Mathematical Programming

, Volume 162, Issue 1–2, pp 165–199 | Cite as

On the linear convergence of the alternating direction method of multipliers

  • Mingyi Hong
  • Zhi-Quan LuoEmail author
Full Length Paper


We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth \(\ell _1\) regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function.


Linear convergence Alternating directions of multipliers Error bound Dual ascent 

Mathematics Subject Classification

49 90 



The authors are grateful to Xiangfeng Wang and Dr. Min Tao of Nanjing University for their constructive comments.


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© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.School of Science and EngineeringThe Chinese University of Hong KongShenzhenChina
  3. 3.Department of Industrial and Manufacturing Systems EngineeringIowa State UniversityAmesUSA

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