Skip to main content
Springer Nature Link
Log in

Inertial iterative algorithms for common solution of variational inequality and system of variational inequalities problems

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

References

  1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Topological Fixed Point Theory and Its Applications. Springer, New York, NY (2009)

    MATH  Google Scholar 

  2. Alfuraidan, M.R., Ansari, Q.H.: Fixed Point Theory and Graph Theory, Foundations and Integrative Approaches (2016)

  3. 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 

  4. 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 

  5. Bauschke, H.H., Combttes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert space. Springer, Berlin (2011)

    Book  Google Scholar 

  6. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  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. Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 3, 459–470 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Ceng, L.C., Sahu, D.R., Yao, J.C.: A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space. J. Inequal. Appl. 460 (2014)

  11. Censor, Y., Gabali, A., Reich, S.: Stong convergence of subgradient extragradient methods for variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011)

    Article  MathSciNet  Google Scholar 

  12. Chen, C., Ma, S., Yang, J.: A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J. Optim. 25(4), 2120–2142 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Dong, Q.L., 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 

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

    Article  MathSciNet  Google Scholar 

  16. He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004)

    Article  MathSciNet  Google Scholar 

  17. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)

    Article  MathSciNet  Google Scholar 

  20. Khobotov, E.N.: A modification of the extragradient method for solving variational inequalities and certain optimization problems, (Russian)Zh. Vychisl. Mat. i Mat. Fiz., 27, 1462–1473, 1597, (1989)

  21. Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851–862 (2003)

    Article  MathSciNet  Google Scholar 

  22. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  23. Lorenz, D.A., Pock, T.: An inertial forword-backword algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Marcotte, P.: Applications of Khobotov’s algorithm to variational and network equlibrium problems. Inf. Syst. Oper. Res. 29, 258–270 (1991)

    MATH  Google Scholar 

  27. Nesterov, Y.E.: A method for unconstrained convex minimization problem with the rate of convergence O(1/k2). Sov. Math. Dokl. 27(2), 372–376 (1983)

    Google Scholar 

  28. Ochs, P., Chen, Y., Brox, T., Pock, T.: Inertial proximal algorithm for non-convex optimization. SIAM J. Imaging Sci. 7(2), 1388–1419 (2014)

    Article  MathSciNet  Google Scholar 

  29. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4, 1–17 (1964)

    Article  Google Scholar 

  30. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 6, 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  31. Sadiq Basha, S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119–129 (2000)

    Article  MathSciNet  Google Scholar 

  32. Sahu, D.R., Kang, S.M., Kumar, A.: Convergence analysis of parallel S-iteration process for system of generalized variational inequalities. J. Funct. Spaces, Article ID 5847096, 10 pages (2017)

  33. Sahu, D.R.: Altering points and applications. Nonlinear Stud. 21(2), 349–365 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Sahu, D.R., Kumar, A., Wen, C.F.: S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalities. Filomat 33(6), 1727–1746 (2019)

    Article  MathSciNet  Google Scholar 

  35. 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 

  36. Thong, D.V., Vuong, P.T.: Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities. Optimization (2019). https://doi.org/10.1080/0233193402.19.1616191

    Article  MathSciNet  MATH  Google Scholar 

  37. Tian, M., Tong, M.: Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems. J. Ineq. Appl. 1(7), 1–19 (2019)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Verma, R.U.: Projection methods, algorithms and a new system of nonlinear variational inequalities. Comput. Math. Appl. 41, 1025–1031 (2001)

    Article  MathSciNet  Google Scholar 

  40. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)

    MATH  Google Scholar 

  41. Zhang, J., Qu, B., Xiu, N.: Some projection-like methods for the generalized Nash equilibria. Comput. Optim. Appl. 45, 89–109 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    D. R. Sahu & Amit Kumar Singh

Authors
  1. D. R. Sahu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amit Kumar Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amit Kumar Singh.

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

Sahu, D.R., Singh, A.K. Inertial iterative algorithms for common solution of variational inequality and system of variational inequalities problems. J. Appl. Math. Comput. 65, 351–378 (2021). https://doi.org/10.1007/s12190-020-01395-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-020-01395-8

Keywords

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy