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A note on the Douglas–Rachford splitting method for optimization problems involving hypoconvex functions

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  • Ke Guo

    (China West Normal University)

  • Deren Han

    (Nanjing Normal University)

Abstract

Recently, the convergence of the Douglas–Rachford splitting method (DRSM) was established for minimizing the sum of a nonsmooth strongly convex function and a nonsmooth hypoconvex function under the assumption that the strong convexity constant $$\beta $$ β is larger than the hypoconvexity constant $$\omega $$ ω . Such an assumption, implying the strong convexity of the objective function, precludes many interesting applications. In this paper, we prove the convergence of the DRSM for the case $$\beta =\omega $$ β = ω , under relatively mild assumptions compared with some existing work in the literature.

Suggested Citation

  • Ke Guo & Deren Han, 2018. "A note on the Douglas–Rachford splitting method for optimization problems involving hypoconvex functions," Journal of Global Optimization, Springer, vol. 72(3), pages 431-441, November.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:3:d:10.1007_s10898-018-0660-z
    DOI: 10.1007/s10898-018-0660-z
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    References listed on IDEAS

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    1. Heinz H. Bauschke & Warren L. Hare & Walaa M. Moursi, 2016. "On the Range of the Douglas–Rachford Operator," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 884-897, August.
    2. Heinz H. Bauschke & Valentin R. Koch & Hung M. Phan, 2016. "Stadium Norm and Douglas-Rachford Splitting: A New Approach to Road Design Optimization," Operations Research, INFORMS, vol. 64(1), pages 201-218, February.
    3. Christian Kanzow & Yekini Shehu, 2017. "Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 67(3), pages 595-620, July.
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    Cited by:

    1. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    2. Chih-Sheng Chuang & Hongjin He & Zhiyuan Zhang, 2022. "A unified Douglas–Rachford algorithm for generalized DC programming," Journal of Global Optimization, Springer, vol. 82(2), pages 331-349, February.
    3. Zhongming Wu & Chongshou Li & Min Li & Andrew Lim, 2021. "Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 79(3), pages 617-644, March.
    4. Zhili Ge & Zhongming Wu & Xin Zhang & Qin Ni, 2023. "An extrapolated proximal iteratively reweighted method for nonconvex composite optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 821-844, August.

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