Science China Mathematics

, Volume 56, Issue 10, pp 2179–2186

A proximal point algorithm revisit on the alternating direction method of multipliers

  • XingJu Cai
  • GuoYong Gu
  • BingSheng He
  • XiaoMing Yuan
Articles

DOI: 10.1007/s11425-013-4683-0

Cite this article as:
Cai, X., Gu, G., He, B. et al. Sci. China Math. (2013) 56: 2179. doi:10.1007/s11425-013-4683-0

Abstract

The alternating direction method of multipliers (ADMM) is a benchmark for solving convex programming problems with separable objective functions and linear constraints. In the literature it has been illustrated as an application of the proximal point algorithm (PPA) to the dual problem of the model under consideration. This paper shows that ADMM can also be regarded as an application of PPA to the primal model with a customized choice of the proximal parameter. This primal illustration of ADMM is thus complemental to its dual illustration in the literature. This PPA revisit on ADMM from the primal perspective also enables us to recover the generalized ADMM proposed by Eckstein and Bertsekas easily. A worst-case O(1/t) convergence rate in ergodic sense is established for a slight extension of Eckstein and Bertsekas’s generalized ADMM.

Keywords

alternating direction method of multipliers convergence rate convex programming proximal point algorithm 

MSC(2010)

65K10 90C25 90C33 

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • XingJu Cai
    • 1
  • GuoYong Gu
    • 1
  • BingSheng He
    • 1
  • XiaoMing Yuan
    • 2
  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Department of MathematicsHong Kong Baptist UniversityHong KongChina

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