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On set-valued optimization problems with variable ordering structure

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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.

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References

  1. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkäuser, Basel (1990)

    MATH  Google Scholar 

  2. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122(2), 301–347 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Chen, G.-Y., Huang, X., Yang, X.: Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)

  6. Durea, M.: First and second-order optimality conditions for set-valued optimization problems. Rendiconti del Circolo Matematico di Palermo 53, 451–468 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems: applications to variational systems. Nonlinear Anal. Theory Methods Appl. 75, 1246–1259 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

  13. Eichfelder, G.: Variable ordering structures in vector optimization, Habilitation Thesis, University Erlangen-Nürnberg (2011)

  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)

  15. Eichfelder, G., Ha, T.X.D.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. John, R.: The concave nontransitive consumer. J. Global Optim. 20(3–4), 297–308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  18. Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13, 1737–1785 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Kuroiwa, D.: The natural criteria in set-valued optimization. RIMS Kokyuroku 1031, 85–90 (1998)

    MATH  MathSciNet  Google Scholar 

  21. Li, S.J., Liao, C.M.: Second-order differentiability of generalized perturbation maps. J. Glob. Optim. 52, 243–252 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, S.J., Meng, K.W., Penot, J.-P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, S., Penot, J.-P., Xue, X.: Codifferential calculus. Set-Valued Var. Anal. 19, 505–536 (2011)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  26. Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nieuwenhuis, J.W.: Supremal points and generalized duality. Math. Operationsforsch. Stat. Ser. Optim. 11, 41–59 (1980)

    MATH  MathSciNet  Google Scholar 

  28. Penot, J.-P.: Differentiability of relations and differential stability perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  29. Penot, J.-P.: Cooperative behavior of functions, sets and relations. Math. Methods Oper. Res. 48, 229–246 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tanino, T.: On the supremum of a set in a multi-dimensional space. J. Math. Anal. Appl. 130, 386–397 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tanino, T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167, 84–97 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The work of M. Durea was supported by the ERC-Like grant of the Romanian National Authority for Scientific Research 1ERC/02.07.2012. The work of R. Strugariu was supported by the grant POSDRU/159/1.5/S/133652.

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Authors and Affiliations

  1. Faculty of Mathematics, “Al. I. Cuza” University, Bd. Carol I, nr. 11, 700506 , Iaşi, Romania

    Marius Durea

  2. Department of Mathematics and Informatics, “Gh. Asachi” Technical University, Bd. Carol I, nr. 11, 700506 , Iaşi, Romania

    Radu Strugariu

  3. Institute for Mathematics, Martin-Luther-University Halle-Wittenberg, 06099 , Halle (Saale), Germany

    Christiane Tammer

Authors
  1. Marius Durea
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  2. Radu Strugariu
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  3. Christiane Tammer
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Correspondence to Christiane Tammer.

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Durea, M., Strugariu, R. & Tammer, C. On set-valued optimization problems with variable ordering structure. J Glob Optim 61, 745–767 (2015). https://doi.org/10.1007/s10898-014-0207-x

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