Mathematical Programming

, Volume 173, Issue 1–2, pp 37–77 | Cite as

Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights

  • Caihua Chen
  • Min Li
  • Xin LiuEmail author
  • Yinyu Ye
Full Length Paper Series A


In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for “nice” problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks.


Nonseparable convex minimization Alternating direction method of multipliers Block coordinate descent method Iterate convergence Random permutation 

Mathematics Subject Classification

65K05 90C26 



Caihua Chen was supported by the National Natural Science Foundation of China [Grant No. 11401300, 71732003, 71673130]. Min Li was supported by the National Natural Science Foundation of China [Grant No.11771078, 71390335, 71661147004]. Xin Liu was supported by the National Natural Science Foundation of China [Grant No. 11622112, 11471325, 91530204, 11331012, 11461161005, 11688101], the National Center for Mathematics and Interdisciplinary Sciences, CAS, the Youth Innovation Promotion Association, CAS, and Key Research Program of Frontier Sciences, CAS. Yinyu Ye was supported by the AFOSR Grant [Grant No. FA9550-12-1-0396]. The authors would like to thank Dr. Ji Liu from University of Rochester and Dr. Ruoyu Sun from Stanford University for the helpful discussions on the block coordinate descent method. The authors would also like to thank the associate editor and two anonymous referees for their detailed and valuable comments and suggestions.


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Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.International Center of Management Science and Engineering, School of Management and EngineeringNanjing UniversityNanjingChina
  2. 2.State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  4. 4.Department of Management Science and Engineering, School of EngineeringStanford UniversityStanfordUSA

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