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On Existence and Solution Methods for Strongly Pseudomonotone Equilibrium Problems

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Abstract

We study the equilibrium problems with strongly pseudomonotone bifunctions in real Hilbert spaces. We show the existence of a unique solution. We then propose a strongly convergent generalized projection method for equilibrium problems with strongly pseudomonotone bifunctions. The proposed method uses only one projection without requiring Lipschitz continuity. Application to variational inequalities is discussed.

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References

  1. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2010)

    Google Scholar 

  2. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Bigi, G., Passacantando, M.: Descent and penalization techniques for equilibrium with nonlinear constraints. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0473-7

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

    MATH  MathSciNet  Google Scholar 

  7. Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19, 195–206 (2004)

    Article  Google Scholar 

  8. Bello Cruz, J.Y., Santos, P.S.M., Scheimberg, S.: A two-phase algorithm for a variational inequality formulation of equilibrium problems. J. Optim. Theory Appl. 159, 562–575 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  10. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities, vol. 3, pp 103–113. Academic Press, New York (1972)

    Google Scholar 

  11. Farouq, N.El.: Pseudomonotone variational inequalities: Convergence of proximal methods. J. Optim. Theory Appl. 109, 311–326 (2001)

  12. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74, 6121–6129 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Glob. Optim. 58, 341–350 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Konnov, I.V.: Regularization method for nonmonone equilibrium problems. J. Nonlinear Convex Anal. 10, 93–101 (2009)

    MATH  MathSciNet  Google Scholar 

  17. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976)

    MATH  Google Scholar 

  18. Lorenzo, D., Passacantando, M., Sciandrone, M.: A convergent inexact solution method for equilibrium problems. Optim. Methods Softw. 29, 979–991 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15, 91–100 (1999)

    MATH  MathSciNet  Google Scholar 

  21. Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. 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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pappalardo, M., Mastroeni, G., Pasacantando, M.: Merit functions: a bridge between optimization and equilibria. 4OR 12, 1–33 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  26. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52, 139–159 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. Vuong, P.T., Strodiot, J.-J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalites and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank the referees for their useful remarks and comments that helped us very much in revising the paper.

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Authors and Affiliations

  1. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Le Dung Muu

  2. Academy of Finance, 8 Phan Huy Chu, Hoan Kiem, Hanoi, Vietnam

    Nguyen Van Quy

Authors
  1. Le Dung Muu
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  2. Nguyen Van Quy
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Corresponding author

Correspondence to Le Dung Muu.

Additional information

This paper is dedicated to Professor Nguyen Khoa Son on the occasion of his 65th birthday.

This work is supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED).

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Muu, L.D., Quy, N.V. On Existence and Solution Methods for Strongly Pseudomonotone Equilibrium Problems. Vietnam J. Math. 43, 229–238 (2015). https://doi.org/10.1007/s10013-014-0115-x

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  • DOI: https://doi.org/10.1007/s10013-014-0115-x

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