A Hybrid Genetic Algorithm for Solving a Class of Nonlinear Bilevel Programming Problems

  • Hecheng Li
  • Yuping Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4247)


In this paper, a special nonlinear bilevel programming problem (BLPP), in which the follower’s problem is a convex quadratic programming in y, is transformed into an equivalent single-level programming problem by using Karush-Kuhn-Tucker(K-K-T) condition. To solve the equivalent problem effectively, firstly, a genetic algorithm is incorporated with Lemke algorithm. For x fixed, the optimal solution y of the follower’s problem can be obtained by Lemke algorithm, then (x,y) is a feasible or approximately feasible solution of the transformed problem and considered as a point in the population; secondly, based on the best individuals in the population, a special crossover operator is designed to generate high quality individuals; finally, a new hybrid genetic algorithm is proposed for solving this class of bilevel programming problems. The simulation on 20 benchmark problems demonstrates the effectiveness of the proposed algorithm.


Crossover Operator Benchmark Problem Hybrid Genetic Algorithm Bilevel Programming Bilevel Programming Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Von Stackelberg, H.: The Theory of the Market Economy. William Hodge, Lond, UK (1952)Google Scholar
  2. 2.
    Marcotte, P.: Network optimization with continuous parameters. Trans. Sci. 17, 181–197 (1983)CrossRefGoogle Scholar
  3. 3.
    Suh, S., Kim, T.: Solving nonlinear bilevel programming models of equilibrium network design problem: A comparative review. Ann. Oper. Res. 34, 203–218 (1992)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Milier, T., Friesz, T., Robin, R.: Heuristic algorithms for delivered price spatially competitive net work facility location problems. Ann. Oper. Res. 34, 177–202 (1992)CrossRefGoogle Scholar
  5. 5.
    Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194–1217 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Amouzegar, M.A., Moshirvaziri, K.: A penalty method for linear bilevel programming problems. In: Migdalas, A., Pardalos, M., Varbrand, P. (eds.) Multilevel optimization: Algorithms, complexity and Applications, ch. 11, Kluwer, Norwell (1997)Google Scholar
  7. 7.
    Bard, J., Moore, J.: A branch and bound algorithm for the bilevel programming problem. SIAM J. Sci. Stat. Comput. 11(5), 281–292 (1990)MATHMathSciNetGoogle Scholar
  8. 8.
    Candler, W., Townsley, R.: A linear bi-level programming problem. Comput. Oper. Res. 9(1), 59–76 (1982)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Amouzegar, M.A.: A global optimization method for nonlinear bilevel programming problems. IEEE Trans. Syst., Man, Cybern. B, Cybern. 29(6), 771–777 (1999)CrossRefGoogle Scholar
  10. 10.
    Vicente, L., Savard, G., Judice, J.: Descent approach for quadratic bilevel programming. J. Optim. Theory Appl. 81(2), 379–399 (1994)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Al-Khayyal, F., Horst, R., Pardalos, P.: Global optimization of concave function subject to quadratic constraints: An application in nonlinear bilevel programming. Ann. Oper. Res. 34, 125–147 (1992)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Oduguwa, V., Roy, R.: Bi-level optimization using genetic algorithm. In: Proc. IEEE Int. Conf. Artificial Intelligence Systems, pp. 123–128 (2002)Google Scholar
  13. 13.
    Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, Chichester (1979)MATHGoogle Scholar
  14. 14.
    Bard, J.F.: Practical Bilevel Optimization. Kluwer, Norwell (1998)MATHGoogle Scholar
  15. 15.
    Outrata, J.V.: On the numerical solution of a class of Stackelberg problem. Zeitschrift Fur Operational Reseach 34, 255–278 (1990)MATHMathSciNetGoogle Scholar
  16. 16.
    Shimizu, K., Aiyoshi, E.: A new computational method for Syackelberg and min-max problems by use of a penalty method. IEEE Trans. Autom. Control AC-26(2), 460–466 (1981)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Aiyoshi, E., Shimuzu, K.: A solution method for the static constrained Stackelberg problem via penalty method. IEEE Trans. Autom. Control AC-29(12), 1111–1114 (1984)CrossRefGoogle Scholar
  18. 18.
    Bard, J.F.: Covex two-level optimization. Math. Programming 40, 15–27 (1988)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hecheng Li
    • 1
  • Yuping Wang
    • 2
  1. 1.Department of Mathematics ScienceXidian UniversityXi’anChina
  2. 2.School of Computer Science and TechnologyXidian UniversityXi’anChina

Personalised recommendations