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Malitsky-Tam forward-reflected-backward splitting method for nonconvex minimization problems

Author

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  • Xianfu Wang

    (University of British Columbia)

  • Ziyuan Wang

    (University of British Columbia)

Abstract

We extend the Malitsky-Tam forward-reflected-backward (FRB) splitting method for inclusion problems of monotone operators to nonconvex minimization problems. By assuming the generalized concave Kurdyka-Łojasiewicz (KL) property of a quadratic regularization of the objective, we show that the FRB method converges globally to a stationary point of the objective and enjoys the finite length property. Convergence rates are also given. The sharpness of our approach is guaranteed by virtue of the exact modulus associated with the generalized concave KL property. Numerical experiments suggest that FRB is competitive compared to the Douglas-Rachford method and the Boţ-Csetnek inertial Tseng’s method.

Suggested Citation

  • Xianfu Wang & Ziyuan Wang, 2022. "Malitsky-Tam forward-reflected-backward splitting method for nonconvex minimization problems," Computational Optimization and Applications, Springer, vol. 82(2), pages 441-463, June.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:2:d:10.1007_s10589-022-00364-0
    DOI: 10.1007/s10589-022-00364-0
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Radu Ioan Bot & Dang-Khoa Nguyen, 2020. "The Proximal Alternating Direction Method of Multipliers in the Nonconvex Setting: Convergence Analysis and Rates," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 682-712, May.
    3. Radu Ioan Boţ & Ernö Robert Csetnek & Szilárd Csaba László, 2016. "An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 3-25, February.
    4. Guoyin Li & Tianxiang Liu & Ting Kei Pong, 2017. "Peaceman–Rachford splitting for a class of nonconvex optimization problems," Computational Optimization and Applications, Springer, vol. 68(2), pages 407-436, November.
    5. Chen Chen & Ting Kei Pong & Lulin Tan & Liaoyuan Zeng, 2020. "A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection," Journal of Global Optimization, Springer, vol. 78(1), pages 107-136, September.
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