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A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems

Author

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  • Kai Tu

    (Beijing University of Technology)

  • Haibin Zhang

    (Beijing University of Technology)

  • Huan Gao

    (Hunan First Normal University)

  • Junkai Feng

    (Beijing University of Technology)

Abstract

In this paper, we propose a hybrid Bregman alternating direction method of multipliers for solving the linearly constrained difference-of-convex problems whose objective can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous concave function. At each iteration, we choose either subgradient step or proximal step to evaluate the concave part. Moreover, the extrapolation technique was utilized to compute the nonsmooth convex part. We prove that the sequence generated by the proposed method converges to a critical point of the considered problem under the assumption that the potential function is a Kurdyka–Łojasiewicz function. One notable advantage of the proposed method is that the convergence can be guaranteed without the Lischitz continuity of the gradient function of concave part. Preliminary numerical experiments show the efficiency of the proposed method.

Suggested Citation

  • Kai Tu & Haibin Zhang & Huan Gao & Junkai Feng, 2020. "A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems," Journal of Global Optimization, Springer, vol. 76(4), pages 665-693, April.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:4:d:10.1007_s10898-019-00828-4
    DOI: 10.1007/s10898-019-00828-4
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    References listed on IDEAS

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    1. Zhaosong Lu & Xiaorui Li, 2018. "Sparse Recovery via Partial Regularization: Models, Theory, and Algorithms," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1290-1316, November.
    2. Zhongming Wu & Min Li & David Z. W. Wang & Deren Han, 2017. "A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-27, December.
    3. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
    4. Kristian Bredies & Dirk A. Lorenz & Stefan Reiterer, 2015. "Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 78-112, April.
    5. Chenxi Chen & Yunmei Chen & Yuyuan Ouyang & Eduardo Pasiliao, 2018. "Stochastic Accelerated Alternating Direction Method of Multipliers with Importance Sampling," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 676-695, November.
    6. W. Geremew & N. M. Nam & A. Semenov & V. Boginski & E. Pasiliao, 2018. "A DC programming approach for solving multicast network design problems via the Nesterov smoothing technique," Journal of Global Optimization, Springer, vol. 72(4), pages 705-729, December.
    7. J. Souza & P. Oliveira, 2015. "A proximal point algorithm for DC fuctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 63(4), pages 797-810, December.
    8. Tianxiang Liu & Ting Kei Pong & Akiko Takeda, 2019. "A refined convergence analysis of $$\hbox {pDCA}_{e}$$ pDCA e with applications to simultaneous sparse recovery and outlier detection," Computational Optimization and Applications, Springer, vol. 73(1), pages 69-100, May.
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    Cited by:

    1. Shota Takahashi & Mituhiro Fukuda & Mirai Tanaka, 2022. "New Bregman proximal type algorithms for solving DC optimization problems," Computational Optimization and Applications, Springer, vol. 83(3), pages 893-931, December.

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