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Journal of Global Optimization

, Volume 52, Issue 3, pp 627–639 | Cite as

An extragradient algorithm for solving bilevel pseudomonotone variational inequalities

  • P. N. Anh
  • J. K. Kim
  • L. D. Muu
Article

Abstract

We present an extragradient-type algorithm for solving bilevel pseudomonone variational inequalities. The proposed algorithm uses simple projection sequences. Under mild conditions, the convergence of the iteration sequences generated by the algorithm is obtained.

Keywords

Bilevel variational inequality Pseudomonotonicity Lipschitz continuity Global convergence Extragradient algorithm 

Mathematics Subject Classification (2000)

65 K10 90 C25 

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References

  1. 1.
    Anh P.N.: An interior-quadratic proximal method for solving monotone generalized variational inequalities. East–West J. Math. 10, 81–100 (2008)Google Scholar
  2. 2.
    Anh P.N.: An interior proximal method for solving pseudomonotone nonlipschitzian multivalued variational inequalities. Nonlinear Anal. Forum 14, 27–42 (2009)Google Scholar
  3. 3.
    Anh P.N., Hien N.D.: Coupling the LQ regularization method and the outer approximation method for solving pseudomonotone variational inequalities. East–West J. Math. 12, 49–58 (2010)Google Scholar
  4. 4.
    Anh P.N., Muu L.D., Strodiot J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. ACTA Mathematica Vietnamica 34, 67–79 (2009)Google Scholar
  5. 5.
    Anh P.N., Muu L.D., Hien N.V., Strodiot J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Optim. Theory Appl. 124, 285–306 (2005)CrossRefGoogle Scholar
  6. 6.
    Baiocchi C., Capelo A.: Variational and Quasivariational Inequalities, Applications to Free Boundary Problems. Wiley, NewYork (1984)Google Scholar
  7. 7.
    Bakushinskii A.B., Polyak P.T.: On the solution of variational inequalities. Sovjet Mathematics Dokl. 219, 1038–1041 (1974) (in Russian)Google Scholar
  8. 8.
    Bakushinskii A.B., Goncharskii A.V.: Ill-Posed Problems: Theory and Applications. Kluwer, Dordrecht (1994)CrossRefGoogle Scholar
  9. 9.
    Daniele P., Giannessi F., Maugeri A.: Equilibrium Problems and Variational Models. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  10. 10.
    Facchinei F., Pang J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, NewYork, NY (2003)Google Scholar
  11. 11.
    Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic Publishers, Dordrecht (2004)CrossRefGoogle Scholar
  12. 12.
    Hung P.G., Muu L.D.: The Tikhonov regularization method extended to pseudomonotone equilibrium problems. Nonlinear Anal. Theory Methods Appl. 74, 6121–6129 (2011)CrossRefGoogle Scholar
  13. 13.
    Kalashnikov V.V., Kalashnikova N.I.: Solving two-level variational inequality. J. Glob. Optim. 8, 289–294 (1996)CrossRefGoogle Scholar
  14. 14.
    Konnov I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)Google Scholar
  15. 15.
    Konnov I.V., Kum S.: Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Anal. Theory Methods Appl. 47, 561–572 (2001)CrossRefGoogle Scholar
  16. 16.
    Konnov I.V., Ali M.S.S.: Descent methods for monotone equilibrium problems in Banach spaces. J. Comput. Appl. Math. 188, 165–179 (2006)CrossRefGoogle Scholar
  17. 17.
    Konnov I.V.: Regularization methods for nonmonone equilibrium problems. J. Nonlinear Convex Anal. 10, 93–101 (2009)Google Scholar
  18. 18.
    Konnov I.V., Dyabilkin D.A.: Nonmonotone equilibrium problems: coercivity conditions and weak regularization. J. Glob. Optim. 49, 575–587 (2011)CrossRefGoogle Scholar
  19. 19.
    Luo Z.Q., Pang J.S., Ralph D.: Mathematical Programs with Equilibrum Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  20. 20.
    Muu L.D., Quoc T.D., Nguyen V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)CrossRefGoogle Scholar
  21. 21.
    Moudafi A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)CrossRefGoogle Scholar
  22. 22.
    Solodov M.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14, 227–237 (2007)Google Scholar
  23. 23.
    Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)CrossRefGoogle Scholar
  24. 24.
    Tikhonov A.N., Arsenin V.Ya.: Solutions of Ill-Posed Problems. Wiley, New York (1977)Google Scholar
  25. 25.
    Xu M.H., Li M., Yang C.C.: Neural networks for a class of bi-level variational inequalities. J. Glob. Optim. 44, 535–552 (2009)CrossRefGoogle Scholar
  26. 26.
    Yao, Y., Liou, Y.C., Yao, J.C.: An extragradient method for fixed point problems and variational inequality problems. J. Inequal. Appl. doi: 10.1155/2007/38752, ID 38752 (2007)
  27. 27.
    Yao, Y., Noor, M.A., Liou, Y.C.: A new hybrid iterative algorithm for variational inequalities. Appl. Math. Comput. doi: 10.1016/j.amc.2010.01.087 (2010)

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Department of MathematicsKyungnam UniversityMasan, KyungnamKorea
  2. 2.Institute of MathematicsHanoiVietnam

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