Advertisement

Journal of Global Optimization

, Volume 50, Issue 1, pp 93–105 | Cite as

Semicontinuity of the solution map of quasivariational inequalities

  • D. Aussel
  • J. Cotrina
Article

Abstract

We investigate continuity properties (closedness and lower semicontinuity) of the solution map of a quasivariational inequality which is subjet to perturbations. Perturbations are here considered both on the set-valued operator and on the constraint map defining the quasivariational inequality. Two concepts of solution map will be considered.

Keywords

Quasivariational inequality Quasimonotonicity Lower semicontinuity Closedness 

Mathematics Subject Classification (2000)

Primary: 49J40 Secondary: 90C31 49K40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ait Mansour M., Aussel D.: Quasimonotone variational inequalities and quasiconvex programming: quantitative stability. Pac. J. Optim. 2, 611–626 (2006)Google Scholar
  2. 2.
    Ait Mansour M., Aussel D.: Quasimonotone variational inequalities and quasiconvex programming: qualitative stability. J. Convex Anal. 15, 459–472 (2008)Google Scholar
  3. 3.
    Anh L.Q., Khanh P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks. I. Upper semicontinuities. Set-Valued Anal. 16, 267–279 (2008)CrossRefGoogle Scholar
  4. 4.
    Anh L.Q., Khanh P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks. II. Lower semicontinuities applications. Set-Valued Anal. 16, 943–960 (2008)CrossRefGoogle Scholar
  5. 5.
    Aubin J.-P., Frankowska H.: Set-Valued Analysis. Birkhuser Boston, Inc., Boston (1990)Google Scholar
  6. 6.
    Aussel D., Ye J.J.: Quasiconvex minimization on locally finite union of convex sets. J. Optim. Th. Appl. 139, 1–16 (2008)CrossRefGoogle Scholar
  7. 7.
    Aussel D., Hadjisavvas N.: On quasimonotone variational inequalities. J. Optim. Th. Appl. 121, 445–450 (2004)CrossRefGoogle Scholar
  8. 8.
    Aussel D., Hadjisavvas N.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)CrossRefGoogle Scholar
  9. 9.
    Facchinei F., Kanzow C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)CrossRefGoogle Scholar
  10. 10.
    Facchinei F., Pang J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II. Springer, New York (2003)Google Scholar
  11. 11.
    Harker P.T.: Generalized Nash games and quasivariational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)CrossRefGoogle Scholar
  12. 12.
    Khanh P.Q., Luu L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Global Optim. 32, 569–580 (2005)CrossRefGoogle Scholar
  13. 13.
    Khanh P.Q., Luu L.M.M.: Lower semicontinuity and upper semicontinuity of the solution sets and approximate solution sets of parametric multivalued quasivariational inequalities. J. Optim. Th. Appl. 133, 329–339 (2007)CrossRefGoogle Scholar
  14. 14.
    Mosco U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)CrossRefGoogle Scholar
  15. 15.
    Pang J.-S., Fukushima M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Lab. PROMESUniversité de PerpignanPerpignanFrance
  2. 2.IMCA, Instituto de Matemática y Ciencias AfinesUniversidad Nacional de IngenieríaLa MolinaPerú

Personalised recommendations