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Continuity properties of solution maps of parametric lexicographic equilibrium problems

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, L.Q., Khanh, P.Q.: Continuity of solution maps of parametric quasiequilibrium problems. J. Global Optim. 46, 247–259 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  4. Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic and sequential equilibrium problems. J. Global Optim. 46, 551–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dempe, S.: Foundations of Bilevel Programming. Kluwer, New York (2002)

    MATH  Google Scholar 

  7. Edgeworth, E.Y.: Mathematical Psychics. Kegan Paul, London (1881)

    Google Scholar 

  8. Guerraggio, A., Luc, D.T.: Optimality conditions for \(C^{1, 1}\) vector optimization problems. J. Optim. Theory Appl. 109, 615–629 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Kluwer, London (1997)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khanh, P.Q.: Proper solutions of vector optimization problems. J. Optim. Theory Appl. 74, 105–130 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Kimura, K., Yao, J.C.: Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems. J. Global Optim. 41, 187–202 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Konnov, I.V.: On lexicographic vector equilibrium problems. J. Optim. Theory Appl. 118, 681–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morgan, J., Scalzo, V.: Pseudocontinuity in optimization and nonzero sum games. J. Optim. Theory Appl. 120, 181–197 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morgan, J., Scalzo, V.: Discontinuous but well-posed optimization problems. SIAM J. Optim. 17, 861–870 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pareto, V.: Cours d’économie politique. F. Rouge, Lausanne (1896)

    Google Scholar 

  22. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1984)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

    L. Q. Anh

  2. Department of Mathematics, University of Science of Hochiminh City, Hochiminh City, Vietnam

    T. Q. Duy

  3. Department of Mathematics, International University, Vietnam National University Hochiminh City, Hochiminh City, Vietnam

    P. Q. Khanh

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  1. L. Q. Anh
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  2. T. Q. Duy
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  3. P. Q. Khanh
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Correspondence to T. Q. Duy.

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Anh, L.Q., Duy, T.Q. & Khanh, P.Q. Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity 20, 61–80 (2016). https://doi.org/10.1007/s11117-015-0341-9

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  • DOI: https://doi.org/10.1007/s11117-015-0341-9

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