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Douglas–Rachford splitting and ADMM for nonconvex optimization: accelerated and Newton-type linesearch algorithms

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  • Andreas Themelis

    (Kyushu University)

  • Lorenzo Stella

    (Amazon)

  • Panagiotis Patrinos

    (KU Leuven)

Abstract

Although the performance of popular optimization algorithms such as the Douglas–Rachford splitting (DRS) and the ADMM is satisfactory in convex and well-scaled problems, ill conditioning and nonconvexity pose a severe obstacle to their reliable employment. Expanding on recent convergence results for DRS and ADMM applied to nonconvex problems, we propose two linesearch algorithms to enhance and robustify these methods by means of quasi-Newton directions. The proposed algorithms are suited for nonconvex problems, require the same black-box oracle of DRS and ADMM, and maintain their (subsequential) convergence properties. Numerical evidence shows that the employment of L-BFGS in the proposed framework greatly improves convergence of DRS and ADMM, making them robust to ill conditioning. Under regularity and nondegeneracy assumptions at the limit point, superlinear convergence is shown when quasi-Newton Broyden directions are adopted.

Suggested Citation

  • Andreas Themelis & Lorenzo Stella & Panagiotis Patrinos, 2022. "Douglas–Rachford splitting and ADMM for nonconvex optimization: accelerated and Newton-type linesearch algorithms," Computational Optimization and Applications, Springer, vol. 82(2), pages 395-440, June.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:2:d:10.1007_s10589-022-00366-y
    DOI: 10.1007/s10589-022-00366-y
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    References listed on IDEAS

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    1. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    2. Bo Jiang & Tianyi Lin & Shiqian Ma & Shuzhong Zhang, 2019. "Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis," Computational Optimization and Applications, Springer, vol. 72(1), pages 115-157, January.
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    Cited by:

    1. Ziyuan Wang & Andreas Themelis & Hongjia Ou & Xianfu Wang, 2024. "A Mirror Inertial Forward–Reflected–Backward Splitting: Convergence Analysis Beyond Convexity and Lipschitz Smoothness," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1127-1159, November.
    2. Jianghua Yin & Chunming Tang & Jinbao Jian & Qiongxuan Huang, 2024. "A partial Bregman ADMM with a general relaxation factor for structured nonconvex and nonsmooth optimization," Journal of Global Optimization, Springer, vol. 89(4), pages 899-926, August.

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