Journal of Scientific Computing

, Volume 76, Issue 1, pp 69–88 | Cite as

Global Convergence of Unmodified 3-Block ADMM for a Class of Convex Minimization Problems

  • Tianyi Lin
  • Shiqian Ma
  • Shuzhong Zhang


The alternating direction method of multipliers (ADMM) has been successfully applied to solve structured convex optimization problems due to its superior practical performance. The convergence properties of the 2-block ADMM have been studied extensively in the literature. Specifically, it has been proven that the 2-block ADMM globally converges for any penalty parameter \(\gamma >0\). In this sense, the 2-block ADMM allows the parameter to be free, i.e., there is no need to restrict the value for the parameter when implementing this algorithm in order to ensure convergence. However, for the 3-block ADMM, Chen et al. (Math Program 155:57–79, 2016) recently constructed a counter-example showing that it can diverge if no further condition is imposed. The existing results on studying further sufficient conditions on guaranteeing the convergence of the 3-block ADMM usually require \(\gamma \) to be smaller than a certain bound, which is usually either difficult to compute or too small to make it a practical algorithm. In this paper, we show that the 3-block ADMM still globally converges with any penalty parameter \(\gamma >0\) if the third function \(f_3\) in the objective is smooth and strongly convex, and its condition number is in [1, 1.0798), besides some other mild conditions. This requirement covers an important class of problems to be called regularized least squares decomposition (RLSD) in this paper.


ADMM Global convergence Convex minimization Regularized least squares decomposition 



The research of Shiqian Ma is supported in part by a startup package in Department of Mathematics at UC Davis. The research of Shuzhong Zhang is supported in part by NSF Grant with Grant No. CMMI-1462408.


  1. 1.
    Boley, D.: Local linear convergence of the alternating direction method of multipliers on quadratic or linear programs. SIAM J. Optim. 23(4), 2183–2207 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)CrossRefMATHGoogle Scholar
  3. 3.
    Cai, X., Han, D., Yuan, X.: On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function. Comput. Optim. Appl. 66(1), 39–73 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155, 57–79 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, C., Shen, Y., You, Y.: On the convergence analysis of the alternating direction method of multipliers with three blocks. Abstr. Appl. Anal. Article ID 183961 (2013)Google Scholar
  6. 6.
    Davis, D., Yin, W.: A three-operator splitting scheme and its optimization applications, pp. 15–13. Technical Report, UCLA CAM Report (2015)Google Scholar
  7. 7.
    Deng, W., Lai, M., Peng, Z., Yin, W.: Parallel multi-block ADMM with \(o(1/k)\) convergence. J. Sci. Comput. 71(2), 712–736 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Deng, W., Yin, W.: On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput. 66(3), 889–916 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam (1983)MATHGoogle Scholar
  11. 11.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems. North-Holland, Amsterdam (1983)Google Scholar
  12. 12.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)CrossRefMATHGoogle Scholar
  13. 13.
    Han, D., Yuan, X.: A note on the alternating direction method of multipliers. J. Optim. Theory Appl. 155(1), 227–238 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    He, B., Hou, L., Yuan, X.: On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming. SIAM J. Optim. 25(4), 2274–2312 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22, 313–340 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    He, B., Tao, M., Yuan, X.: A splitting method for separable convex programming. IMA J. Numer. Anal. 35(1), 394–426 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    He, B., Tao, M., Yuan, X.: Convergence rate and iteration complexity on the alternating direction method of multipliers with a substitution procedure for separable convex programming. Math. Oper. Res. 42(3), 662–691 (2017)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    He, B., Yuan, X.: On the \({O}(1/n)\) convergence rate of Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hong, M., Chang, T.-H., Wang, X., Razaviyayn, M., Ma, S., Luo, Z.-Q.: A block successive upper bound minimization method of multipliers for linearly constrained convex optimization. arXiv preprint arXiv:1401.7079 (2014)
  20. 20.
    Hong, M., Luo, Z.-Q.: On the linear convergence of the alternating direction method of multipliers. Math. Program. 162(1), 165–199 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hong, M., Luo, Z.-Q., Razaviyayn, M.: Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J. Optim. 26(1), 337–364 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, G., Pong, T.: Global convergence of splitting methods for nonconvex composite optimization. SIAM J. Optim. 25, 2434–2460 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li, M., Sun, D., Toh, K.-C.: A convergent 3-block semi-proximal ADMM for convex minimization problems with one strongly convex block. Asia Pac. J. Oper. Res. 32(3), 1550024 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li, X., Ng, M.K., Yuan, X.: Median filtering-based methods for static background extraction from surveillance video. Numer. Linear Algebra Appl. 22(5), 845–865 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lin, T., Ma, S., Zhang, S.: On the global linear convergence of the ADMM with multiblock variables. SIAM J. Optim. 25(3), 1478–1497 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lin, T., Ma, S., Zhang, S.: On the sublinear convergence rate of multi-block ADMM. J. Oper. Res. Soc. China 3(3), 251–274 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lin, T., Ma, S., Zhang, S.: Iteration complexity analysis of multi-block ADMM for a family of convex minimization without strong convexity. J. Sci. Comput. 69, 52–81 (2016)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lin, Z., Chen, M., Wu, L., Ma, Y.: The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. UIUC Technical Report UILU-ENG-09-2215. arXiv:1009.5055v2 (2009)
  29. 29.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ma, S., Johnson, D., Ashby, C., Xiong, D., Cramer, C.L., Moore, J.H., Zhang, S., Huang, X.: SPARCoC: a new framework for molecular pattern discovery and cancer gene identification. PLoS ONE 10(3), e0117135 (2015)CrossRefGoogle Scholar
  31. 31.
    Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23, 475–507 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Peng, Y., Ganesh, A., Wright, J., Xu, W., Ma, Y.: RASL: robust alignment by sparse and low-rank decomposition for linearly correlated images. IEEE Trans. Pattern Anal. Mach. Intell. 34(11), 2233–2246 (2012)CrossRefGoogle Scholar
  33. 33.
    Sun, D., Toh, K.-C., Yang, L.: A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM J. Optim. 25, 882–915 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Sun, R., Luo, Z.-Q., Ye, Y.: On the expected convergence of randomly permuted ADMM. (2015)
  35. 35.
    Tao, M., Yuan, X.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21, 57–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc B 58(1), 267–288 (1996)MathSciNetMATHGoogle Scholar
  37. 37.
    Wang, X., Hong, M., Ma, S., Luo, Z.-Q.: Solving multiple-block separable convex minimization problems using two-block alternating direction method of multipliers. Pac. J. Optim. 11(4), 645–667 (2015)MathSciNetMATHGoogle Scholar
  38. 38.
    Waters, A., Sankaranarayanan, A., Baraniuk, R.: Sparcs: recovering low-rank and sparse matrices from compressive measurements. In: NIPS (2011)Google Scholar
  39. 39.
    Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203–230 (2010)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wright, J., Ganesh, A., Min, K., Ma, Y.: Compressive principal component pursuit. Inf. Inference 2(1), 32–68 (2013)Google Scholar
  41. 41.
    Yang, J., Zhang, Y.: Alternating direction algorithms for \(\ell _1\) problems in compressive sensing. SIAM J. Sci. Comput. 33(1), 250–278 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zhou, Z., Li, X., Wright, J., Candès, E.J., Ma, Y.: Stable principal component pursuit. In: Proceedings of International Symposium on Information Theory (2010)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchUC BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsUC DavisDavisUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of MinnesotaMinneapolisUSA

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