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Approximate bregman proximal gradient algorithm for relatively smooth nonconvex optimization

Author

Listed:
  • Shota Takahashi

    (The University of Tokyo)

  • Akiko Takeda

    (The University of Tokyo
    RIKEN)

Abstract

In this paper, we propose the approximate Bregman proximal gradient algorithm (ABPG) for solving composite nonconvex optimization problems. ABPG employs a new distance that approximates the Bregman distance, making the subproblem of ABPG simpler to solve compared to existing Bregman-type algorithms. The subproblem of ABPG is often expressed in a closed form. Similarly to existing Bregman-type algorithms, ABPG does not require the global Lipschitz continuity for the gradient of the smooth part. Instead, assuming the smooth adaptable property, we establish the global subsequential convergence under standard assumptions. Additionally, assuming that the Kurdyka–Łojasiewicz property holds, we prove the global convergence for a special case. Our numerical experiments on the $$\ell _p$$ ℓ p regularized least squares problem, the $$\ell _p$$ ℓ p loss problem, and the nonnegative linear system show that ABPG outperforms existing algorithms especially when the gradient of the smooth part is not globally Lipschitz or even locally Lipschitz continuous.

Suggested Citation

  • Shota Takahashi & Akiko Takeda, 2025. "Approximate bregman proximal gradient algorithm for relatively smooth nonconvex optimization," Computational Optimization and Applications, Springer, vol. 90(1), pages 227-256, January.
    • Handle: RePEc:spr:coopap:v:90:y:2025:i:1:d:10.1007_s10589-024-00618-z
    DOI: 10.1007/s10589-024-00618-z
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    References listed on IDEAS

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    1. Xue Gao & Xingju Cai & Xiangfeng Wang & Deren Han, 2023. "An alternating structure-adapted Bregman proximal gradient descent algorithm for constrained nonconvex nonsmooth optimization problems and its inertial variant," Journal of Global Optimization, Springer, vol. 87(1), pages 277-300, September.
    2. Filip Hanzely & Peter Richtárik & Lin Xiao, 2021. "Accelerated Bregman proximal gradient methods for relatively smooth convex optimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 405-440, June.
    3. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. 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.
    5. 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.
    6. Radu-Alexandru Dragomir & Alexandre d’Aspremont & Jérôme Bolte, 2021. "Quartic First-Order Methods for Low-Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 341-363, May.
    7. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
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