Skip to main content
SpringerLink
Log in

Modified subgradient extragradient method for variational inequality problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

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

Similar content being viewed by others

References

  1. Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14, 773–782 (2004)

    Article  MathSciNet  Google Scholar 

  2. 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 

  3. Attouch, H., Goudon, X., Redont, P.: The heavy ball with friction. I. The continuous dynamical system. Commun. Contemp. Math. 2, 1–34 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Attouch, H., Czarnecki, M.O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differ. Equ. 179, 278–310 (2002)

    Article  MathSciNet  Google Scholar 

  5. Bot, R.I., Csetnek, E.R., Laszlo, S.C.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J. Comput. Optim. 4, 3–25 (2016)

    Article  MathSciNet  Google Scholar 

  6. Bot, R.I., Csetnek, E.R.: An inertial Tseng’s type proximal algorithm for nonsmooth and nonconvex optimization problems. J. Optim. Theory Appl. 171, 600–616 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bot, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. 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  Google Scholar 

  15. Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)

    Article  MathSciNet  Google Scholar 

  16. Chen, C., Ma, S., Yang, J.: A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J. Optim. 25, 2120–2142 (2015)

    Article  MathSciNet  Google Scholar 

  17. Dong, L.Q., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65, 2217–2226 (2016)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  19. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. (2016). https://doi.org/10.1007/s10589-016-9857-6

    Article  MathSciNet  Google Scholar 

  20. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)

    Article  MathSciNet  Google Scholar 

  21. Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)

    Article  MathSciNet  Google Scholar 

  22. Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111, 823–840 (2017)

    Article  MathSciNet  Google Scholar 

  23. Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malays. Math. Sci. Soc. (2017). https://doi.org/10.1007/s40840-017-0474-z

  24. Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. (2017). https://doi.org/10.1007/s10898-017-0564-3

    Article  MathSciNet  Google Scholar 

  25. 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  Google Scholar 

  26. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  27. 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 

  28. 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 

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

    Article  MathSciNet  Google Scholar 

  30. Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 34, 876–887 (2008)

    Article  MathSciNet  Google Scholar 

  31. Maingé, P.E.: Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set Valued Anal. 15, 67–79 (2007)

    Article  MathSciNet  Google Scholar 

  32. Maingé, P.E.: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)

    Article  MathSciNet  Google Scholar 

  33. Maingé, P.E., Moudafi, A.: Convergence of new inertial proximal methods for DC programming. SIAM J. Optim. 19, 397–413 (2008)

    Article  MathSciNet  Google Scholar 

  34. Maingé, P.E., Gobinddass, M.L.: Convergence of one step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)

    Article  MathSciNet  Google Scholar 

  35. 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 

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

    MathSciNet  MATH  Google Scholar 

  37. Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  39. 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 

  40. Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67, 83–102 (2018)

    Article  MathSciNet  Google Scholar 

  41. Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. 19, 3029–3051 (2017)

    Article  MathSciNet  Google Scholar 

  42. Thong, D.V.: Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J. Fixed Point Theory Appl. 19, 1481–1499 (2017)

    Article  MathSciNet  Google Scholar 

  43. Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numerical Algorithms. (2017). https://doi.org/10.1007/s11075-017-0412-z

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. Xiu, N.H., Zhang, J.Z.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–587 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Professor Pham Ky Anh for drawing our attention to the subject and for many useful discussions.

Author information

Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Duong Viet Thong

  2. Department of Mathematics, College of Air Force, Nha Trang, Vietnam

    Dang Van Hieu

Authors
  1. Duong Viet Thong
    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

Corresponding author

Correspondence to Duong Viet Thong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thong, D.V., Van Hieu, D. Modified subgradient extragradient method for variational inequality problems. Numer Algor 79, 597–610 (2018). https://doi.org/10.1007/s11075-017-0452-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0452-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation