Skip to main content
SpringerLink
Log in

Stability of Local Efficiency in Multiobjective Optimization

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

Abstract

Analyzing the behavior and stability properties of a local optimum in an optimization problem, when small perturbations are added to the objective functions, are important considerations in optimization. The tilt stability of a local minimum in a scalar optimization problem is a well-studied concept in optimization which is a version of the Lipschitzian stability condition for a local minimum. In this paper, we define a new concept of stability pertinent to the study of multiobjective optimization problems. We prove that our new concept of stability is equivalent to tilt stability when scalar optimizations are available. We then use our new notions of stability to establish new necessary and sufficient conditions on when strict locally efficient solutions of a multiobjective optimization problem will have small changes when correspondingly small perturbations are added to the objective functions.

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. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  2. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  3. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    MATH  Google Scholar 

  4. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications Grundlehren Math. Springer, Berlin (2006)

    Google Scholar 

  5. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren Math. Wiss., vol. 317. Springer, Berlin (2006)

    Google Scholar 

  6. Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drusvyatskiy, D., Lewis, A.S.: Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential. SIAM J. Optim. 23, 256–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eberhard, A.C., Wenczel, R.: A study of tilt-stable optimality and sufficient conditions. Nonlinear Anal. 75, 1240–1281 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mordukhovich, B., Outrata, S.: Tilt stability in nonlinear programming under Mangasarim–Fromouvitz constraint qualification. Kybernetika 49, 446–464 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with application to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mordukhovich, B.S.: Lipschitzian stability of constraint systems and generalized equations. Nonlinear Anal. 22, 173–206 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, SIAM Volume in Applied Mathematics, vol. 58, pp. 32–46 (1992)

  13. Ben-Tal, A., Ghaoui, L.-E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  MATH  Google Scholar 

  14. Bertsimas, D., Brown, D.-B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bokrantz, R., Fredriksson, A.: On solutions to robust multiobjective optimization problems that are optimal under convex scalarization. arXiv:1308.4616v2 (2014)

  16. Deb, K., Gupta, H.: Introducing robustness in multi-objective optimization. Evol. Comput. 14(4), 463–494 (2006)

    Article  Google Scholar 

  17. Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multiobjective optimization problems. Eur. J. Oper. Res. 239(1), 17–31 (2014)

    Article  MATH  Google Scholar 

  18. Fliege, J., Werner, R.: Robust multiobjective optimization applications in portfolio optimization. Eur. J. Oper. Res. 234(2), 422–433 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Georgiev, P.-G., Luc, D.-T., Pardalos, P.: Robust aspects of solutions in deterministic multiple objective linear programming. Eur. J. Oper. Res. 229(1), 29–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zamani, M., Soleimani-damaneh, M., Kabgani, A.: Robustness in nonsmooth nonlinear multi-objective programming. Eur. J. Oper. Res. 247(2), 370–378 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jeyakumar, V., Lee, G.M., Li, G.: Characterizing robust solution sets of convex programs under data uncertainty. J. Optim. Theory Appl. 164, 407–435 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bot, R.I., Jeyakumar, V., Li, G.Y.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Book Math, vol. 20. Springer, New York (2005)

    Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Grundlehren Math. Springer, Berlin (2006)

    Google Scholar 

  25. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Wiley, Hoboken (2016)

    MATH  Google Scholar 

  26. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  27. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to Prof. Boris Mordukhovich for his serious discussions on the earlier draft of this manuscript to enrich this manuscript. Moreover, the authors are grateful for the editor and the two anonymous referees for their valuable and constructive comments to improve this manuscript. They would also like to extend their thankfulness to Maxie Schmidt and Karim Rezaei for the linguistic editing of the final draft of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran

    Sanaz Sadeghi & S. Morteza Mirdehghan

Authors
  1. Sanaz Sadeghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Morteza Mirdehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Morteza Mirdehghan.

Additional information

Communicated by Fabián Flores-Bazán.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sadeghi, S., Mirdehghan, S.M. Stability of Local Efficiency in Multiobjective Optimization. J Optim Theory Appl 178, 591–613 (2018). https://doi.org/10.1007/s10957-018-1312-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1312-7

Keywords

Mathematics Subject Classification

Search

Navigation