SCIENCE CHINA Information Sciences, Volume 61, Issue 12: 122101(2018) https://doi.org/10.1007/s11432-017-9367-6

Convergence of multi-blockBregman ADMM for nonconvex composite problems

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  • ReceivedSep 12, 2017
  • AcceptedFeb 26, 2018
  • PublishedJun 21, 2018


The alternating direction method with multipliers (ADMM) isone of the most powerful and successful methods for solving variouscomposite problems. The convergence of the conventional ADMM (i.e.,2-block) for convex objective functions has been stated for along time, and its convergence for nonconvex objective functionshas, however, been established very recently. The multi-block ADMM,a natural extension of ADMM, is a widely used scheme and has alsobeen found very useful in solving various nonconvex optimizationproblems. It is thus expected to establish the convergence ofthe multi-block ADMM under nonconvex frameworks. In this paper, wefirst justify the convergence of 3-block Bregman ADMM.We next extend these results to the $N$-block case ($N~\geq~3$),which underlines the feasibility of multi-block ADMM applications innonconvex settings. Finally, we present a simulation study and areal-world application to support the correctness of the obtainedtheoretical assertions.


This work was supported by National Natural Science Foundation of China (Grant No. 61603235), and Program for Science and Technology Innovation Talents in Universities of Henan Province (Grant No. 15HASTIT013). We thank all anonymous reviewers for their thoughtful and constructive comments that greatly improve the analysis and writing of the manuscript.


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  • Figure 1

    (Color online) Convergence results for (a) the noiseless case and (b) Gaussian noise with the standard deviation $\sigma=0.01$.

  • Figure 2

    Background subtraction results in the real-world video clips. (a) Lobby; (b) Bootstrap; (c) Hall; (d) ShoppingMall.

    (ttr ttspr $\text{RelErr}$ $\text{Rank}({L}^{*})$ $\text{Rank}(\hat{L})$ $\|{S}^{*}\|_0$ $\|\hat{S}\|_0$
    Noiseless case ($\sigma=~0$)(1, 0.05) 4.8674E$-$06 1 1500500
    (1, 0.1) 5.0446E$-$06 1 1 10001000
    (5, 0.05) 2.2342E$-$06 5 5500500
    (5, 0.1) 2.4366E$-$0655 1000 1000
    (10, 0.05) 1.5039E$-$06 1010 500 500
    (10, 0.1) 1.8572E$-$0610 10 10001000
    (20, 0.05) 1.2889E$-$06 20 20500500
    (20, 0.1) 1.6974E$-$0620 20 10001000
    Gauss noise $(\sigma=~0.01)$(1, 0.05) 0.0049 1 15001723
    (1, 0.1) 0.0060 1 1 10003797
    (5, 0.05) 0.00255 55001541
    (5, 0.1) 0.0033 55 1000 3551
    (10, 0.05) 0.0022 1010 500 1318
    (10, 0.1) 0.0024 10 10 10003183
    (20, 0.05) 0.0020 20 205001110
    (20, 0.1) 0.0024 20 20 10003612

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