Skip to main content
SpringerLink
Log in

Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications, Springer, Berlin, 2018), we establish strong Karush–Kuhn–Tucker type necessary optimality conditions for the weak efficiency of a nonsmooth nonconvex multiobjective programming problem with set, inequality and equality constraints. We also provide several new definitions for the Mordukhovich-pseudoconvexity and Mordukhovich-quasiconvexity with extended-real-valued functions, and then provide sufficient optimality conditions for weak efficiency to such problem in terms of Mordukhovich subdifferentials.

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. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (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), 301–247 (2010)

    Article  MathSciNet  Google Scholar 

  3. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global. Optim. 43(4), 533–552 (2009)

    Article  MathSciNet  Google Scholar 

  4. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135(2), 179–203 (2007)

    Article  MathSciNet  Google Scholar 

  5. Burachik, R.S., Rizvi, M.M.: On weak and strong KuhnTucker conditions for smooth multiobjective optimization. J. Optim. Theory Appl. 155, 477–491 (2012)

    Article  MathSciNet  Google Scholar 

  6. Constantin, E.: First-order necessary conditions in locally Lipschitz multiobjective optimization. Optimization 67, 1447–1460 (2018)

    Article  MathSciNet  Google Scholar 

  7. Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)

    Article  MathSciNet  Google Scholar 

  8. Goberna, M.A., Lopéz, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  9. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)

    Book  Google Scholar 

  10. Jiménez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449–471 (2008)

    Article  MathSciNet  Google Scholar 

  11. Jiménez, B., Novo, V.: A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization. J. Math. Anal. Appl. 270, 340–356 (2002)

    Article  MathSciNet  Google Scholar 

  12. Khan, A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)

    Book  Google Scholar 

  13. Liu, J.J., Zhao, K.Q., Yang, X.M.: Optimality and regularity conditions using Mordukhovich’s subdifferential. J. Nonlinear Convex Anal. 13, 43–53 (2017)

    MATH  Google Scholar 

  14. Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)

    Article  MathSciNet  Google Scholar 

  15. Luu, D.V.: Necessary efficiency conditions for vector equilibrium problems with general inequality constraints via convexificators. Bull. Braz. Math. Soc. New Ser. 50, 685–704 (2019)

    Article  MathSciNet  Google Scholar 

  16. Luu, D.V.: Necessary conditions for efficiency in terms of the Michel–Penot subdifferentials. Optimization 61, 1099–1117 (2012)

    Article  MathSciNet  Google Scholar 

  17. Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems. J. Glob. Optim. 70, 437–453 (2018)

    Article  MathSciNet  Google Scholar 

  18. Luu, D.V., Su, T.V.: Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints. RAIRO - Oper. Res. bf 52, 543–559 (2018)

    Article  MathSciNet  Google Scholar 

  19. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)

    MATH  Google Scholar 

  20. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)

    Book  Google Scholar 

  21. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II: Applications. Springer, Berlin (2006)

    Book  Google Scholar 

  22. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Berlin (2018)

    Book  Google Scholar 

  23. Rockafellar, R.T.: Convex Analysis Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  24. Su, T.V., Hien, N.D.: Studniarski’s derivatives and efficiency conditions for constrained vector equilibrium problems with applications. Optimization (2019). https://doi.org/10.1080/02331934.2019.1702985

    Article  MATH  Google Scholar 

  25. Su, T.V., Hang, D.D.: Optimality conditions for the efficient solutions of vector equilibrium problems with constraints in terms of directional derivatives and applications. Bull. Iran. Math. Soc. 45(6), 1619–1650 (2019)

    Article  MathSciNet  Google Scholar 

  26. Zhao, K.Q.: Strong Kuhn-Tucker optimality in nonsmooth multiobjective optimization problems. Pac. J. Optim. 11, 483–494 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express many thanks to anonymous referees for careful reading of the manuscript, which improved the paper in its present form.

Author information

Authors and Affiliations

  1. Department of Mathematics, Quang Nam University, 102 Hung Vuong, Tamky, Vietnam

    Tran Van Su

  2. Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

    Nguyen Duc Hien

  3. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Nguyen Duc Hien

Authors
  1. Tran Van Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Duc Hien
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tran Van Su.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This manuscript does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Su, T., Hien, N.D. Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials. Optim Lett 15, 1175–1194 (2021). https://doi.org/10.1007/s11590-020-01620-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01620-0

Keywords

Search

Navigation