Skip to main content
SpringerLink
Log in

Relaxed Forward–Backward Splitting Methods for Solving Variational Inclusions and Applications

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we revisit the modified forward–backward splitting method (MFBSM) for solving a variational inclusion problem of the sum of two operators in Hilbert spaces. First, we introduce a relaxed version of the method (MFBSM) where it can be implemented more easily without the prior knowledge of the Lipschitz constant of component operators. The algorithm uses variable step-sizes which are updated at each iteration by a simple computation. Second, we establish the convergence and the linear rate of convergence of the proposed algorithm. Third, we propose and analyze the convergence of another relaxed algorithm which is a combination between the first one with the inertial method. Finally, we give several numerical experiments to illustrate the convergence of some new algorithms and also to compare them with others.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)

    Article  MathSciNet  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  Google Scholar 

  3. Davis, D., Yin, W.T.: A three-operator splitting scheme and its optimization applications. Set Valued Var. Anal. 25, 829–858 (2017)

    Article  MathSciNet  Google Scholar 

  4. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)

    MATH  Google Scholar 

  5. Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966)

    Article  MathSciNet  Google Scholar 

  6. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified forward–backward splitting method for variational inclusions. 4OR 19, 127–151 (2021)

    Article  MathSciNet  Google Scholar 

  7. Hieu, D.V., Anh, P.K., Muu, L.D., Strodiot, J.J.: Iterative regularization methods with new stepsize rules for solving variational inclusions. J. Appl. Math. Comput. (2021). https://doi.org/10.1007/s12190-021-01534-9

    Article  Google Scholar 

  8. Hieu, D.V., Cho, Y.J., Xiao, Y.-B., Kumam, P.: Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces. Vietnam J. Math. (2020). https://doi.org/10.1007/s10013-020-00447-7

    Article  Google Scholar 

  9. Hieu, D.V., Reich, S., Anh, P.K., Ha, N.H.: A new proximal-like algorithm for solving split variational inclusion problems. Numer. Algorithm (2021). https://doi.org/10.1007/s11075-021-01135-4

    Article  Google Scholar 

  10. Hieu, D.V., Vy, L.V., Quy, P.K.: Three-operator splitting algorithm for a class of variational inclusion problems. Bull. Iran. Math Soc. 46, 1055–1071 (2020)

    Article  MathSciNet  Google Scholar 

  11. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  12. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  13. 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 

  14. 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 

  15. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  16. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)

    Article  MathSciNet  Google Scholar 

  17. Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)

    MathSciNet  Google Scholar 

  18. Ryu, E.K., Vu, B.C.: Finding the Forward–Douglas–Rachford–Forward method. J. Optim. Theory Appl. 184, 858–876 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Zong, C., Tang, Y., Cho, Y.J.: Convergence analysis of an inexact three-operator splitting algorithm. Symmetry (2018). https://doi.org/10.3390/sym10110563

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referee for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The research of the second author is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06. The first and third authors were supported by Thailand Science Research and Innovation under the Project IRN62W0007.

Author information

Authors and Affiliations

  1. School of Science, University of Phayao, Phayao, 56000, Thailand

    Prasit Cholamjiak

  2. TIMAS - Thang Long University, Hanoi, Vietnam

    Dang Van Hieu

  3. Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea

    Yeol Je Cho

  4. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Yeol Je Cho

Authors
  1. Prasit Cholamjiak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dang Van Hieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yeol Je Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Van Hieu.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the authors.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cholamjiak, P., Van Hieu, D. & Cho, Y.J. Relaxed Forward–Backward Splitting Methods for Solving Variational Inclusions and Applications. J Sci Comput 88, 85 (2021). https://doi.org/10.1007/s10915-021-01608-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01608-7

Keywords

Mathematics Subject Classification

Search

Navigation