Skip to main content
SpringerLink
Log in

A subgradient proximal method for solving a class of monotone multivalued variational inequality problems

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

Abstract

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

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

Similar content being viewed by others

References

  1. Anh, P.N., Kuno, T.: A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities. In: Bock, H.G., et al. (eds.) Modeling, Simulation and Optimization of Complex Processes. https://doi.org/10.1007/978-3-642-25707-0-1, pp 1–11. Springer-Verlag Berlin, Heidelberg (2012)

  2. Anh, P.N., Le Thi, H.A.: Outer-interior proximal projection methods for multivalued variational inequalities. Acta Math. Vietnamica 42, 61–79 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anh, P.N., Le Thi, H.A.: Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces. Math. Meth. Appl. Sc. 43(6), 3260–3279 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anh, P.N., Kim, J.K.: An interior proximal cutting hyperplane method for multivalued variational inequalities. J. Nonl. Convex Anal. 11, 491–502 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Anh, P.N., Kim, J.K., Binh, D.T., Phuc, D.H.: A proximal point-type algorithm for solving nonLipschitzian multivalued variational inequalities. Vietnam J. Math. 38, 413–423 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient method for solving bilevel variational inequalities. J. Glob. Optim. 52, 627–639 (2012)

    Article  MATH  Google Scholar 

  7. Anh, P.N., Muu, L.D.: Contraction mapping fixed point algorithms for multivalued mixed variational inequalities on network. In: Dempe, S., Vyacheslav, K. (eds.) Optimization with Multivalued Mappings. Springer (2006)

  8. Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math. Vietnamica 34, 67–79 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Opt. Theory Appl. 124, 285–306 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math. Vietnamica 34, 67–79 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Baiocchi, C., Capelo, A.: Variational and quasivariational inequalities, Applications to Free Boundary Problems. NewYork. Cambridge University Press (1984)

  12. Beck, A.: First-order methods in optimization, MOS-SIAM series on optimization. Chapter 6 (2017)

  13. Bnouhachem, A.: An LQP method for psedomonotone variational inequalities. J. Glob. Optim. 36, 351–363 (2006)

    Article  MATH  Google Scholar 

  14. Ceng, L.C., Cubiotti, P., Yao, J.C.: An implicit iterative scheme for monotone variational inequalities and fixed point problems. Nonlinear Anal. 69, 2445–2457 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ceng, L.C., Latif, A., Ansari, Q.H.: Hybrid extragradient method for hierarchical variationa inequalities. Fixed Point Theory Appl. 2014:222, 1–35 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Ceng, L.C., Liou, Y.C., Wen, C.F.: Hybrid extragradient viscosity method for general system of variational inequalities. J. Ineq. Appl. 2015:150, 1–43 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Cho, Y.J., Argyros, I.K., Petrot, N.: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 60, 2292–2301 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cohen, G.: Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium problems and variational models. Kluwer Academic Publishers. Springer US, Dordrecht (2003)

    Book  MATH  Google Scholar 

  20. Dong, Q.L., Lu, Y.Y., Yang, J., He, S.: Approximately solving multivalued variational inequalities by using a projection and contraction algorithm. Num. Alg. 76, 799–812 (2017)

    Article  MATH  Google Scholar 

  21. Dong, Q.L., Gibali, A., YJang, Y.: A modified subgradient extragradient method for solving the variational inequality problem. Num. Alg. 3, 927–940 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Meth. Softw. 4, 75–83 (1994)

    Article  Google Scholar 

  23. Fang, C.J., Chen, S.L.: A subgradient extragradient algorithmfor solving multivalued variationalinequality. Appl. Math. Comput. 229, 123–130 (2014)

    MathSciNet  Google Scholar 

  24. Fortin, M., Slowinski, R.: Augmented Lagrangian methods: applications to the numerical solution of boundary value problems. North-Holland, Amsterdam. NEW YORK, OXFORD (1983)

  25. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium problems: nonsmooth optimization and variational inequality models. Kluwer (2004)

  26. Glowinski, R.: Numerical methods for nonlinear variational problems. Springer, Berlin (1980)

    MATH  Google Scholar 

  27. Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. He, B.S., Yang, H., Zhang, C. -S.: A modified augmented Lagrangian method for a class of monotone variational inequalities. Eur. J. Oper. Res. 159, 35–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. He, B.S., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23, 151–161 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. He, S.N., Dong, Q.L., Tian, H.L.: Relaxed projection and contraction methods for solving Lipschitz continuous monotone variational inequalities. Rev. R. Acad. Cienc. Exactas Fisicas Naturales Ser. Mat. 113(3), 2773–2791 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. 73, 913–932 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Huang, N.J.: A new method for a class of nonlinear set-valued variational inequalities. Z. Angew. Math. Mech. 78, 427–430 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer-Verlag, Berlin (2000)

    MATH  Google Scholar 

  35. Korpelevich, G.M.: Extragradient method for finding saddle points and other problems. Ekon. Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  36. Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163(2), 399–412 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Maingé, P.E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Malitsky, Y.: Proximal extrapolated gradient methods for variational inequalities. Optim. Methods Softw. 33(1), 140–164 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nagurney, A.: Network economics: a variational inequality approach. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  41. Shehu, Y., Dong, Q -L, Jiang, D.: Single projection method for pseudomonotone variational inequality in Hilbert spaces. Optimization 68, 385–409 (2018)

    Article  MATH  Google Scholar 

  42. Sodolov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Contr. Optim. 37(3), 765–776 (1999)

    Article  MathSciNet  Google Scholar 

  43. Ronald, E., Bruck, J.R.: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert Space. J. Math. Anal. Appl. 61, 159–164 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  44. Taji, K., Fukushima, M.: A new merit function and a successive quadratic programming algorithm for variational inequality problems. SIAM J. Optim. 6, 704–713 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tang, G.J., Huang, N.J.: Strong convergence of an inexact projected subgradient method for mixed variational inequalities. Optimization 63, 601–615 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tseng, P.: On linear convergence of iterative methods for the variational inequality problem. J. Comput. Appl. Math. 60, 237–252 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xia, F.Q., Huang, N.J.: A projection proximal point algorithm for solving generalized variational inequalities. J. Optim. Theory Appl. 150, 98–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, M.H.: Viscosity method for hierarchical fixed point approach to variational inequalities. Taiwan. J. Math. 14(2), 463–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Yura, M.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very grateful to two anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.

Author information

Authors and Affiliations

  1. Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    P.N. Anh

  2. Electric Power University, Hanoi, Vietnam

    T.V. Thang

  3. University of Transport Technology, Hanoi, Vietnam

    H.T.C. Thach

Authors
  1. P.N. Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T.V. Thang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H.T.C. Thach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P.N. Anh.

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

Anh, P., Thang, T. & Thach, H. A subgradient proximal method for solving a class of monotone multivalued variational inequality problems. Numer Algor 89, 409–430 (2022). https://doi.org/10.1007/s11075-021-01119-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01119-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation