Journal of Combinatorial Optimization

, Volume 22, Issue 4, pp 594–608 | Cite as

Optimality conditions for a bilevel matroid problem

  • Diana Fanghänel


In bilevel programming there are two decision makers, the leader and the follower, who act in a hierarchy. In this paper we deal with a weighted matroid problem where each of the decision makers has a different set of weights. The independent set of the matroid that is chosen by the follower determines the payoff to both the leader and the follower according to their different weights. The leader can increase his payoff by changing the weights of the follower, thus influencing the follower’s decision, but he has to pay a penalty for this. We want to find an optimum strategy for the leader. This is a bilevel programming problem with continuous variables in the upper level and a parametric weighted matroid problem in the lower level. We analyze the structure of the lower level problem. We use this structure to develop local optimality criteria for the bilevel problem that can be verified in polynomial time.


Bilevel programming Combinatorial optimization Matroids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bard JF (1998) Practical bilevel optimization—algorithms and applications. Kluwer Academic, Dordrecht MATHGoogle Scholar
  2. Cook W, Cunningham W, Pulleyblank W, Schrijver A (1998) Combinatorial optimization. Wiley, New York MATHGoogle Scholar
  3. Dempe S (2002) Foundations of bilevel programming. Kluwer Academic, Dordrecht MATHGoogle Scholar
  4. Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52:333–359 MathSciNetMATHCrossRefGoogle Scholar
  5. Dempe S, Kalashnikov V, Ríos-Mercado RZ (2005) Discrete bilevel programming: application to a natural gas cash-out problem. Eur J Oper Res 166(2):469–488 MATHCrossRefGoogle Scholar
  6. Fanghänel D (2006a) Optimality criteria for bilevel programming problems using the radial subdifferential. In: Dempe S, Kalashnikov V (eds) Optimization with multivalued mappings: theory, application and algorithms. Springer, Berlin, pp 73–95 CrossRefGoogle Scholar
  7. Fanghänel D (2006b) Zwei Ebenen Optimierung mit diskreter unterer Ebene. Dissertation, TU Bergakademie Freiberg. Available via
  8. Fanghänel D, Dempe S (2009) Bilevel programming with discrete lower level. Optimization 58(8):1029–1047 MathSciNetMATHCrossRefGoogle Scholar
  9. Fujishige S (2005) Submodular functions and optimization, 2nd edn. Annals of Discrete Mathematics, vol 58. Elsevier, Amsterdam MATHGoogle Scholar
  10. Heuberger C (2004) Inverse combinatorial optimization: a survey on problems, methods, and constraints. J Comb Optim 8:329–361 MathSciNetMATHCrossRefGoogle Scholar
  11. Kalashnikov V, Ríos-Mercado RZ (2001) A penalty-function approach to a mixed integer bilevel programming problem. In: Zozaya C et al (eds) Proceedings of the 3rd international meeting on computer science, Aquascalientes, Mexico, vol 2, pp 1045–1054 Google Scholar
  12. Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin MATHGoogle Scholar
  13. Vicente LN, Savard G, Judice JJ (1994) Descent approaches for quadratic bilevel programming. J Optim Theory Appl 81:379–399 MathSciNetMATHCrossRefGoogle Scholar
  14. Vicente LN, Savard G, Judice JJ (1996) The discrete linear bilevel programming problem. J Optim Theory Appl 89:597–614 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.FB Elektrotechnik/InformatikUniversität KasselKasselGermany

Personalised recommendations