Advertisement

Journal of Optimization Theory and Applications

, Volume 178, Issue 2, pp 591–613 | Cite as

Stability of Local Efficiency in Multiobjective Optimization

  • Sanaz Sadeghi
  • S. Morteza Mirdehghan
Article

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.

Keywords

Multiobjective programming Variational analysis Tilt stability Weighted sum scalarization 

Mathematics Subject Classification

90C29 90C31 49K40 

Notes

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.

References

  1. 1.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  4. 4.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications Grundlehren Math. Springer, Berlin (2006)Google Scholar
  5. 5.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren Math. Wiss., vol. 317. Springer, Berlin (2006)Google Scholar
  6. 6.
    Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eberhard, A.C., Wenczel, R.: A study of tilt-stable optimality and sufficient conditions. Nonlinear Anal. 75, 1240–1281 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mordukhovich, B., Outrata, S.: Tilt stability in nonlinear programming under Mangasarim–Fromouvitz constraint qualification. Kybernetika 49, 446–464 (2013)MathSciNetzbMATHGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mordukhovich, B.S.: Lipschitzian stability of constraint systems and generalized equations. Nonlinear Anal. 22, 173–206 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    Ben-Tal, A., Ghaoui, L.-E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bertsimas, D., Brown, D.-B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bokrantz, R., Fredriksson, A.: On solutions to robust multiobjective optimization problems that are optimal under convex scalarization. arXiv:1308.4616v2 (2014)
  16. 16.
    Deb, K., Gupta, H.: Introducing robustness in multi-objective optimization. Evol. Comput. 14(4), 463–494 (2006)CrossRefGoogle Scholar
  17. 17.
    Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multiobjective optimization problems. Eur. J. Oper. Res. 239(1), 17–31 (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fliege, J., Werner, R.: Robust multiobjective optimization applications in portfolio optimization. Eur. J. Oper. Res. 234(2), 422–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zamani, M., Soleimani-damaneh, M., Kabgani, A.: Robustness in nonsmooth nonlinear multi-objective programming. Eur. J. Oper. Res. 247(2), 370–378 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bot, R.I., Jeyakumar, V., Li, G.Y.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Book Math, vol. 20. Springer, New York (2005)Google Scholar
  24. 24.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Grundlehren Math. Springer, Berlin (2006)Google Scholar
  25. 25.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Wiley, Hoboken (2016)zbMATHGoogle Scholar
  26. 26.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar
  27. 27.
    Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran

Personalised recommendations