pp 1–24 | Cite as

Necessary optimality conditions for a semivectorial bilevel optimization problem using the kth-objective weighted-constraint approach

  • Nazih Abderrazzak Gadhi
  • Mohammed El idrissiEmail author
  • Khadija Hamdaoui


In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.


Constraint qualification Limiting subdifferential Limiting normal cone Optimality condition Semivectorial bilevel program Weakly efficient solution 

Mathematics Subject Classification

Primary 90C29 90C26 90C70 Secondary 49K99 



Our sincere acknowledgements to the anonymous referees for their insightful remarks and suggestions. This work has been supported by the Alexander von Humboldt-foundation.


  1. 1.
    Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators. J. Glob. Optim. 34, 535–549 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bard, J.F.: Some properities of the bilevel programming problem. J. Optim. Theory Appl. 68, 371–378 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonnel, K.: Optimality conditions for the semivectorial bilevel optimization problem. Pac. J. Optim. 2, 447–468 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burachik, R.S., Kaya, C.Y., Rizvi, M.M.: A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets. J. Optim. Theory Appl. 162, 428–446 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, vol. 5, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990) Google Scholar
  6. 6.
    Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problem. Optimization 25, 341–354 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dempe, S., Zemkoho, A.: The generalized Mangasarian–Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148, 46–68 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dempe, S., Gadhi, N.: Necessary optimality conditions for bilevel set optimization problems. J. Glob. Optim. 39, 529–542 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dempe, S., Gadhi, N.: A new equivalent single-level problem for bilevel problems. Optimization 63, 789–798 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dempe, S., Gadhi, N., Zemkoho, A.: New optimality conditions for the semivectoriel bilevel optimization problem. J. Optim. Theory Appl. 157, 54–74 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dempe, S., Mehlitz, P.: Semivectorial bilevel programming versus scalar bilevel programming. Optimization (2019).
  12. 12.
    Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56, 577–604 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gadhi, N., El idrissi, M.: An equivalent one level optimization problem to a semivectorial bilevel problem. Positivity 22, 261–274 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Henrion, R., Outrata, J.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258, 110–130 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lafhim, L.: New optimality conditions and a scalarization approach for a nonconvex semi-vectorial bilevel optimization problem. Positivity (2019). CrossRefGoogle Scholar
  17. 17.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)CrossRefGoogle Scholar
  19. 19.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  20. 20.
    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)CrossRefGoogle Scholar
  21. 21.
    Mordukhovich, B.S., Nam, N.M.: Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30, 800–816 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mordukhovich, B.S., Nam, N.M., Phan, H.M.: Variational analysis of marginal functions with applications to bilevel programming. J. Optim. Theory Appl. 152, 557–586 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Outrata, J.V.: On necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 76, 306–320 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rochafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  27. 27.
    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Nazih Abderrazzak Gadhi
    • 1
  • Mohammed El idrissi
    • 1
    Email author
  • Khadija Hamdaoui
    • 1
  1. 1.Department of MathematicsLSO, Dhar El Mehrez, Sidi Mohamed Ben Abdellah UniversityFesMorocco

Personalised recommendations