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On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function

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Abstract

The alternating direction method of multipliers (ADMM) is a benchmark for solving a two-block linearly constrained convex minimization model whose objective function is the sum of two functions without coupled variables. Meanwhile, it is known that the convergence is not guaranteed if the ADMM is directly extended to a multiple-block convex minimization model whose objective function has more than two functions. Recently, some authors have actively studied the strong convexity condition on the objective function to sufficiently ensure the convergence of the direct extension of ADMM or the resulting convergence when the original scheme is appropriately twisted. We focus on the three-block case of such a model whose objective function is the sum of three functions, and discuss the convergence of the direct extension of ADMM. We show that when one function in the objective is strongly convex, the penalty parameter and the operators in the linear equality constraint are appropriately restricted, it is sufficient to guarantee the convergence of the direct extension of ADMM. We further estimate the worst-case convergence rate measured by the iteration complexity in both the ergodic and nonergodic senses, and derive the globally linear convergence in asymptotical sense under some additional conditions.

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References

  1. Boley, D.: Local linear convergence of ADMM on quadratic or linear programs. SIAM J. Optim. 23(4), 2183–2207 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. In: Jordan, M. (ed.) Foundations and Trends in Machine Learning, vol. 3,pp. 1–122 (2011)

  3. Chamboulle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekaran, V., Parrilo, P.A., Willsky, A.S.: Latent variable graphical model selection via convex optimization. Ann. Stat. 40, 1935–1967 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C.H., He, B.S., Ye, Y.Y., Yuan, X.M.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155, 57–79 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, C.H., Shen, Y., You, Y.F.: On the convergence analysis of the alternating direction method of multipliers with three blocks. Abstr. Appl. Anal. 2013 (2013), Article ID 183961

  7. Corman, E., Yuan, X.M.: A generalized proximal point algorithm and its convergence rate. SIAM J. Optim. 24(4), 1614–1638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deng, W., Yin, W.T.: On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput. 66(3), 889–916 (2016)

    Article  MathSciNet  Google Scholar 

  9. Deng, W., Lai, M.J., Peng, Z.M., Yin, W.T.: Parallel multi-block ADMM with \(o(1/k)\) convergence. Manuscript (2014)

  10. Eckstein, J., Yao, W.: Understanding the convergence of the alternating direction method of multipliers: theoretical and computational perspectives. Pac. J. Optim. 11, 619–644 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  MATH  Google Scholar 

  12. Glowinski, R., Marrocco, A.: Sur l’approximation paréléments finis d’ordre un et larésolution parpénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. Revue Fr. Autom. Inf. Rech. Opér., Anal. Numér. 2, 41–76 (1975)

    MathSciNet  Google Scholar 

  13. Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. Model. Simul. Optim. Sci. Technol. Comput. Methods Appl. Sci. 34, 59–82 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Han, D.R., Yuan, X.M.: A note on the alternating direction method of multipliers. J. Optim. Theory Appl. 155, 227–238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Han, D.R., Yuan, X.M.: Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J. Numer. Anal. 51, 3446–3457 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Han, D.R., Yuan, X.M., Zhang, W.X.: An augmented-Lagrangian-based parallel splitting method for separable convex programming with applications to image processing. Math. Comput. 83, 2263–2291 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. He, B.S., Tao, M., Yuan, X.M.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22, 313–340 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, B.S., Tao, M., Yuan, X.M.: Convergence rate and iteration complexity on the alternating direction method of multipliers with a substitution procedure for separable convex programming. Math. Oper. Res. (under revision)

  19. He, B.S., Tao, M., Yuan, X.M.: A splitting method for separable convex programming. IMA J. Numer. Anal. 35, 394–426 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hong, M.Y., Luo, Z.Q.: On the linear convergence of alternating direction method of multipliers. Math. Program. (to appear)

  22. Li, M., Sun, D.F., 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, 1550024 (2015). 19 pages

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin, T.Y., Ma, S.Q., Zhang, S.Z.: On the sublinear convergence rate of multi-block ADMM. J. Oper. Res. Soc. China 3, 251–274 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin, T.Y., Ma, S.Q., Zhang, S.Z.: On the global linear convergence of the ADMM with multi-block variables. SIAM J. Optim. 25, 1478–1497 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martinet, B.: Regularization d’inequations variationelles par approximations successives. Revue Francaise d’Informatique et de Recherche Opérationelle 4, 154–159 (1970)

    MathSciNet  MATH  Google Scholar 

  26. Moreau, J.J.: Proximité et dualit ’e dans un espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)

    MathSciNet  MATH  Google Scholar 

  27. McLachlan, G.J.: Discriminant Analysis and Statistical Pattern Recognition, vol. 544. Wiley, New York (2004)

    MATH  Google Scholar 

  28. Mohan, K., London, P., Fazel, M., Witten, D., Lee, S.: Node-based learning of multiple gaussian graphical models. J. Mach. Learn. Res. 15, 445–488 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Peng, Y.G., Ganesh, A., Wright, J., Xu, W.L., Ma, Y.: Robust alignment by sparse and low-rank decomposition for linearly correlated images. IEEE Trans. Pattern Anal. Mach. Intell. 34, 2233–2246 (2012)

    Article  Google Scholar 

  30. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)

    Google Scholar 

  31. Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14, 206–214 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  33. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tao, M., Yuan, X.M.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21, 57–81 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, W.H., Han, D.R.: Linear convergence of alternating direction method of multipliers for a class of convex optimization problems. SIAM J. Numer. Anal. 54(2), 625–640 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zheng, X.Y., Ng, K.F.: Metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization. SIAM J. Optim. 24, 154–174 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, X., Yang, C., Yu, W.: Moving object detection by detecting contiguous outliers in the Low-Rank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35, 597–610 (2013)

    Article  Google Scholar 

  38. Zhou, Z., Li, X., Wright, J., Candes, E.J., Ma, Y.: Stable principal component pursuit. In: Proceedings of International Symposium on Information Theory. Austin (2010)

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Acknowledgments

Xingju Cai is supported by the NSFC Grant 11401315 and the NSF from Jiangsu province BK20140914. Deren Han is supported by a project funded by PAPD of Jiangsu Higher Education Institutions and the NSFC Grants 11371197 and 11431002. Xiaoming Yuan was supported by the NSFC/RGC Joint Research Scheme: N_PolyU504/14.

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Authors and Affiliations

  1. School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Xingju Cai & Deren Han

  2. Department of Mathematics, Hong Kong Baptist University, Hong Kong, People’s Republic of China

    Xiaoming Yuan

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  1. Xingju Cai
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  2. Deren Han
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  3. Xiaoming Yuan
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Correspondence to Xiaoming Yuan.

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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, 39–73 (2017). https://doi.org/10.1007/s10589-016-9860-y

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  • DOI: https://doi.org/10.1007/s10589-016-9860-y

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