# Generalized alternating direction method of multipliers: new theoretical insights and applications

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

Research output: Contribution to journalArticle

34 Scopus citations

### 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 $${\mathcal {O}}(1/k)$$O(1/k) convergence rate measured by the iteration complexity ($$k$$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.

Original language English (US) 149-187 39 Mathematical Programming Computation 7 2 https://doi.org/10.1007/s12532-015-0078-2 Published - Jun 18 2015

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Software

### Keywords

• Alternating direction method of multipliers
• Convergence rate
• Convex optimization
• Discriminant analysis
• Statistical learning
• Variable selection