IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v204y2025i1d10.1007_s10957-024-02560-w.html
   My bibliography  Save this article

Forward-Primal-Dual-Half-Forward Algorithm for Splitting Four Operators

Author

Listed:
  • Fernando Roldán

    (Universidad de Concepción)

Abstract

In this article, we propose a splitting algorithm to find zeros of the sum of four maximally monotone operators in real Hilbert spaces. In particular, we consider a Lipschitz operator, a cocoercive operator, and a linear composite term. In the case where the Lipschitz operator is absent, our method reduces to the Condat-Vũ algorithm. On the other hand, when the linear composite term is absent, the algorithm reduces to the Forward-Backward-Half-Forward algorithm (FBHF). Additionally, in each case, the set of step-sizes that guarantee the weak convergence of those methods are recovered. Therefore, our algorithm can be seen as a combination of Condat-Vũ and FBHF. Moreover, we propose extensions and applications of our method in multivariate monotone inclusions and saddle point problems. Finally, we present a numerical experiment on image deblurring problems.

Suggested Citation

  • Fernando Roldán, 2025. "Forward-Primal-Dual-Half-Forward Algorithm for Splitting Four Operators," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-26, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02560-w
    DOI: 10.1007/s10957-024-02560-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02560-w
    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/s10957-024-02560-w?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. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
    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. Yunda Dong, 2021. "Weak convergence of an extended splitting method for monotone inclusions," Journal of Global Optimization, Springer, vol. 79(1), pages 257-277, January.
    2. Puya Latafat & Panagiotis Patrinos, 2017. "Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators," Computational Optimization and Applications, Springer, vol. 68(1), pages 57-93, September.
    3. Xin Jiang & Lieven Vandenberghe, 2022. "Bregman primal–dual first-order method and application to sparse semidefinite programming," Computational Optimization and Applications, Springer, vol. 81(1), pages 127-159, January.
    4. Cécile Bastidon & Myriam Bontonou & Pierre Borgnat & Pablo Jensen & Patrice Abry & Antoine Parent, 2024. "Learning smooth graphs with sparse temporal variations to explore long-term financial trends," Post-Print hal-04731912, HAL.
    5. 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.
    6. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    7. Sun, Shilin & Wang, Tianyang & Yang, Hongxing & Chu, Fulei, 2022. "Damage identification of wind turbine blades using an adaptive method for compressive beamforming based on the generalized minimax-concave penalty function," Renewable Energy, Elsevier, vol. 181(C), pages 59-70.
    8. Fu, Penghui & Tan, Zhiqiang, 2024. "Block-wise primal-dual algorithms for large-scale doubly penalized ANOVA modeling," Computational Statistics & Data Analysis, Elsevier, vol. 194(C).
    9. David Degras, 2021. "Sparse group fused lasso for model segmentation: a hybrid approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 625-671, September.
    10. Yawei Shi & Liang Ran & Jialong Tang & Xiangzhao Wu, 2022. "Distributed Optimization Algorithm for Composite Optimization Problems with Non-Smooth Function," Mathematics, MDPI, vol. 10(17), pages 1-17, September.
    11. Xiaoliang Wang & Liping Pang & Qi Wu & Mingkun Zhang, 2021. "An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization," Mathematics, MDPI, vol. 9(8), pages 1-27, April.
    12. Luis Briceño-Arias & Sergio López Rivera, 2019. "A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 907-924, March.
    13. Walaa M. Moursi & Lieven Vandenberghe, 2019. "Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 179-198, October.
    14. Patrick R. Johnstone & Pierre Moulin, 2017. "Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 67(2), pages 259-292, June.
    15. Ernest K. Ryu & Bằng Công Vũ, 2020. "Finding the Forward-Douglas–Rachford-Forward Method," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 858-876, March.
    16. Adil Salim & Laurent Condat & Konstantin Mishchenko & Peter Richtárik, 2022. "Dualize, Split, Randomize: Toward Fast Nonsmooth Optimization Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 102-130, October.
    17. Fernando Roldán, 2024. "Forward-reflected-backward and shadow-Douglas–Rachford with partial inverse for solving monotone inclusions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(3), pages 723-752, December.
    18. Luis Briceño-Arias & Fernando Roldán, 2023. "Primal-dual splittings as fixed point iterations in the range of linear operators," Journal of Global Optimization, Springer, vol. 85(4), pages 847-866, April.
    19. Ke-wei Ding & Jörg Fliege & Phan Tu Vuong, 2025. "Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem," Computational Optimization and Applications, Springer, vol. 90(1), pages 151-192, January.
    20. Daniel O’Connor & Lieven Vandenberghe, 2017. "Total variation image deblurring with space-varying kernel," Computational Optimization and Applications, Springer, vol. 67(3), pages 521-541, July.

    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:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02560-w. 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.