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On Linear Convergence of Non-Euclidean Gradient Methods without Strong Convexity and Lipschitz Gradient Continuity

Author

Listed:
  • Heinz H. Bauschke

    (University of British Columbia Okanagan)

  • Jérôme Bolte

    (Université Toulouse 1 Capitole)

  • Jiawei Chen

    (Southwest University)

  • Marc Teboulle

    (Tel Aviv University)

  • Xianfu Wang

    (University of British Columbia Okanagan)

Abstract

The gradient method is well known to globally converge linearly when the objective function is strongly convex and admits a Lipschitz continuous gradient. In many applications, both assumptions are often too stringent, precluding the use of gradient methods. In the early 1960s, after the amazing breakthrough of Łojasiewicz on gradient inequalities, it was observed that uniform convexity assumptions could be relaxed and replaced by these inequalities. On the other hand, very recently, it has been shown that the Lipschitz gradient continuity can be lifted and replaced by a class of functions satisfying a non-Euclidean descent property expressed in terms of a Bregman distance. In this note, we combine these two ideas to introduce a class of non-Euclidean gradient-like inequalities, allowing to prove linear convergence of a Bregman gradient method for nonconvex minimization, even when neither strong convexity nor Lipschitz gradient continuity holds.

Suggested Citation

  • Heinz H. Bauschke & Jérôme Bolte & Jiawei Chen & Marc Teboulle & Xianfu Wang, 2019. "On Linear Convergence of Non-Euclidean Gradient Methods without Strong Convexity and Lipschitz Gradient Continuity," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1068-1087, September.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:3:d:10.1007_s10957-019-01516-9
    DOI: 10.1007/s10957-019-01516-9
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    References listed on IDEAS

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    1. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    2. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    3. 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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    Cited by:

    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. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    3. Hui Zhang & Yu-Hong Dai & Lei Guo & Wei Peng, 2021. "Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 61-81, February.
    4. Yin Liu & Sam Davanloo Tajbakhsh, 2023. "Stochastic Composition Optimization of Functions Without Lipschitz Continuous Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 239-289, July.
    5. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    6. 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.
    7. Jing Zhao & Qiao-Li Dong & Michael Th. Rassias & Fenghui Wang, 2022. "Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems," Journal of Global Optimization, Springer, vol. 84(4), pages 941-966, December.
    8. Emanuel Laude & Peter Ochs & Daniel Cremers, 2020. "Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 724-761, March.

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