Skip to main content

Advertisement

Springer Nature Link
Log in

A new low-cost feasible projection algorithm for pseudomonotone variational inequalities

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

Abstract

In this paper, we design a low-cost feasible projection algorithm for variational inequalities by replacing the projection onto the feasible set with the projection onto a ball. In each iteration, it only needs to calculate the value of the mapping once, and the projection onto the ball contained in the feasible set (which has an explicit expression), so the algorithm is easier to implement and feasible. The convergence of the algorithm is proved when the Slater condition holds for the feasible set and the mapping is pseudomonotone, Lipschitz continuous. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

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

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

The authors confirm that all data generated or analyzed in the development of this work are adequately included or referenced in the article itself.

References

  1. Auslender, A., Shefi, R., Teboulle, M.: A moving balls approximation method for a class of smooth constrained minimization problems. SIAM Journal on Optimization. 20(6), 3232–3259 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Archive for Rational Mechanics and Analysis 209(3), 797–834 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauschke, H. H., Combettes, P. L.: Convex analysis and monotone operator theory in Hilbert spaces (Vol. 408). New York: Springer, (2011)

  4. Bot, R.I., Csetnek, E.R., Vuong, P.T.: The Forward-Backward-Forward method from discrete and continuous perspective for pseudo-monotone variational inequalities in hilbert spaces. European Journal of Operational Research. 287(1), 49–56 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, New York (2012)

    MATH  Google Scholar 

  6. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optimization Methods and Software. 26(4–5), 827–845 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. Journal of Optimization Theory and Applications. 148(2), 318–335 (2011)

    Article  MathSciNet  MATH  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(9), 1119–1132 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, J.X., Ye, M.L.: A new modified two-subgradient extragradient algorithm for solving variational inequality problems. Journal of Mathematical Research with Applications. 42(4), 402–412 (2022)

    MathSciNet  MATH  Google Scholar 

  10. Dafermos, S., Nagurney, A.: A network formulation of market equilibrium problems and variational inequalities. Operations Research Letters. 3, 247–250 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dautray, R., Lions, J. L.:Mathematical analysis and numerical methods for science and technology: volume 3 spectral theory and applications. Springer Science & Business Media. (2012)

  12. Feng, L. M., Zhang, Y. L., He, Y. R.: A new feasible moving ball projection algorithm for pseudomonotone variational inequality. Accepted by Optimization Letters

  13. Freund, R..M.., Grigas, P., Mazumder, R.: An extended Frank-Wolfe method with “in-face’’ directions, and its application to low-rank matrix completion. SIAM Journal on Optimization. 27, 319–346 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gibali, A., Thong, D.V.: A new low-cost double projection method for solving variational inequalities. Optimization and Engineering 21(4), 1613–1634 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and non-expansive mappings. Marcel Dekker, New York and Basel (1984)

    MATH  Google Scholar 

  16. Goldstein, A.A.: Convex programming in Hilbert space. Bulletin of the American Mathematical Society 70(5), 709–710 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  17. Harchaoui, Z., Juditsky, A., Nemirovski, A.: Conditional gradient algorithms for norm-regularized smooth convex optimization. Mathematical Programming 152, 75–112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, S.N., Xu, H.K.: Uniqueness of supporting hyperplanes and an alternative to solutions of variational inequalities. Journal of Global Optimization. 57(4), 1375–1384 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Heinemann, C., Sturm, K.: Shape optimization for a class of semilinear variational inequalities with applications to damage models. SIAM Journal on Mathematical Analysis 48(5), 3579–3617 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hung, N. V., Tam, V. M.: Error bound analysis of the D-gap functions for a class of elliptic variational inequalities with applications to frictional contact mechanics. Zeitschrift für angewandte Mathematik und Physik, 72, Article number:173, (2021)

  21. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevichs method for variational inequalities with a new search strategy. Optimization 42(4), 309–321 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. Journal of Global Optimization 58(2), 341–350 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kolobov, V. I., Reich, S., Zalas, R.: Finitely convergent iterative methods with overrelaxations revisited. Journal of Fixed Point Theory and Applications, 23, Article number: 57, (2021)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Malitsky, Y.V.: Projected refected gradient methods for monotone variational inequalities. SIAM Journal on Optimization 25(1), 502–520 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Malitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybernetics and Systems Analysis 50(2), 271–277 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. Society for Industrial and Applied Mathematics (2000)

  29. Popov, L.D.: A modification of the Arrow-Hurwicz method for search of saddle points. Mathematical notes of the Academy of Sciences of the USSR 28(5), 845–848 (1980)

    MATH  Google Scholar 

  30. Rockafellar, R.T., Sun, J.: Solving Lagrangian variational inequalities with applications to stochastic programming. Mathematical Programming 181(2), 435–451 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. Journal of optimization theory and applications 176(2), 399–409 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numerical Algorithms 81(1), 269–291 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yao, Y., Iyiola, O.S., Shehu, Y.: Subgradient extragradient method with double inertial steps for variational inequalities. Journal of Scientific Computing 90, 1–29 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author appreciates the valuable comments of anonymous referees which helped to improve the quality of this paper.

Funding

The first author was supported partly by the National Natural Science Foundation of China (11901414). The third author was supported partly by the National Natural Science Foundation of China (11871359).

Author information

Authors and Affiliations

  1. Department of Mathematics, Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, 610066, Sichuan, People’s Republic of China

    Yongle Zhang, Limei Feng & Yiran He

Authors
  1. Yongle Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Limei Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yiran He
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Yongle Zhang and Limei Feng wrote the main manuscript text, Yiran He provided the source of the problem, and Limei Feng prepared all figures. All authors reviewed the manuscript.

Corresponding author

Correspondence to Yongle Zhang.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

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

Zhang, Y., Feng, L. & He, Y. A new low-cost feasible projection algorithm for pseudomonotone variational inequalities. Numer Algor 94, 1031–1054 (2023). https://doi.org/10.1007/s11075-023-01622-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01622-w

Keywords

Search

Navigation