Skip to main content

Advertisement

SpringerLink
Log in

A new Bregman projection method with a self-adaptive process for solving variational inequality problem in reflexive Banach spaces

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

Abstract

In this paper, we mainly propose a new Bregman projection method with a different self-adaptive process for solving variational inequalities in a real reflexive Banach space. Exactly, we obtain that the iterative sequence generated by our new algorithm converges strongly to an element of solution set for the variational inequality problem. Our algorithm is interesting and easy to implement in numerical experiments because it has only one projection and does not need to know the Lipschitz constant of the considered operator in advance. The results obtained in this paper can be considered as an improvement and supplement of many recent ones in the field.

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

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.

References

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Iusem, A.N., Nasri, M.: Korpelevich method for variational inequality problems in Banach spaces. J. Glob. Optim. 50, 59–76 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827–845 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonom. i Mat. Metody. 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  10. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms 78, 1045–1060 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. 80, 1438–1462 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Thong, D.V., Shehu, Y., Iyiola, O.S.: Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings. Numer. Algorithms 84, 795–823 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thong, D.V., Triet, N.A., Li, X.H., Dong, Q.L.: Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer. Algorithms 83, 1123–1143 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Thong, D.V., Vinh, N.T., Cho, Y.J.: A strong convergence theorem for Tseng’s extragradient method for solving variational inequality problems. Optim. Lett. 14, 1157–1175 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, Th.. M.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, Th.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12, 87–102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A new double-projection method for solving variational inequalities in Banach spaces. J. Optim. Theory Appl. 178, 219–239 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shehu, Y., Iyiola, O.S., Enyi, C.D.: An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces. Numer. Algorithms 72, 835–864 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shehu, Y., Iyiola, O.S.: Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J. Fixed Point Theory Appl. 19, 2483–2510 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shehu, Y., Iyiola, O.S.: Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer. Algorithms 79, 529–553 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shehu, Y., Dong, Q.-L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68, 385–409 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shehu, Y., Gibali, A., Sagratella, S.: Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces. J. Optim. Theory Appl. 184, 877–894 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shehu, Y., Liu, L.L., Mu, X.W., Dong, Q.-L.: Analysis of versions of relaxed inertial projection and contraction method. Appl. Numer. Math. 165, 1–21 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shehu, Y., Iyiola, O.S., Thong, D.V., Van, N.T.C.: An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems. Math. Methods Oper. Res. 93, 213–242 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shehu, Y., Gibali, A.: New inertial relaxed method for solving split feasibilities. Optim. Lett. 15, 2109–2126 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shehu, Y.: Single projection algorithm for variational inequalities in Banach spaces with application to contact problem. Acta Math. Sci. Ser. B 40, 1045–1063 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gibali, A., Liu, L.W., Tang, Y.C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. 12, 817–830 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gibali, A., Kfer, K.H., Reem, D., Sss, P.: A generalized projection-based scheme for solving convex constrained optimization problems. Comput. Optim. Appl. 70, 737–762 (2018)

    Article  MathSciNet  Google Scholar 

  31. Gibali, A., Thong, D.V., Tuan, P.A.: Two simple projection-type methods for solving variational inequalities. Anal. Math. Phys. 9, 2203–2225 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gibali, A., Thong, D.V.: A new low-cost double projection method for solving variational inequalities. Optim. Eng. 21, 1613–1634 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang, Y., Gibali, A.: New self-adaptive step size algorithms for solving split variational inclusion problems and its applications. Numer. Algorithms 83, 305–331 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jolaoso, L.O., Shehu, Y.: Single Bregman projection method for solving variational inequalities in reflexive Banach spaces. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2020.1869947

    Article  MATH  Google Scholar 

  35. Bauschke, H. H., Borwein, J. M.: Combettes P. L. Essential smoothness, essential strict convexity and Legendre functions in Banach space. Commun. Contemp. Math. 3, 615–647 (2001). 26, 16 pp (2020)

  36. Cai, D., Dong, Q.-L., Peng, Y.: Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators. J. Optim. Theory Appl. 188, 447–472 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  37. Thong, D.V., Vuong, P.T.: Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities. Optimization 68(11), 2207–2226 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Applied Optimization, vol. 40. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  39. Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  40. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Harker, P.T., Pang, J.-S.: A damped-Newton method for the linear complementarity problem. In: Allgower, G., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, vol. 26, pp. 265–284. AMS, Providence, RI (1990)

    Google Scholar 

  43. Jolaoso, L.O., Aphane, M.: Weak and strong convergence Bregman extragradient schemes for solving pseudo-monotone and non-Lipschitz variational inequalities. J Inequal Appl 2020, 195 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the NSF of China (Grant No. 12171435).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Shaotao Hu & Yuanheng Wang

  2. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Ping Jing

  3. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Qiao-Li Dong

Authors
  1. Shaotao Hu
    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. Ping Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qiao-Li Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanheng Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Hu, S., Wang, Y., Jing, P. et al. A new Bregman projection method with a self-adaptive process for solving variational inequality problem in reflexive Banach spaces. Optim Lett 17, 935–954 (2023). https://doi.org/10.1007/s11590-022-01909-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-022-01909-2

Keywords

Search

Navigation