[go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Convergence Analysis of a New Forward-Reflected-Backward Algorithm for Four Operators Without Cocoercivity

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we propose a new splitting algorithm to find the zero of a monotone inclusion problem that features the sum of three maximal monotone operators and a Lipschitz continuous monotone operator in Hilbert spaces. We prove that the sequence of iterates generated by our proposed splitting algorithm converges weakly to the zero of the considered inclusion problem under mild conditions on the iterative parameters. Several splitting algorithms in the literature are recovered as special cases of our proposed algorithm. Another interesting feature of our algorithm is that one forward evaluation of the Lipschitz continuous monotone operator is utilized at each iteration. Numerical results are given to support the theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Code Availability

The Matlab codes employed to run the numerical experiments are available on request.

References

  1. Aragón Artacho, F.J., Campoy, R., Tam, M.K.: Strengthened splitting methods for computing resolvents. Comput. Optim. Appl. 80, 549–585 (2021)

    Article  MathSciNet  Google Scholar 

  2. Aragón Artacho, F.J., Campoy, R., Tam, M.K., Torregroa-Belén, D.: Distributed forward-backward methods for ring networks. Comput. Optim. Appl. 86, 845–870 (2023)

    Article  MathSciNet  Google Scholar 

  3. Aragón Artacho, F.J., Censor, Y., Gibali, A.: The cyclic Douglas–Rachford algorithm with \(r\)-sets-Douglas–Rachford operators. Optim. Methods Softw. 34, 875–889 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)

    Book  Google Scholar 

  5. Borwein, J.M., Tam, M.K.: A cyclic Douglas–Rachford iteration scheme. J. Optim. Theory Appl. 160, 1–29 (2014)

    Article  MathSciNet  Google Scholar 

  6. Briceño-Arias, L.M.: Forward-Douglas–Rachford splitting and forward-partial inverse method for solving monotone inclusions. Optimization 64, 1239–1261 (2015)

    Article  MathSciNet  Google Scholar 

  7. Cai, J.F., Candes, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)

    Article  MathSciNet  Google Scholar 

  8. Combettes, P.L., Pesquet, J.C.: A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1, 564–574 (2007)

    Article  Google Scholar 

  9. Davis, D., Yin, W.T.: A three-operator splitting scheme and its optimization applications. Set-Valued Var. Anal. 25, 829–858 (2019)

    Article  MathSciNet  Google Scholar 

  10. Goldstein, A.A.: Convex programming in Hilbert space. Bull. Am. Math. Soc. 70, 709–710 (1964)

    Article  MathSciNet  Google Scholar 

  11. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

    Article  MathSciNet  Google Scholar 

  12. Malitsky, Y., Tam, M.K.: A forward-backward splitting method for monotone inclusions without cocoercivity. SIAM J. Optim. 30, 1451–1472 (2020)

    Article  MathSciNet  Google Scholar 

  13. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  Google Scholar 

  14. Raguet, H., Fadili, J., Peyré, G.: A generalized forward-backward splitting. SIAM J. Imag. Sci. 6, 1199–1226 (2013)

    Article  MathSciNet  Google Scholar 

  15. Rieger, J., Tam, M.K.: Backward-forward-reflected-backward splitting for three operator monotone inclusion. Appl. Math. Comput. 381, 125248 (2020)

    MathSciNet  Google Scholar 

  16. Ryu, E.K.: Uniqueness of DRS as the 2 oprator resolvent-splitting and impossibility of 3 operator resolvent-splitting. Math. Program. 182, 233–273 (2020)

    Article  MathSciNet  Google Scholar 

  17. Tseng, P.: A modified forward-backward splitting method for maximal monotone mapping. SIAM J. Control. Optim. 38, 431–446 (2000)

    Article  MathSciNet  Google Scholar 

  18. Zong, C.X., Tang, Y.C., Zhang, G.F.: Solving monotone inclusions involving the sum of three maximally monotone operators and cocoercive operator with applications. Set-Val. Var. Anal. (2023). https://doi.org/10.1007/s11228-023-00677-0

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, China

    Yu Cao, Yuanheng Wang, Habib ur Rehman & Yekini Shehu

  2. Center for General Education, China Medical University, Taichung, Taiwan

    Jen-Chih Yao

  3. Academy of Romanian Scientists, Bucharest, Romania

    Jen-Chih Yao

Authors
  1. Yu Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Habib ur Rehman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yekini Shehu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

YW and J-CY conceived and designed the analysis; YC and YS wrote the manuscript, HR prepared all the figures and tables.

Corresponding author

Correspondence to Yekini Shehu.

Ethics declarations

Conflict of interest

The authors declare no Conflict of interest.

Consent for publication

All the authors gave consent for the publication of identifiable details to be published in the journal and article.

Ethical Approval and Consent to participate

All the authors gave the ethical approval and consent to participate in this article.

Additional information

Communicated by Aviv Gibali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, Y., Wang, Y., ur Rehman, H. et al. Convergence Analysis of a New Forward-Reflected-Backward Algorithm for Four Operators Without Cocoercivity. J Optim Theory Appl 203, 256–284 (2024). https://doi.org/10.1007/s10957-024-02501-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-024-02501-7

Keywords

Mathematics Subject Classification

Search

Navigation