IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v87y2023i1d10.1007_s10898-023-01300-0.html
   My bibliography  Save this article

An alternating structure-adapted Bregman proximal gradient descent algorithm for constrained nonconvex nonsmooth optimization problems and its inertial variant

Author

Listed:
  • Xue Gao

    (Hebei University of Technology)

  • Xingju Cai

    (Nanjing Normal University)

  • Xiangfeng Wang

    (East China Normal University)

  • Deren Han

    (Beihang University)

Abstract

We consider the nonconvex nonsmooth minimization problem over abstract sets, whose objective function is the sum of a proper lower semicontinuous biconvex function of the entire variables and two smooth nonconvex functions of their private variables. Fully exploiting the problem structure, we propose an alternating structure-adapted Bregman proximal (ASABP for short) gradient descent algorithm, where the geometry of the abstract set and the function is captured by employing generalized Bregman function. Under the assumption that the underlying function satisfies the Kurdyka–Łojasiewicz property, we prove that each bounded sequence generated by ASABP globally converges to a critical point. We then adopt an inertial strategy to accelerate the ASABP algorithm (IASABP), and utilize a backtracking line search scheme to find “suitable” step sizes, making the algorithm efficient and robust. The global O(1/K) sublinear convergence rate measured by Bregman distance is also established. Furthermore, to illustrate the potential of ASABP and its inertial version (IASABP), we apply them to solving the Poisson linear inverse problem, and the results are promising.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-023-01300-0
    DOI: 10.1007/s10898-023-01300-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-023-01300-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-023-01300-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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).
    2. 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.
    3. 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.
    4. 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.
    5. Xue Gao & Xingju Cai & Deren Han, 2020. "A Gauss–Seidel type inertial proximal alternating linearized minimization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 76(4), pages 863-887, April.
    6. 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.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    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. 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.
    4. 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.
    5. 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.
    6. Zehui Jia & Jieru Huang & Xingju Cai, 2021. "Proximal-like incremental aggregated gradient method with Bregman distance in weakly convex optimization problems," Journal of Global Optimization, Springer, vol. 80(4), pages 841-864, August.
    7. 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.
    8. Abbaszadehpeivasti, Hadi, 2024. "Performance analysis of optimization methods for machine learning," Other publications TiSEM 3050a62d-1a1f-494e-99ef-7, Tilburg University, School of Economics and Management.
    9. 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.
    10. Szilárd Csaba László, 2023. "A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 387-427, July.
    11. 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.
    12. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    13. Francesco Rinaldi & Damiano Zeffiro, 2023. "Avoiding bad steps in Frank-Wolfe variants," Computational Optimization and Applications, Springer, vol. 84(1), pages 225-264, January.
    14. Kely D. V. Villacorta & Paulo R. Oliveira & Antoine Soubeyran, 2014. "A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 865-889, March.
    15. 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.
    16. Glaydston Carvalho Bento & João Xavier Cruz Neto & Antoine Soubeyran & Valdinês Leite Sousa Júnior, 2016. "Dual Descent Methods as Tension Reduction Systems," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 209-227, October.
    17. Bolte, Jérôme & Le, Tam & Pauwels, Edouard & Silveti-Falls, Antonio, 2022. "Nonsmooth Implicit Differentiation for Machine Learning and Optimization," TSE Working Papers 22-1314, Toulouse School of Economics (TSE).
    18. Alexander Y. Kruger & Nguyen H. Thao, 2015. "Quantitative Characterizations of Regularity Properties of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 41-67, January.
    19. Fornasier, Massimo & Maly, Johannes & Naumova, Valeriya, 2021. "Robust recovery of low-rank matrices with non-orthogonal sparse decomposition from incomplete measurements," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    20. W. Ackooij & S. Demassey & P. Javal & H. Morais & W. Oliveira & B. Swaminathan, 2021. "A bundle method for nonsmooth DC programming with application to chance-constrained problems," Computational Optimization and Applications, Springer, vol. 78(2), pages 451-490, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-023-01300-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.