Skip to main content
SpringerLink
Log in

A self adaptive inertial algorithm for solving variational inequalities over the solution set of the split variational inequality problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The purpose of this paper is to investigate a new inertial self-adaptive iterative algorithm for solving variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Our algorithm is accelerated by the inertial technique and eliminates the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator. Two applications in the network bandwidth allocation problem and the image classification problem are shown, and some numerical experiments are presented to show the advantages of the proposed algorithm.

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

Similar content being viewed by others

Data Availibility Statement

The MNIST dataset that supports the findings of this study is available from https://cs.nyu.edu/%7Eroweis/data.html.

References

  1. Anh, P.K., Anh, T.V., Muu, L.D.: On bilevel split pseudomonotone variational inequality problems with applications. Acta. Math. Vietnam. 42(3), 413–429 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cegielski, A., Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities defined over the solution set of a split convex feasibility problem. Numer. Funct. Anal. Optim. 41, 1089–1108 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ceng, L.C., Ansari, Q.H., Yao, J.C.: An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl. 64, 633–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ceng, L.C., Petruşel, A., Qin, X., Yao, J.C.: Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization 70(5–6), 1337–1358 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its application. Inverse Probl. 21(6), 2071–2084 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  10. Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiplesets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer Algorithms 59, 301–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Combettes, P.L., Pesquet, J.C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Problems 24, 065014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cuong, T.L., Anh, T.V., Van, T.H.M.: A self-adaptive step size algorithm for solving variational inequalities with the split feasibility problem with multiple output sets constraints. Num. Funct. Anal. Optim. 43(9), 1009–1026 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York and Basel (1984)

    MATH  Google Scholar 

  15. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Hlavácek, I., Haslinger, J., Necas, J., Lovísek, J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1998)

    MATH  Google Scholar 

  18. Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Iiduka, H.: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733–1742 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iiduka, H.: Fixed point optimization algorithms for distributed optimization in network systems. SIAM J. Optim. 23, 1–26 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaneko, M., Popovski, P., Dahl, J.: Proportional fairness in multi-carrier system: upper bound and approximation algorithms. IEEE Commun. Lett. 10(6), 462–464 (2006)

    Article  Google Scholar 

  22. Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, X., Cui, Y.: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. 73, 76–83 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Low, S., Lapsley, D.E.: Optimization flow control, i: Basic algorithm and convergence. IEEE/ACM Trans. Netw. 7(6), 861–874 (1999)

    Article  Google Scholar 

  25. Maillé, P., Toka, L.: Managing a peer-to-peer data storage system in a selfish society. IEEE J. Sel. Areas Commun. 26(7), 1295–1301 (2008)

    Article  Google Scholar 

  26. Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47(3), 1499–1515 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Petrot, N., Prangprakhon, M., Promsinchai, P., Nimana, N.: A dynamic distributed conjugate gradient method for variational inequality problem over the common fixed-point constraints. Numer. Algor. 93, 639–668 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  28. Reich, S., Truong, M.T., Mai, T.N.H.: The split feasibility problem with multiple output sets in Hilbert spaces. Optim. Lett. 14, 2335–2353 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reich, S., Tuyen, T.M.: Two projection algorithms for solving the split common fixed point problem. J. Optim. Theory Appl. 186, 148–168 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Reich, S., Tuyen, T.M.: Two new self-adaptive algorithms for solving the split common null point problem with multiple output sets in Hilbert spaces. J. Fixed Point Theory Appl. 23, 1–19 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sharma, S., Teneketzis, D.: An externalities-based decentralized optimal power allocation algorithm for wireless networks. IEEE/ACM Trans. Netw. 17, 1819–1831 (2009)

    Article  Google Scholar 

  32. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publisher, Yokohama (2009)

    MATH  Google Scholar 

  33. Thuy, N.T.T.: A strong convergence theorem for an iterative method for solving the split variational inequalities in Hilbert spaces. Vietnam J. Math. 50(1), 69–86 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  34. Thuy, N.T.T., Hieu, P.T.: A hybrid method for solving variational inequalities over the common fixed point sets of infinite families of nonexpansive mappings in Banach spaces. Optimization 69, 2155–2176 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thuy, N.T.T., Hieu, P.T., Strodiot, J.J.: Regularization methods for accretive variational inequalities over the set of common fixed points of nonexpansive semigroups. Optimization 65(8), 1553–1567 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Thuy, N.T.T., Nghia, N.T.: A parallel algorithm for generalized multiple-set split feasibility with application to optimal control problems. Vietnam J Math. 26(5), 1069–1092 (2022)

    MathSciNet  MATH  Google Scholar 

  37. Thuy, N.T.T., Nghia, N.T.: A new iterative method for solving the multipleset split variational inequality problem in Hilbert spaces. Optimization 72, 1549–1575 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  38. Thuy, N.T.T., Nghia, N.T.: A hybrid projection method for solving the multiple-sets splitfeasibility problem. Comput. Appl. Math. 42, 292 (2023). https://doi.org/10.1007/s40314-023-02416-5

    Article  MATH  Google Scholar 

  39. Thuy, N.T.T., Nghia, N.T.: An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints. Optimization (2023). https://doi.org/10.1080/02331934.2023

    Article  Google Scholar 

  40. Wang, F.: The split feasibility problem with multiple output sets for demicontractive mappings. J. Optim. Theory Appl. 195(3), 837–853 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(1), 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Chapter 8, pp. 473–504. Elsevier Science Publishers, Amsterdam (2001)

    Google Scholar 

  43. Yang, P., Yao, Y., Liou, Y.C., Chen, R.: Hybrid algorithms of nonexpansive semigroups for variational inequalities. J. Appl. Math. (2012). https://doi.org/10.1155/2012/634927

    Article  MathSciNet  MATH  Google Scholar 

  44. Yao, Y., Noor, M.A., Liou, Y.C.: A new hybrid iterative algorithm for variational inequalities. Appl. Math. Comput. 216, 822–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zeidler, E.: Nonlinear Functional Analysis ans Its Applications III Variational Methods and Optimization. Springer, Singapore (1985)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees and the editor for their valuable comments and suggestions which improve the presentation of this manuscript.

Funding

No funding was provided to conduct the research.

Author information

Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam

    Nguyen Thi Thu Thuy & Tran Thanh Tung

Authors
  1. Nguyen Thi Thu Thuy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tran Thanh Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to this work.

Corresponding author

Correspondence to Nguyen Thi Thu Thuy.

Ethics declarations

Conflicts of interest

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

Ethical approval

Not applicable.

Additional information

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

Thuy, N.T.T., Tung, T.T. A self adaptive inertial algorithm for solving variational inequalities over the solution set of the split variational inequality problem. Optim Lett (2023). https://doi.org/10.1007/s11590-023-02080-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11590-023-02080-y

Keywords

Search

Navigation