Skip to main content
SpringerLink
Log in

Stability of Parametric Quasivariational Inequality of the Minty Type

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

Abstract

In this paper, stability of a parametric quasivariational inequality of the Minty type is studied via various sufficient conditions characterizing upper and lower semicontinuity of the solution sets as well as the approximate solution sets. Sufficient conditions ensuring upper semicontinuity of the approximate solution sets of an optimization problem with quasivariational inequality constraints are also presented.

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. Dafermos, S.: Sensitivity analysis in variational inequalities. Math. Oper. Res. 13, 421–434 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Tobin, R.L.: Sensitivity analysis for variational inequalities. J. Optim. Theory Appl. 48, 191–204 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Zhao, J.: The lower semicontinuity of optimal solution sets. J. Math. Anal. Appl. 207, 240–254 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kien, B.T.: On the lower semicontinuity of optimal solution sets. Optimization 54, 123–130 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jianghua, F., Renyou, Z.: Stability analysis for variational inequality in reflexive Banach space. Nonlinear Anal. 69, 2566–2574 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Minty, G.J.: On the generalization of a direct method of the calculus of variations. Bull. Am. Math. Soc. 73, 315–321 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds.) New Trends in Mathematical Programming. Applied Optimization, vol. 13, pp. 93–99. Kluwer Academic, Massachusetts (1998)

    Google Scholar 

  8. Crespi, G., Guerraggio, A., Rocca, M.: Minty variational inequality and optimization: scalar and vector case. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity. Generalized Monotonicity and Applications. Nonconvex Optimization and its Applications, vol. 77, pp. 193–211. Springer, New York (2005)

    Chapter  Google Scholar 

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

    Google Scholar 

  10. John, R.: Variational inequalities and pseudomonotone functions: some characterizations. In: Crouzeix, J.P., Martinez-Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity, pp. 291–301. Kluwer, Dordrecht (1998)

    Google Scholar 

  11. Bensoussan, A., Lions, J.L.: Controle impulsionel et inequations quasivariationelles d’evolution. C. R. Acad. Sci., Paris, Sér. A 276, 1333–1338 (1973)

    MathSciNet  MATH  Google Scholar 

  12. Chan, D., Pang, J.S.: The generalized quasivariational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  14. Yao, J.C.: The generalized quasivariational inequality problem with applications. J. Math. Anal. Appl. 158, 139–160 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kien, B.T., Wong, N.C., Yao, J.C.: On the solution existence of generalized quasivariational inequalities with discontinuous multifunctions. J. Optim. Theory Appl. 135, 515–530 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mosco, U.: Implicit variational problems and quasivariational inequalities. In: Nonlin. Oper. Calc. Var. Proc. Summer School, Bruxelles, 1975. Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)

    Google Scholar 

  17. Gong, L.: Global stability result for the generalized quasivariational inequality problem. J. Optim. Theory Appl. 70, 365–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khanh, P.Q., Luu, L.M.: Lower semicontinuity and upper semicontinuity of the solution sets and approximate solution sets of parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133, 329–339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Dover, Mineola (2006). Reprint of the 1984 original

    MATH  Google Scholar 

  20. Muu, L.D.: Stability property of a class of variational inequalities. Math. Operationsforsch. Stat. Ser. Optim. 15, 347–351 (1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi, 110021, India

    C. S. Lalitha

  2. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Guneet Bhatia

Authors
  1. C. S. Lalitha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guneet Bhatia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guneet Bhatia.

Additional information

Communicated by J.-C. Yao.

Research of C.S. Lalitha was supported by R&D Doctoral Research Programme funds for University faculty.

The authors are grateful to Professor J.-C. Yao for his valuable comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lalitha, C.S., Bhatia, G. Stability of Parametric Quasivariational Inequality of the Minty Type. J Optim Theory Appl 148, 281–300 (2011). https://doi.org/10.1007/s10957-010-9755-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9755-5

Keywords

Search

Navigation