Skip to main content
SpringerLink
Log in

Implementation of Augmented Lagrangian Methods for Equilibrium Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Actual implementation of augmented Lagrangian algorithms requires a solution of the subproblem generated at each iterate, which is the most challenging task. In this paper, we propose two approaches to make the augmented Lagrangian algorithms, introduced in Iusem and Nasri (RAIRO Oper Res 44:5–26, 2010) for equilibrium problems, computer amenable. The first algorithm that we suggest here incorporates the Newton method and the other one benefits from the Shor subgradient method to solve the subproblems that are produced when the augmented Lagrangian algorithms are applied to the equilibrium problem. We also illustrate our findings by numerical results which are obtained when our algorithms are implemented to solve quadratic equilibrium problems and certain generalized Nash equilibrium problem, including the river basin pollution problem, a particular case of the equilibrium problem. Moreover, we compare our numerical results with those presented in Matioli et al. (Comput Optim Appl 52:281–292, 2012) and Tran et al. (Optimization 57:749–776, 2008) for the same test problems.

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. Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Annal. 142, 305–310 (1961)

    Article  MATH  Google Scholar 

  2. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  3. Brezis, H., Nirenberg, L., Stampacchia, S.: A remark on ky fan minimax principle. Boll. Unione Matematica Ital. 6, 293–300 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Bianchi, M., Pini, R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124, 79–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Iusem, A.N., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116, 259–273 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227, 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. Theory Methods Appl 52, 621–635 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Nasri, M., Sosa, W.: Generalized Nash games and equilibrium problems. Optimization 60, 1161–1170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  11. Flam, S., Antipin, A.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Burachik, R.S., Kassay, G.: On a generalized proximal point method for solving equilibrium problems in Banach spaces. Nonlinear Anal. Theory, Methods Appl. 75, 6456–6464 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Iusem, A.N., Nasri, M.: Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 28, 1279–1308 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Iusem, A.N., Sosa, W.: A proximal point method for equilibrium problem in Hilbert spaces. Optimization 59, 1259–1274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Iusem, A.N., Nasri, M.: Augmented lagrangian methods for equilibrium problems. RAIRO Oper. Res. 44, 5–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Muu, L., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. Theory, Methods Appl. 18, 1159–1166 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Tran, D., Dung, M., Nguyen, V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Van, N., Strodiot, J., Nguyen, V.: A bundle method for solving equilibrium problems. Math. Program. 116, 529–552 (2009)

    Article  MathSciNet  Google Scholar 

  21. Muu, L., Quoc, T.: Regularization algorithms for solving monotone ky fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Quoc, T., Muu, L.: Implementable quadratic regularization methods for solving pseudomonotone equilibrium problems. East-West J. Math. 6, 101–123 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Van, N., Strodiot, J., Nguyen, V.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44, 175–192 (2009)

    Article  MATH  Google Scholar 

  24. Minoux, M.: Mathematical Programming Theory and Algorithms. Wiley, New York (1986)

    MATH  Google Scholar 

  25. Polyak, B.: Introduction to Optimization. Optimization Software Inc., Publications Division, New York (1987)

    Google Scholar 

  26. Shor, N.: Convergence rate of the gradient descent method with dilatation of the space. Kibernetika 6, 102–108 (1970)

    Google Scholar 

  27. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matioli, L.C., Sosa, W., Yuan, J.Y.: A numerical algorithm for finding solutions of a generalized nash equilibrium problem. Comput. Optim. Appl. 52, 281–292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2007)

    Google Scholar 

  30. Facchinei, F., Fischer, A., Piccialli, V.: On generalized nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mashreghi, J., Nasri, M.: Hybrid Lagrange multiplier approaches for solving infinite dimensional equilibrium problems with cone constraints. J. Nonlinear Convex Anal. 13, 331–349 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Mashreghi, J., Nasri, M.: A proximal augmented Lagrangian method for equilibrium problems. Appl. Anal. 91, 157–172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The work of the second author was partially supported by CAPES that is greatly appreciated.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, McGill University, Montreal, QC, Canada

    Mostafa Nasri

  2. Centro Politécnico, Universidade Federal do Paraná, UFPR PPGM/PPGMNE, Curitiba, Pr, 81530-900, Brasil

    Luiz Carlos Matioli

  3. Faculdade de Administração, Ciências, Educação e Letras, FACEL and PPGMNE, Curitiba, Pr, Brasil

    Euda Mara Da Silva Ferreira

  4. Universidade Tecnológica Federal do Paraná - UTFPR PPGMNE, Curitiba, Pr, Brasil

    Adilson da Silveira

Authors
  1. Mostafa Nasri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luiz Carlos Matioli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Euda Mara Da Silva Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Adilson da Silveira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mostafa Nasri.

Additional information

Communicated by Michael Patriksson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nasri, M., Matioli, L.C., Da Silva Ferreira, E.M. et al. Implementation of Augmented Lagrangian Methods for Equilibrium Problems. J Optim Theory Appl 168, 971–991 (2016). https://doi.org/10.1007/s10957-015-0846-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0846-1

Keywords

Mathematics Subject Classification

Search

Navigation