Journal of Optimization Theory and Applications

, Volume 179, Issue 1, pp 137–162 | Cite as

Robustness in Deterministic Vector Optimization

  • Morteza Rahimi
  • Majid Soleimani-damanehEmail author


In this paper, robust efficient solutions of a vector optimization problem, whose image space is ordered by an arbitrary ordering cone, are defined. This is done from different points of view, including set based and norm based. The relationships between these solution concepts are established. Furthermore, it is shown that, for a general vector optimization problem, each norm-based robust efficient solution is a strictly efficient solution; each isolated efficient solution is a norm-based robust efficient solution; and, under appropriate assumptions, each norm-based robust efficient solution is a Henig properly efficient solution. Various necessary and sufficient conditions for characterizing norm-based robust solutions of a general vector optimization problem, in terms of the tangent and normal cones and the nonascent directions, are presented. An optimization problem for calculating a robustness radius is provided, and then, the largest robustness radius is determined. Moreover, some alterations of objective functions preserving weak/strict/Henig proper/robust efficiency are studied.


Vector optimization Nonsmooth optimization Robust solutions Robustness radius 

Mathematics Subject Classification

90C29 90C31 49J52 



The authors would like to express their gratitude to the editor in chief of JOTA, handling editor, and two anonymous referees for their helpful comments on the earlier versions of the paper. The research of the second author was in part supported by a grant from the Iran National Science Foundation (INSF) (No. 95849588).


  1. 1.
    Ben-Tal, A., Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series Application of Mathematics. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust optimization methodology and applications. Math. Progr. 92, 453–480 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2013)zbMATHGoogle Scholar
  5. 5.
    Mordukhovich, B.S., Outrata, J.V., Sarabi, M.E.: Full stability of locally optimal solutions in second-order cone programs. SIAM J. Optim. 24, 1581–1613 (2014)MathSciNetCrossRefzbMATHGoogle 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.
    Soyster, A.: Technical Note-Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bokrantz, R., Fredriksson, A.: On solutions to robust multiobjective optimization problems that are optimal under convex scalarization. arXiv:1308.4616v2 (2013)
  9. 9.
    Deb, K., Gupta, H.: Introducing robustness in multi-objective optimization. Evolut. Comput. 14, 463–494 (2006)CrossRefGoogle Scholar
  10. 10.
    Ehrgott, M., Ide, J., Schöbel, A.: A Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239, 17–31 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Georgiev, P.G., Luc, D.T., Pardalos, P.: Robust aspects of solutions in deterministic multiple objective linear programming. Eur. J. Oper. Res. 229, 29–36 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goberna, M.A., Jeyakumar, V., Li, G., López, M.A.: Robust linear semi-infinite programming duality under uncertainty. Math. Progr. 139, 185–203 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goberna, M.A., Jeyakumar, V., Li, G., López, M.A.: Robust solutions to multi-objective linear programs with uncertain data. Eur. J. Oper. Res. 242, 730–743 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ide, J., Köbis, E.: Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations. Math. Methods Oper. Res. 80, 99–127 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ide, J., Köbis, E., Kuroiwa, D., Schöbel, A., Tammer, C.: The relationship between multi-objective robustness concepts and set-valued optimization. Fixed Point Theory Appl. 2014, 83 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ide, J., Schöbel, A.: Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spectr. 38, 235–271 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuroiwa, D., Lee, G.M.: On robust multi-objective optimization. Vietnam J. Math. 40, 305–317 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pourkarimi, L., Soleimani-damaneh, M.: Robustness in deterministic multi-objective linear programming with respect to the relative interior and angle deviation. Optimization 65, 1983–2005 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zamani, M., Soleimani-damaneh, M., Kabgani, A.: Robustness in nonsmooth nonlinear multi-objective programming. Eur. J. Oper. Res. 247, 370–378 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fliege, J., Werner, R.: Robust multiobjective optimization applications in portfolio optimization. Eur. J. Oper. Res. 234, 422–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar
  22. 22.
    Henig, M.: proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khaledian, K., Khorram, E., Soleimani-damaneh, M.: Strongly proper efficient solutions: efficient solutions with bounded trade-offs. J. Optim. Theory Appl. 168, 864–883 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuhn, H., Tucker, A.: Proceedings of the second Berkeley symposium on mathematical statistics and probability. In: Neyman, J. (ed.) Nonlinear Programming, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  25. 25.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)Google Scholar
  27. 27.
    Markowitz, H.M.: Portfolio Selection-efficient Diversification Of Investments. Wiley, New York (1959)Google Scholar
  28. 28.
    Pac, A.B., Pinar, M.C.: Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity. Top 22(3), 875–891 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Korhonen, P., Soleimani-damaneh, M., Wallenius, J.: Dual cone approach to convex-cone dominance in multiple criteria decision making. Eur. J. Oper. Res. 249, 1139–1143 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goroohi Sardou, I., Khodayar, M.E., Khaledian, K., Soleimani-damaneh, M., Ameli, M.T.: Energy and reserve market clearing with microgrid aggregators. IEEE Trans. Smart Grid 7, 2703–2712 (2016)CrossRefGoogle Scholar
  31. 31.
    Franklin, J.N.: Methods of Mathematical Economics: Linear and Nonlinear Programming. Cambridge University Press, Cambridge (2003)Google Scholar
  32. 32.
    Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization. Appl. Math. 51, 5–36 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rockafellar, R.T.: Extensions of subgradient calculus with applications to optimization. Nonlinear Anal. Theory Methods Appl. 9, 665–698 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
  35. 35.
    Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University, Princeton (1972)Google Scholar
  36. 36.
    Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, Berlin (1998)zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

Personalised recommendations