IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v91y2025i3d10.1007_s10589-025-00684-x.html
   My bibliography  Save this article

On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems

Author

Listed:
  • Behzad Azmi

    (University of Konstanz)

  • Marco Bernreuther

    (University of Stuttgart)

Abstract

This paper provides a comprehensive study of the nonmonotone forward–backward splitting (FBS) method for solving a class of nonsmooth composite problems in Hilbert spaces. The objective function is the sum of a Fréchet differentiable (not necessarily convex) function and a proper lower semicontinuous convex (not necessarily smooth) function. These problems appear, for example, frequently in the context of optimal control of nonlinear partial differential equations (PDEs) with nonsmooth sparsity-promoting cost functionals. We discuss the convergence and complexity of FBS equipped with the nonmonotone linesearch under different conditions. In particular, R-linear convergence will be derived under quadratic growth-type conditions. We also investigate the applicability of the algorithm to problems governed by PDEs. Numerical experiments are also given that justify our theoretical findings.

Suggested Citation

  • Behzad Azmi & Marco Bernreuther, 2025. "On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems," Computational Optimization and Applications, Springer, vol. 91(3), pages 1263-1308, July.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00684-x
    DOI: 10.1007/s10589-025-00684-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-025-00684-x
    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/s10589-025-00684-x?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. Yunier Bello-Cruz & Guoyin Li & Tran Thai An Nghia, 2022. "Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 167-190, July.
    2. Christian Kanzow & Theresa Lechner, 2021. "Correction to: Globalized inexact proximal Newton-type methods for nonconvex composite functions," Computational Optimization and Applications, Springer, vol. 80(2), pages 679-680, November.
    3. Behzad Azmi & Karl Kunisch, 2020. "Analysis of the Barzilai-Borwein Step-Sizes for Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 819-844, June.
    4. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    6. Pierre Frankel & Guillaume Garrigos & Juan Peypouquet, 2015. "Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 874-900, June.
    7. Yunier Bello-Cruz & Guoyin Li & Tran T. A. Nghia, 2021. "On the Linear Convergence of Forward–Backward Splitting Method: Part I—Convergence Analysis," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 378-401, February.
    8. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Christian Kanzow & Patrick Mehlitz, 2022. "Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 624-646, November.
    10. Christian Kanzow & Theresa Lechner, 2021. "Globalized inexact proximal Newton-type methods for nonconvex composite functions," Computational Optimization and Applications, Springer, vol. 78(2), pages 377-410, 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. Simeon vom Dahl & Christian Kanzow, 2024. "An inexact regularized proximal Newton method without line search," Computational Optimization and Applications, Springer, vol. 89(3), pages 585-624, December.
    2. Hao Wang & Hao Zeng & Jiashan Wang, 2022. "An extrapolated iteratively reweighted $$\ell _1$$ ℓ 1 method with complexity analysis," Computational Optimization and Applications, Springer, vol. 83(3), pages 967-997, December.
    3. Yassine Nabou & Ion Necoara, 2024. "Efficiency of higher-order algorithms for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 87(2), pages 441-473, March.
    4. Juan José Maulén & Juan Peypouquet, 2023. "A Speed Restart Scheme for a Dynamics with Hessian-Driven Damping," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 831-855, November.
    5. Ching-pei Lee & Stephen J. Wright, 2019. "Inexact Successive quadratic approximation for regularized optimization," Computational Optimization and Applications, Springer, vol. 72(3), pages 641-674, April.
    6. Christian Kanzow & Theresa Lechner, 2022. "COAP 2021 Best Paper Prize," Computational Optimization and Applications, Springer, vol. 83(3), pages 723-726, December.
    7. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    8. Jean-François Aujol & Charles Dossal & Hippolyte Labarrière & Aude Rondepierre, 2025. "FISTA Restart Using an Automatic Estimation of the Growth Parameter," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-27, August.
    9. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.
    10. Lei Yang, 2024. "Proximal Gradient Method with Extrapolation and Line Search for a Class of Non-convex and Non-smooth Problems," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 68-103, January.
    11. Huynh Ngai & Ta Anh Son, 2022. "Generalized Nesterov’s accelerated proximal gradient algorithms with convergence rate of order o(1/k2)," Computational Optimization and Applications, Springer, vol. 83(2), pages 615-649, November.
    12. Olivier Fercoq & Zheng Qu, 2020. "Restarting the accelerated coordinate descent method with a rough strong convexity estimate," Computational Optimization and Applications, Springer, vol. 75(1), pages 63-91, January.
    13. Xiaoya Zhang & Wei Peng & Hui Zhang, 2022. "Inertial proximal incremental aggregated gradient method with linear convergence guarantees," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 187-213, October.
    14. Yunier Bello-Cruz & Guoyin Li & Tran Thai An Nghia, 2022. "Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 167-190, July.
    15. Helmut Gfrerer, 2025. "On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization," Computational Optimization and Applications, Springer, vol. 91(1), pages 67-124, May.
    16. Ruyu Liu & Shaohua Pan & Yuqia Wu & Xiaoqi Yang, 2024. "An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization," Computational Optimization and Applications, Springer, vol. 88(2), pages 603-641, June.
    17. 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.
    18. Kaiwen Ma & Nikolaos V. Sahinidis & Sreekanth Rajagopalan & Satyajith Amaran & Scott J Bury, 2021. "Decomposition in derivative-free optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 269-292, October.
    19. A. Scagliotti & P. Colli Franzone, 2022. "A piecewise conservative method for unconstrained convex optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 251-288, January.
    20. Ren Jiang & Zhifeng Ji & Wuling Mo & Suhua Wang & Mingjun Zhang & Wei Yin & Zhen Wang & Yaping Lin & Xueke Wang & Umar Ashraf, 2022. "A Novel Method of Deep Learning for Shear Velocity Prediction in a Tight Sandstone Reservoir," Energies, MDPI, vol. 15(19), pages 1-20, September.

    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:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00684-x. 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.