘子交替方向法（ADMM）受到了广泛领域的密切关注。Eckstein和Bertsekas提出的广义ADMM（GADMM）是一种高效且简单的ADMM加速方案。在本文中，我们深入研究了GADMM的线性化版本，其中它的一个子问题通过线性化策略进行近似。这个线性化版本对于来自不同领域的许多应用特别有效。从理论上讲，我们证明了在遍历和非遍历意义上，以迭代复杂度（表示迭代计数器）衡量的GADMM线性化版本的最坏情况𝒪(1/)收敛速率。在数值上，我们通过统计学习中的一些相当新颖且核心的应用展示了GADMM线性化版本的效率。还开发了用于这些应用的Matlab代码包。
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