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Acta Mathematica Vietnamica

, Volume 38, Issue 4, pp 529–540 | Cite as

Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities

  • Bui Van Dinh
  • Le Dung MuuEmail author
Article

Abstract

We propose algorithms for finding the projection of a given point onto the solution set of the pseudomonotone variational inequality problem. This problem arises in the Tikhonov regularization method for pseudomonotone variational inequality. Since the solution set of the variational inequality is not given explicitly, the available methods of mathematical programming and variational inequalities cannot be applied directly.

Keywords

Bilevel variational inequality Pseudomonotonicity Projection method Armijo linesearch Convergence 

Mathematics Subject Classification (2010)

49M37 90C26 65K15 

Notes

Acknowledgements

We would like to thank the Associate Editor and the referees for their useful remarks and comments that helped us very much in revising the paper. This work is supported by the Vietnam National Foundation for Science Technology Development (NAFOSTED) under Grant 101-02-2011.19.

References

  1. 1.
    Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Glob. Optim. 52, 627–639 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) CrossRefzbMATHGoogle Scholar
  3. 3.
    Ceng, L.C., Ansari, Q.H., Yao, J.-C.: Iterative methods for triple hierarchical variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 489–512 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, G.: Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ding, X.P.: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel equilibrium problems in Banach spaces. J. Optim. Theory Appl. 146, 347–357 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. (2011). doi: 10.1155/2011/646452 Google Scholar
  8. 8.
    Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
  9. 9.
    Giannessi, F., Maugeri, A., Pardalos, P.M. (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Nonconvex Optim. Appl., vol. 58. Kluwer Academic, Dordrecht (2001) Google Scholar
  10. 10.
    Giuseppe, M., Xu, H.-K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61–78 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hao, N.T.: The Tikhonov regularization algorithm for pseudomonotone variational inequalities. Acta Math. Vietnam. 31, 283–289 (2006) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74, 6121–6129 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Iusem, A.N., Swaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kalashnikov, V.V., Klashnikova, N.I.: Solving two-level variational inequality. J. Glob. Optim. 8, 289–294 (1996) CrossRefzbMATHGoogle Scholar
  15. 15.
    Kinderlehrer, D., Stampachia, G.: An Introduction to Variational Inequalities and Their Applications. Pure Appl. Math., vol. 88. Academic Press, New York (1980) zbMATHGoogle Scholar
  16. 16.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Lecture Notes in Economics and Mathematical Systems, vol. 495. Springer, Berlin (2001) CrossRefzbMATHGoogle Scholar
  17. 17.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Èkon. Mat. Metody 12, 747–756 (1976) zbMATHGoogle Scholar
  18. 18.
    Luo, J.Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) CrossRefGoogle Scholar
  19. 19.
    Maingé, P.-E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010) CrossRefzbMATHGoogle Scholar
  20. 20.
    Marino, G., Xu, H.-K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61–78 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Quoc, T.D., Dung, L.M., Van Nguyen, H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Noor, M.A.: Extragradient methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 117, 475–488 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tam, N.N., Yao, J.C., Yen, N.D.: Solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problem over the fixed point of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yao, Y., Liou, Y.-C., Kang, S.M.: Minimization of equilibrium problems, variational inequality problems and fixed point problems. J. Glob. Optim. 48, 643–656 (2010) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013

Authors and Affiliations

  1. 1.Le Quy Don Technical UniversityHanoiVietnam
  2. 2.Institute of MathematicsVASTHanoiVietnam

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