Skip to main content

Advertisement

SpringerLink
Log in

On Gap Functions for Multivalued Stampacchia Variational Inequalities

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Gap functions have proved to be efficient tools to study single-valued variational inequalities. This approach allows us to reformulate the problem into an optimization problem.

New notions of gap functions are defined for set-valued variational inequalities. We prove finiteness and error bounds properties, i.e. upper estimates for the distance to the solution set of the variational inequality.

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.

Similar content being viewed by others

References

  1. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics, vol. 88. Academic Press, Inc. [Harcourt Brace Jovanovich], New York (1980)

    MATH  Google Scholar 

  2. Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities. Springer, Berlin (2003)

    Google Scholar 

  3. John, R.: A note on Minty variational inequalities and generalized monotonicity. In: Generalized Convexity and Generalized Monotonicity (Karlovassi, 1999), pp. 240–246. Lecture Notes in Econom. and Math. Systems, vol. 502. Springer, Berlin (2001)

    Google Scholar 

  4. Giannessi, F.: On Minty variational principle. In: New Trends in Mathematical Programming. Appl. Optim., vol. 13, pp. 93–99. Kluwer Academic, Boston (1998)

    Google Scholar 

  5. Marcotte, P., Zhu, D.L.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9, 179–189 (1999) [Erratum: Weak sharp solutions of variational inequalities. SIAM J. Optim. 10, 942 (2000)]

    Article  MathSciNet  Google Scholar 

  6. Crouzeix, J.-P., Marcotte, P., Zhu, D.: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math. Program. 88, 521–539 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Marcotte, P., Zhu, D.L.: A cutting plane method for solving quasimonotone variational inequalities. Comput. Optim. Appl. 20, 317–324 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eberhard, A.C., Crouzeix, J.-P.: Existence of Closed Graph, Maximal, cyclic pseudo-monotone relations and revealed preference theory. J. Ind. Manag. Optim. 3, 233–255 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Solodov, M.V.: Merit functions and error bounds for generalized variational inequalities. J. Math. Anal. Appl. 287, 405–414 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang, J., Wan, C., Xiu, N.: The dual gap function for variational inequalities. Appl. Math. Optim. 48, 129–148 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Auslender, A.: Résolution numérique d’inégalités variationnelles. RAIRO R2, 67–72 (1973)

  12. Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Burachik, R., Iusem, A.: A generalized proximal point algorithm for the variational inequality problem in Hilbert space. SIAM J. Optim. 8, 197–216 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dietrich, H.: A smooth dual gap function solution to a class of quasivariational inequalities. J. Math. Anal. Appl. 235, 380–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)

    MathSciNet  MATH  Google Scholar 

  16. Panicucci, B., Pappalardo, M., Passacantando, M.: On finite-dimensional generalized variational inequalities. J. Ind. Manag. Optim. 2, 43–53 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Fukushima, M.: Equivalent differentiable optimization problem and descent methods for asymmetric variational inequalities. Math. Program. 53, 99–110 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fukushima, M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Aussel, D., Hadjisavvas, N.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Borde, J., Crouzeix, J.P.: Continuity properties of the normal cone to the sublevel sets of a quasiconvex function. J. Optim. Theory Appl. 66, 415–429 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. 1. Springer, Berlin (2005)

    MATH  Google Scholar 

  22. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. 2. Springer, Berlin (2005)

    MATH  Google Scholar 

  23. Dutta, J., Tammer, C.: Lagrangian conditions for vector optimization in Banach spaces. Math. Methods Oper. Res. 64, 521–540 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lab. PROMES, Université de Perpignan, Perpignan, France

    D. Aussel

  2. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

    J. Dutta

Authors
  1. D. Aussel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Aussel.

Additional information

Communicated by J.P. Crouzeix.

We would like to thank the two anonymous referees and Professor J.P. Crouzeix for their constructive suggestions which improved the presentation of the paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aussel, D., Dutta, J. On Gap Functions for Multivalued Stampacchia Variational Inequalities. J Optim Theory Appl 149, 513–527 (2011). https://doi.org/10.1007/s10957-011-9801-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9801-y

Keywords

Search

Navigation