Generalized Alternating Direction Method of Multipliers: New Theoretical Insights and Applications.

IF 4.3 1区数学 Q1 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Mathematical Programming Computation Pub Date : 2015-06-01 Epub Date: 2015-02-06 DOI:10.1007/s12532-015-0078-2

Ethan X Fang, Bingsheng He, Han Liu, Xiaoming Yuan

引用次数: 109

Abstract

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/k) convergence rate measured by the iteration complexity (k 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.

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广义乘数交替方向法:新的理论见解与应用。

近年来，乘法器的交替方向法(ADMM)受到了广泛的关注。Eckstein和Bertsekas提出的广义ADMM (GADMM)是一种简单有效的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是由线性化策略近似的。这种线性化的版本对于来自不同领域的许多应用程序特别有效。从理论上讲，我们展示了线性化版本的GADMM在遍历和非遍历意义上由迭代复杂度(k表示迭代计数器)测量的最坏情况的态(1/k)收敛速率。在数值上，我们通过统计学习中一些相当新的核心应用证明了这种线性化版本的GADMM的效率。还为这些应用程序开发了Matlab中的代码包。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Programming Computation OPERATIONS RESEARCH & MANAGEMENT SCIENCE-

CiteScore

10.80

自引率

4.80%

发文量

期刊介绍： Mathematical Programming Computation (MPC) publishes original research articles advancing the state of the art of practical computation in Mathematical Optimization and closely related fields. Authors are required to submit software source code and data along with their manuscripts (while open-source software is encouraged, it is not required). Where applicable, the review process will aim for verification of reported computational results. Topics of articles include: New algorithmic techniques, with substantial computational testing New applications, with substantial computational testing Innovative software Comparative tests of algorithms Modeling environments Libraries of problem instances Software frameworks or libraries. Among the specific topics covered in MPC are linear programming, convex optimization, nonlinear optimization, stochastic optimization, integer programming, combinatorial optimization, global optimization, network algorithms, and modeling languages. MPC accepts manuscript submission from its own editorial board members in cases in which the identities of the associate editor, reviewers, and technical editor handling the manuscript can remain fully confidential. To be accepted, manuscripts submitted by editorial board members must meet the same quality standards as all other accepted submissions; there is absolutely no special preference or consideration given to such submissions.