Mathematical Programming

, Volume 136, Issue 1, pp 65–89 | Cite as

Bilevel optimization: on the structure of the feasible set

  • H. Th. Jongen
  • V. ShikhmanEmail author
Full Length Paper Series B


We consider bilevel optimization from the optimistic point of view. Let the pair (x, y) denote the variables. The main difficulty in studying such problems lies in the fact that the lower level contains a global constraint. In fact, a point (x, y) is feasible if y solves a parametric optimization problem L(x). In this paper we restrict ourselves to the special case that the variable x is one-dimensional. We describe the generic structure of the feasible set M. Moreover, we discuss local reductions of the bilevel problem as well as corresponding optimality criteria. Finally, we point out typical problems that appear when trying to extend the ideas to higher dimensional x-dimensions. This will clarify the high intrinsic complexity of the general generic structure of the feasible set M and corresponding optimality conditions for the bilevel problem U.


Bilevel programming Parametric optimization Structure of the feasible set Local reduction Optimality criteria 

Mathematics Subject Classification



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  1. 1.
    Arnold V.I.: Singularity Theory. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  2. 2.
    Bard J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht (1998)zbMATHGoogle Scholar
  3. 3.
    Bonnans F., Shapiro A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)zbMATHGoogle Scholar
  4. 4.
    Bröcker P., Lander L.: Differentiable germs and catastrophes. Cambridge University Press, Cambridge (1975)zbMATHGoogle Scholar
  5. 5.
    Dempe S.: A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25, 341–354 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dempe S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002)zbMATHGoogle Scholar
  7. 7.
    Dempe S., Günzel H., Jongen H.Th.: On reducibility in bilevel problems. SIAM J. Optim. 20(2), 718–727 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dempe S., Dutta J.: Is bilevel programming a special case of a mathematical program with complementarity constraints?. Math. Program. 131, 37–48 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Goresky M., MacPherson R.: Stratified Morse Theory. Springer, Berlin (1988)zbMATHCrossRefGoogle Scholar
  10. 10.
    Hirsch M.W.: Differential Topology. Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  11. 11.
    Kojima M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)Google Scholar
  12. 12.
    Jongen H.Th.: Parametric optimization: critical points and local minima. Lect. Appl. Math. 26, 317–335 (1990)MathSciNetGoogle Scholar
  13. 13.
    Jongen H.Th., Jonker P., Twilt F.: Critical sets in parametric optimization. Math. Program. 34, 333–353 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jongen H.Th., Jonker P., Twilt F.: Nonlinear Optimization in Finite Dimensions. Kluwer, Dordrecht (2000)zbMATHGoogle Scholar
  15. 15.
    Scheel H., Scholtes S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ye J.J., Zhu D.L.: New necessary optimality conditions for bilevel programs by combined MPEC and the value function approach. SIAM J. Optim. 20, 1885–1905 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ye J.J., Zhu D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of Mathematics—CRWTH Aachen UniversityAachenGermany

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