Skip to main content
SpringerLink
Log in

The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we deal with the Tikhonov regularization method for pseudo-monotone equilibrium problems. Under mild conditions of semicontinuity and convexity, we show that strictly pseudo-monotone bifunctions can be also used as regularization bifunctions as well as strongly monotone bifunctions. We extend Berge’s maximum theorem and establish the relationship between quasi-hemivariational inequalities and equilibrium problems. Applications of the Tikhonov regularization method to quasi-hemivariational inequalities are also given.

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. Alleche, B.: Multivalued mixed variational inequalities with locally Lipschitzian and locally cocoercive multivalued mappings. J. Math. Anal. Appl. 399, 625–637 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alleche, B.: On hemicontinuity of bifunctions for solving equilibrium problems. Adv. Nonlinear Anal. 3(2), 69–80 (2014)

    MATH  MathSciNet  Google Scholar 

  3. Alleche, B., Calbrix, J.: On the coincidence of the upper Kuratowski topology with the cocompact topology. Topol. Appl. 93, 207–218 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alleche, B., Rădulescu, V.D.: Equilibrium problem techniques in the qualitative analysis of quasi-hemivariational inequalities. Optimization (2014). doi:10.1080/02331934.2014.917307

    MATH  Google Scholar 

  5. Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization. CRC Press, Taylor & Francis Group (2014)

    MATH  Google Scholar 

  6. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Press, New York (1988)

  7. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90(1), 31–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Glob. Optim. 30, 121–134 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. Cambini, A., Martein, L.: Generalized convexity and optimization. Theory and applications, vol. 616. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2009)

  11. Clarke, F.H.: Optimization and nonsmooth analysis. In: Society for Industrial and Applied Mathematics, Philadelphia, sIAM Edition (1990)

  12. Costea, C., Rădulescu, V.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Glob. Optim. 52, 743–756 (2012)

    Article  MATH  Google Scholar 

  13. Ding, X.P.: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. J. Optim. Theory Appl 146(2), 347–357 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. 2011, 1–14 (2011)

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

    Article  MATH  MathSciNet  Google Scholar 

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

  17. Hadjisavvas, N., Schaible, S.: On strong pseudomonotonicity and (semi)strict quasimonotonicity. J. Optim. Theory Appl. 79, 139–155 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hadjisavvas, N., Schaible, S.: Generalized monotonicity: applications to variational inequalities and equilibrium problems. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1202–1209. Springer, US (2009)

  19. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74(17), 6121–6129 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kassay, G.: On equilibrium problems. In: Chinchuluun, A., Pardalos, P.M., Enkhbat, R., Tseveendorj, I. (eds.) Optimization and Optimal Control: Theory and Applications, vol. 39. Optimization and its Applications. Springer, Berlin, pp. 55–83 (2010)

  21. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities. Academic Press, New York (1980)

    MATH  Google Scholar 

  22. Konnov, I.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  23. Konnov, I.V., Schaible, S.: Duality for equilibrium problems under generalized monotonicity. J. Optim. Theory Appl. 104, 395–408 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lions, J.-L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493–519 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27(4), 411–426 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18(12), 1159–1166 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)

    Google Scholar 

  29. Panagiotopoulos, P.D.: Nonconvex energy functions. Hemivariational inequalities and substationarity principles. Acta. Mech. 42, 160–183 (1983)

  30. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions. Birkhäuser, Boston (1985)

    Book  MATH  Google Scholar 

  31. Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.H.: Handbook of applied analysis. In: Advances in Mechanics and Mathematics, vol. 19. Springer, Dordrecht (2009)

  32. Rockafellar, R.T.: Convex Analysis. Princeton University Press, New Jersey (1970)

    MATH  Google Scholar 

  33. Rădulescu, V.: Qualitative analysis of nonlinear elliptic partial differential equations. In: Contemporary Mathematics and its Applications, vol. 6. Hindawi Publ Corporation, New York (2008)

  34. Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Doklady Akademii Nauk SSSR 151, 501–504 (1963)

    MathSciNet  Google Scholar 

  35. Wangkeeree, R., Preechasilp, P.: Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces. J. Glob. Optim. 57, 1447–1464 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous referees for the careful reading of this paper and for their useful remarks concerning Theorem 3.1.

Author information

Authors and Affiliations

  1. Laboratoire de Mécanique, Physique et Modélisation Mathématique, Université de Médéa, 26000 , Médéa, Algeria

    Boualem Alleche & Madjid Sebaoui

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Vicenţiu D. Rădulescu

Authors
  1. Boualem Alleche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicenţiu D. Rădulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Madjid Sebaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicenţiu D. Rădulescu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alleche, B., Rădulescu, V.D. & Sebaoui, M. The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities. Optim Lett 9, 483–503 (2015). https://doi.org/10.1007/s11590-014-0765-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-014-0765-3

Keywords

Search

Navigation