Journal of Global Optimization

, Volume 61, Issue 4, pp 745–767 | Cite as

On set-valued optimization problems with variable ordering structure

Article

Abstract

In this paper we introduce and investigate an optimality concept for set-valued optimization problems with variable ordering structure. In our approach, the ordering structure is governed by a set-valued map acting between the same spaces as the objective multifunction. Necessary optimality conditions for the proposed problem are derived in terms of Bouligand and Mordukhovich generalized differentiation objects.

Keywords

Nondomination property Pareto optimization Variable ordering structure Openness for sum-multifunction Necessary optimality conditions 

Mathematics Subject Classification

90C30 49J52 49J53 

References

  1. 1.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkäuser, Basel (1990)MATHGoogle Scholar
  2. 2.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122(2), 301–347 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bao, T.Q., Mordukhovich, B.S.: Necessary nondomination conditions in set and vector optimization with variable ordering structures. J. Optim. Theory Appl. doi:10.1007/s10957-013-0332-6
  4. 4.
    Bao, T.Q., Mordukhovich, B.S., Souberyan, A.: Variational analysis in psychological modeling, http://www.optimization-online.org/DB_HTML/2013/11/4112.html
  5. 5.
    Chen, G.-Y., Huang, X., Yang, X.: Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)Google Scholar
  6. 6.
    Durea, M.: First and second-order optimality conditions for set-valued optimization problems. Rendiconti del Circolo Matematico di Palermo 53, 451–468 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Durea, M., Strugariu, R.: Optimality conditions in terms of Bouligand derivatives for Pareto efficiency in set-valued optimization. Optim. Lett. 5, 141–151 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems: applications to variational systems. Nonlinear Anal. Theory Methods Appl. 75, 1246–1259 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Durea, M., Strugariu, R.: Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions. J. Glob. Optim. 56, 587–603 (2013)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Durea, M., Strugariu, R.: Scalarization of constraints system in some vector optimization problems and applications. Optim. Lett. doi:10.1007/s11590-013-0690-x
  12. 12.
    Durea, M., Huynh, V.N., Nguyen, H.T., Strugariu, R.: Metric regularity of composition set-valued mappings: metric setting and coderivative conditions. J. Math. Anal. Appl. 412, 41–62 (2014)Google Scholar
  13. 13.
    Eichfelder, G.: Variable ordering structures in vector optimization, Habilitation Thesis, University Erlangen-Nürnberg (2011)Google Scholar
  14. 14.
    Eichfelder, G.: Variable ordering structures in vector optimization. In: Q.H. Ansari, J.-C. Yao, (eds.) Recent Developments in Vector Optimization, Chapter 4, pp. 95–126. Springer, Heidelberg (2012)Google Scholar
  15. 15.
    Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    John, R.: The concave nontransitive consumer. J. Global Optim. 20(3–4), 297–308 (2001)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    John, R.: Local and global consumer preferences. In: Generalized Convexity and Related Topics, Lecture Notes in Econom. and Math. Systems 583, pp. 315–325. Springer, Berlin (2007)Google Scholar
  18. 18.
    Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13, 1737–1785 (2009)MATHMathSciNetGoogle Scholar
  20. 20.
    Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1031, 85–90 (1998)MATHMathSciNetGoogle Scholar
  21. 21.
    Li, S.J., Liao, C.M.: Second-order differentiability of generalized perturbation maps. J. Glob. Optim. 52, 243–252 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Li, S.J., Meng, K.W., Penot, J.-P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Li, S., Penot, J.-P., Xue, X.: Codifferential calculus. Set-Valued Var. Anal. 19, 505–536 (2011)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation, Vol. I: Basic Theory, Vol. II: Applications, Springer, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), Vol. 330 and 331, Berlin (2006)Google Scholar
  25. 25.
    Ngai, H.V., Nguyen, H.T., Théra, M.: Metric regularity of the sum of multifunctions and applications. J. Optim. Theory Appl. 160, 355–390 (2014)Google Scholar
  26. 26.
    Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Nieuwenhuis, J.W.: Supremal points and generalized duality. Math. Operationsforsch. Stat. Ser. Optim. 11, 41–59 (1980)MATHMathSciNetGoogle Scholar
  28. 28.
    Penot, J.-P.: Differentiability of relations and differential stability perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Penot, J.-P.: Cooperative behavior of functions, sets and relations. Math. Methods Oper. Res. 48, 229–246 (1998)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Tanino, T.: On the supremum of a set in a multi-dimensional space. J. Math. Anal. Appl. 130, 386–397 (1988)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167, 84–97 (1992)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Marius Durea
    • 1
  • Radu Strugariu
    • 2
  • Christiane Tammer
    • 3
  1. 1.Faculty of Mathematics“Al. I. Cuza” UniversityIaşiRomania
  2. 2.Department of Mathematics and Informatics“Gh. Asachi” Technical UniversityIaşiRomania
  3. 3.Institute for MathematicsMartin-Luther-University Halle-WittenbergHalle (Saale)Germany

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