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Positivity

, Volume 20, Issue 1, pp 61–80 | Cite as

Continuity properties of solution maps of parametric lexicographic equilibrium problems

  • L. Q. Anh
  • T. Q. DuyEmail author
  • P. Q. Khanh
Article

Abstract

Inspired by the great importance of equilibrium problems and the lexicographic order, we consider a parametric lexicographic equilibrium problem. Sufficient conditions for the upper semicontinuity, closedness, and continuity of solution maps are established. Many examples are provided to ensure the essentialness of the imposed assumptions. Applications to lexicographic variational inequalities and lexicographic optimization problems are discussed.

Keywords

Lexicographic equilibrium problems Variational inequalities and optimization Continuity Upper semicontinuity Closedness Upper and lower pseudocontinuity Relaxed monotonicity properties 

Mathematics Subject Classification

90C31 49J40 49K40 

Notes

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.44. The authors wish to thank the anonymous referees for their helpful remarks and suggestions that helped to significantly improve the original manuscript.

References

  1. 1.
    Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Anh, L.Q., Khanh, P.Q.: Continuity of solution maps of parametric quasiequilibrium problems. J. Global Optim. 46, 247–259 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  4. 4.
    Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic and sequential equilibrium problems. J. Global Optim. 46, 551–560 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer, New York (2002)zbMATHGoogle Scholar
  7. 7.
    Edgeworth, E.Y.: Mathematical Psychics. Kegan Paul, London (1881)Google Scholar
  8. 8.
    Guerraggio, A., Luc, D.T.: Optimality conditions for \(C^{1, 1}\) vector optimization problems. J. Optim. Theory Appl. 109, 615–629 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Kluwer, London (1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    Jiménez, B., Novo, V.: First and second-order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284, 496–510 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Khanh, P.Q.: Proper solutions of vector optimization problems. J. Optim. Theory Appl. 74, 105–130 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Khanh, P.Q., Tuan, N.D.: First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces. J. Optim. Theory Appl. 130, 289–308 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133, 341–357 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Kim, W.K., Kum, S., Lee, K.H.: Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems. Nonlinear Anal. 71, 2182–2187 (2009)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kimura, K., Yao, J.C.: Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems. J. Global Optim. 41, 187–202 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Konnov, I.V.: On lexicographic vector equilibrium problems. J. Optim. Theory Appl. 118, 681–688 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Küçük, M., Soyertem, M., Küçük, Y.: On constructing total orders and solving vector optimization problems with total orders. J. Global Optim. 50, 235–247 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Makarov, E.K., Rachkovski, N.H.: Unified representation of proper efficiency by means of dilating cones. J. Optim. Theory Appl. 101, 141–165 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Morgan, J., Scalzo, V.: Pseudocontinuity in optimization and nonzero sum games. J. Optim. Theory Appl. 120, 181–197 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Morgan, J., Scalzo, V.: Discontinuous but well-posed optimization problems. SIAM J. Optim. 17, 861–870 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Pareto, V.: Cours d’économie politique. F. Rouge, Lausanne (1896)Google Scholar
  22. 22.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1984)Google Scholar
  23. 23.
    Xu, Y.D., Li, S.J.: On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem. Positivity 17, 341–353 (2013)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Teacher CollegeCantho UniversityCanthoVietnam
  2. 2.Department of MathematicsUniversity of Science of Hochiminh CityHochiminh CityVietnam
  3. 3.Department of MathematicsInternational University, Vietnam National University Hochiminh CityHochiminh CityVietnam

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