Skip to main content
SpringerLink
Log in

Strong Convergence of the Modified Inertial Extragradient Method with Line-Search Process for Solving Variational Inequality Problems in Hilbert Spaces

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

Abstract

The aim of this paper is to give a strong convergence theorem of a new iterative algorithm for solving variational inequalities with pseudomonotone and non-Lipschitzian operators in real Hilbert spaces. The proposed algorithm combines the inertial method and the extragradient method which simplifies and accelerates the process of convergence by establishing a new step size rule. Finally, we give some numerical experiments to show the performance of the proposed algorithm through comparison with related algorithms.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)

    MathSciNet  MATH  Google Scholar 

  2. Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7, 91–140 (1964)

    MathSciNet  MATH  Google Scholar 

  3. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. 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  Google Scholar 

  7. 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  Google Scholar 

  8. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2011)

    Article  MathSciNet  Google Scholar 

  9. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 56, 301–323 (2012)

    Article  MathSciNet  Google Scholar 

  10. Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in refelexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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  Google Scholar 

  13. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)

    Article  MathSciNet  Google Scholar 

  14. Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algorithms 79, 597–610 (2018)

    Article  MathSciNet  Google Scholar 

  15. 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  Google Scholar 

  16. Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algorithm 81, 269–291 (2019)

    Article  MathSciNet  Google Scholar 

  17. Iusem, A.N.: An iterative algorithm for the variational inequality problem. Comput. Appl. Math. 13, 103–114 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Khobotov, E.N.: Modifications of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1987)

    Article  MathSciNet  Google Scholar 

  19. Marcotte, P.: Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems. Inf. Syst. Oper. Res. 29, 258–270 (1991)

    MATH  Google Scholar 

  20. Thong, D.V., Gibali, A.: Extragradient methods for solving non-Lipschitzian pseudo-monotone variational inequalities. J. Fixed Point Theory Appl. 21(20), 19 (2019)

    MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Moudafi, A., Elisabeth, E.: An approximate inertial proximal method using enlargement of a maximal monotone operator. Int. J. Pure Appl. Math. 5, 283–299 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)

    Article  MathSciNet  Google Scholar 

  24. Bot, R.I., Csetnek, E.R.: A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36, 951–963 (2015)

    Article  MathSciNet  Google Scholar 

  25. Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  26. Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)

    Article  MathSciNet  Google Scholar 

  27. Iusem, A.N., Gárciga Otero, R.: Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer. Funct. Anal. Optim. 22, 609–640 (2001)

    Article  MathSciNet  Google Scholar 

  28. Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)

    Article  MathSciNet  Google Scholar 

  29. 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  Google Scholar 

  30. Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)

    Article  MathSciNet  Google Scholar 

  31. Thong, D.V., Hieu, D.V., Rassias, T.M.: Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim. Lett. 14, 115–144 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the NSF of China (Grant No. 11771063, 11971082), the Natural Science Foundation of Chongqing (Grant No.cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504), the Program of Chongqing Innovation Research Group Project in University (Grant No. CXQT19018) and Fundamental Research Funds for the Central Universities (No.3122019142).

Author information

Authors and Affiliations

  1. School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Zhongbing Xie & Gang Cai

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

    Xiaoxiao Li & Qiao-Li Dong

Authors
  1. Zhongbing Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoxiao Li
    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 Gang Cai.

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

Xie, Z., Cai, G., Li, X. et al. Strong Convergence of the Modified Inertial Extragradient Method with Line-Search Process for Solving Variational Inequality Problems in Hilbert Spaces. J Sci Comput 88, 50 (2021). https://doi.org/10.1007/s10915-021-01585-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01585-x

Keywords

Mathematics Subject Classification

Search

Navigation