Skip to main content
SpringerLink
Log in

On Some Methods to Derive Necessary and Sufficient Optimality Conditions in Vector Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The aim of this paper is to address new approaches, in separate ways, to necessary and, respectively, sufficient optimality conditions in constrained vector optimization. In this respect, for the necessary optimality conditions that we derive, we use a kind of vectorial penalization technique, while for the sufficient optimality conditions we make use of an appropriate scalarization method. In both cases, the approaches couple a basic technique (of penalization or scalarization, respectively) with several results in variational analysis and optimization obtained by the authors in the last years. These combinations allow us to arrive to optimality conditions which are, in terms of assumptions made, new.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)

  2. Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions on Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Glob. Optim. 43, 533–552 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Klamroth, K., Tind, J.: Constrained optimization using multiple objective programming. J. Glob. Optim. 37, 325–355 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Günther, C., Tammer, Chr.: Relationships between constrained and unconstrained multi-objective optimization and application in location theory. Math. Methods Oper. Res. 84, 359–387 (2016)

  8. Ye, J.J.: The exact penalty principle. Nonlinear Anal. 75, 1642–1654 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Durea, M., Strugariu, R., Tammer, Chr: On set-valued optimization problems with variable ordering structure. J. Glob. Optim. 61, 745–767 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Durea, M., Panţiruc, M., Strugariu, R.: Minimal time function with respect to a set of directions. Basic properties and applications. Optim. Methods Softw. 31, 535–561 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24, 1044–1049 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ward, D.E.: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl. 80, 551–571 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chankong, V., Haimes, Y.Y.: Multiobjective Decision Making. North-Holland Publishing Co., New York (1983)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Göpfert, A., Riahi, H., Tammer, Chr, Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)

    MATH  Google Scholar 

  17. Dauer, J.P., Saleh, O.A.: A characterization of proper minimal points as solutions of sublinear optimization problems. J. Math. Anal. Appl. 178, 227–246 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Apetrii, M., Durea, M., Strugariu, R.: A new penalization tool in scalar and vector optimizations. Nonlinear Anal. 107, 22–33 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  20. Durea, M., Huyhn, 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Durea, M., Tammer, Chr: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48, 881–908 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  24. Durea, M.: Remarks on strict efficiency in scalar and vector optimization. J. Glob. Optim. 47, 13–27 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nam, N.M., Zălinescu, C.: Variational analysis of directional minimal time functions and applications to location problems. Set-Valued Var. Anal. 21, 405–430 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the two anonymous referees for their remarks, which led to the improvement of this paper. The work of M. Durea and R. Strugariu was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, Project number PN-II-RU-TE-2014-4-0019.

Author information

Authors and Affiliations

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

    Marius Durea

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

    Radu Strugariu

  3. Institute of Mathematics, Martin-Luther-Universität Halle-Wittenberg, 06099, Halle (Saale), Germany

    Christiane Tammer

Authors
  1. Marius Durea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radu Strugariu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christiane Tammer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christiane Tammer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Durea, M., Strugariu, R. & Tammer, C. On Some Methods to Derive Necessary and Sufficient Optimality Conditions in Vector Optimization. J Optim Theory Appl 175, 738–763 (2017). https://doi.org/10.1007/s10957-016-1059-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-1059-y

Keywords

Mathematics Subject Classification

Search

Navigation