Journal of Optimization Theory and Applications

, Volume 167, Issue 3, pp 969–984 | Cite as

On Robust Optimization

Relations Between Scalar Robust Optimization and Unconstrained Multicriteria Optimization


We introduce an unconstrained multicriteria optimization problem and discuss its relation to various well-known scalar robust optimization problems with a finite uncertainty set. Specifically, we show that a unique solution of a robust optimization problem is Pareto optimal for the unconstrained optimization problem. Furthermore, it is demonstrated that the set of weakly Pareto optimal solutions of the unconstrained multicriteria optimization problem contains all solutions of certain scalar robust optimization problems. An example is presented to verify our results. In addition, we show that the set of solutions of a weighted robust optimization problem always contains Pareto optimal solutions of the unconstrained multicriteria optimization problem. Similarly, we indicate that the set of solutions of a strictly robust optimization problem comprises Pareto optimal points of the unconstrained vector-valued problem. By assembling all these results we point out strong relations between unconstrained vector optimization and the more intuitively introduced concepts of scalar robust optimization. Finally, we provide a sufficient condition for an optimal solution of a strictly robust optimization problem.


Robust scalar optimization Multicriteria optimization Scalarization Constrained optimization 



The author would like to thank Kathrin Klamroth, Anita Schöbel, and Christiane Tammer for their ideas and advice in the preparation of this article. The author would also like to express sincere thanks to two anonymous referees for their helpful comments and detailed corrections.


  1. 1.
    Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973) MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton and Oxford (2009) MATHCrossRefGoogle Scholar
  3. 3.
    Kouvelis, P., Yu, G.: Robust Discrete Optimization and Its Applications. Kluwer Academic, Amsterdam (1997) MATHCrossRefGoogle Scholar
  4. 4.
    Sayin, S., Kouvelis, P.: The multiobjective discrete optimization problem: a weighted min–max two-stage optimization approach and a bicriteria algorithm. Manag. Sci. 51, 1572–1581 (2005) MATHCrossRefGoogle Scholar
  5. 5.
    Klamroth, K., Köbis, E., Schöbel, A., Tammer, C.: A unified approach for different kinds of robustness and stochastic programming via nonlinear scalarizing functionals. Optimization 62(5), 649–671 (2013) MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fischetti, M., Monaci, M.: Light robustness. In: Ahuja, R.K., Moehring, R., Zaroliagis, C. (eds.) Robust and Online Large-Scale Optimization, vol. 5868, pp. 61–84. Springer, Berlin (2009) CrossRefGoogle Scholar
  8. 8.
    Ehrgott, M.: Multicriteria Optimization. Springer, New York (2005) MATHGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999) MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Köbis, E., Tammer, C.: Relations between strictly robust optimization problems and a nonlinear scalarization method. AIP Conf. Proc. 1479, 2371–2374 (2012) CrossRefGoogle Scholar
  11. 11.
    Haimes, Y., Lasdon, L., Wismer, D.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1, 296–297 (1971) MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York (1986) MATHGoogle Scholar
  13. 13.
    Chankong, V., Haimes, Y.Y.: Multiobjective Decision Making: Theory and Methodology. Elsevier, New York (1983) MATHGoogle Scholar
  14. 14.
    Klamroth, K., Tind, J.: Constrained Optimization Using Multiple Objective Programming. J. Glob. Optim. 37, 325–355 (2007) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematics, Faculty of Natural Sciences IIMartin-Luther-University Halle-WittenbergHalle (Saale)Germany

Personalised recommendations