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Journal of Global Optimization

, Volume 70, Issue 2, pp 477–495 | Cite as

The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces

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Abstract

In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach.

Keywords

Maximal monotone operator Glowinski–Le Tallec splitting method Equilibrium problem Nash equilibrium Global convergence 

Notes

Acknowledgements

The authors would like to thank the Associate Editor and the two anonymous referees for their useful remks, comments and suggestions that allowed to improve substantially the original version of this paper. This work was mostly carried out when the first author was a PhD student working at the Institute for Computational Science and Technology—Ho Chi Minh City, Vietnam. This research was supported by this Institute and partly, for the first author, by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant 101.01-2017.315 and the Austrian Science Foundation (FWF), Grant P26640-N25.

References

  1. 1.
    Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpansiveness properties of the marginal mappings in generalized variational inequalities involving co-coercive operartors. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol. 77, pp. 89–111. Springer, Boston (2005)Google Scholar
  2. 2.
    Balashov, M.V., Golubev, M.O.: About the Lipschitz property of the metric projection in the Hilbert space. J. Math. Anal. Appl. 394, 545–551 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2010)MATHGoogle Scholar
  4. 4.
    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227, 1–11 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetMATHGoogle Scholar
  6. 6.
    Briceno-Arias, L.: A Douglas–Rachford splitting method for solving equilibrium problems. Nonlinear Anal. 75, 6053–6059 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, G., Rockafellar, R.T.: Convergence and Structure of Forward–backward Splitting Methods. Technical Report, Department of Applied Mathematics, University of Washington (1990)Google Scholar
  8. 8.
    Chen, G., Rockafellar, R.T.: Extended Forward–backward Splitting Methods and Convergence. Technical Report, Department of Applied Mathematics, University of Washington (1990)Google Scholar
  9. 9.
    Chen, G., Rockafellar, R.T.: Forward–Backward Splitting Methods in Lagrangian Optimization. Technical Report, Department of Applied Mathematics, University of Washington (1992)Google Scholar
  10. 10.
    Douglas, J., Rachford, H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eckstein, J., Svaiter, B.F.: A family of projective splitting methods for the sum of two maximal monotone operators. Math. Program. 111, 173–199 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)Google Scholar
  14. 14.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)CrossRefMATHGoogle Scholar
  15. 15.
    Haubruge, S., Strodiot, J.J., Nguyen, V.H.: Convergence analysis and applications of the Glowinski–Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 97, 645–673 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    He, B., Yuan, X.: On the convergence rate of Douglas–Rachford operator splitting method. Math. Program. 153, 715–722 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59, 1259–1274 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Glob. Optim. 58, 341–350 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Krawczyk, J.B., Uryasev, S.: Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess. 5, 63–73 (2000)CrossRefGoogle Scholar
  20. 20.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Luo, Z.Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2, 43–54 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lyashko, S.L., Semenov, V.V., Voitova, T.A.: Low cost modification of Korpelevich’s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631–639 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Marcotte, P., Wu, J.H.: On the convergence of projection methods: application to the decomposition of affine variational inequalities. J. Optim. Theory Appl. 85, 347–362 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Gianessi, F., Maugeri, A. (eds.) Equilibrium Problems and Variational Models, pp. 289–298. Kluwer, Dordrecht (2003)CrossRefGoogle Scholar
  25. 25.
    Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359, 508–513 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Muu, L.D.: Stability property of a class of variational inequalities. Optimization 15, 347–353 (1984)MathSciNetMATHGoogle Scholar
  28. 28.
    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constraint equilibria. Nonlinear Anal. 18, 1159–1166 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving equilibrium problems. Math. Program. 116, 529–552 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44, 175–192 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert Space. J. Math. Anal. Appl. 72, 383–390 (1979)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Peaceman, D.H., Rachford, H.H.: The numerical solution of parabolic elliptic differential equations. SIAM J. Appl. Math. 3, 28–41 (1955)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Phan, H.M.: Linear convergence of the Douglas–Rachford method for two closed sets. Optimization 65, 369–385 (2016)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Scheimberg, S., Santos, P.S.M.: A relaxed projection method for finite dimensional equilibrium problems. Optimization 60, 1193–1208 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Strodiot, J.J., Vuong, P.T., Nguyen, T.T.V.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Glob. Optim. 64, 159–178 (2016)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Tran, D.Q., Le Dung, M., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zeng, L.C., Yao, J.Y.: Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces. J. Optim. Theory Appl. 131, 469–483 (2006)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zhu, C.: Asymptotic convergence nnalysis of the forward-backward splitting algorithm. Math. Oper. Res. 20, 449–464 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Statistics and Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  2. 2.HCMC University of Technology and EducationHo Chi Minh CityVietnam
  3. 3.Institute for Computational Science and Technology - HCMC (ICST)Ho Chi Minh CityVietnam
  4. 4.University of NamurNamurBelgium

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