Double-Layered Hybrid Neural Network Approach for Solving Mixed Integer Quadratic Bilevel Problems

  • Shamshul Bahar Yaakob
  • Junzo Watada
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 68)

Abstract

In this paper we build a double-layered hybrid neural network method to solve mixed integer quadratic bilevel programming problems. Bilevel programming problems arise when one optimization problem, the upper problem, is constrained by another optimization, the lower problem. In this paper, mixed integer quadratic bilevel programming problem is transformed into a double-layered hybrid neural network. We propose an efficient method for solving bilevel programming problems which employs a double-layered hybrid neural network. A two-layered neural network is formulate by comprising a Hopfield network, genetic algorithm, and a Boltzmann machine in order to effectively and efficiently select the limited number of units from those available. The Hopfield network and genetic algorithm are employed in the upper layer to select the limited number of units, and the Boltzmann machine is employed in the lower layer to decide the optimal solution/units from the limited number of units selected by the upper layer.The proposed method leads the mixed integer quadratic bilevel programming problem to a global optimal solution. To illustrate this approach, several numerical examples are solved and compared.

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References

  1. 1.
    Bard, J.: Practical bilevel optimization: Algorithm and applications. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  2. 2.
    Dempe, S.: Foundation of bilevel programming. Kluwer Academic Publishers, London (2002)Google Scholar
  3. 3.
    Ben-Ayed, O., Blair, O.: Computational difficulty of bilevel linear programming. Operations Research 38(3), 556–560 (1990)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Anandalingam, G., Friesz, T.L.: Hierarchical optimization: An introduction. Annals of Operations Research 34(1), 1–11 (1992)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bard, J.F.: Coordination of a multidivisional organization through two levels of management. Omega 11(5), 457–468 (1983)CrossRefGoogle Scholar
  6. 6.
    Carrion., M., Arroyo, J.M., Conejo, A.J.: A bilevel stochastic programming approach for retailer futures market trading. IEEE Transactions on Power Systems 24(3), 1446–1456 (2009)CrossRefGoogle Scholar
  7. 7.
    Shimizu, K., Ishizuka, Y., Bard, J.F.: Nondifferentiable and two-level mathematical programming. Kluwer Academic, Boston (1997)MATHGoogle Scholar
  8. 8.
    Colson, B., Marcotte, P., Savard, G.: Bilevel programming: A survey. International Journal of Operations Research 3(2), 87–107 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Bard, J.F., Moore, J.T.: A branch-and-bound algorithm for the bilevel programming problem. SIAM Journal on Scientific and Statistical Computing 11(2), 281–292 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Al-Khayyal, F., Horst, R., Paradalos, P.: Global optimization on concave functions subject to quadratic constraints: an application in nonlinear bilevel programming. Annals of Operations Research 34(1), 125–147 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Edmunds, T., Bard, J.F.: An algorithm for the mixed-integer nonlinear bilevel programming problem. Annals of Operations Research 34(1), 149–162 (1992)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Savard, G., Gauvin, J.: The steepest descent direction for the nonlinear bilevel programming problem. Operations Research Letters 15(5), 265–272 (1994)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vicente, L., Savarg, G., Judice, J.: Descent approaches for quadratic bilevel programming. Journal of Optimization Theory and Applications 81(2), 379–399 (1994)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hejazi, S.R., Memariani, A., Jahanshanloo, G., Sepehri, M.M.: Bilevel programming solution by genetic algorithms. In: Proceeding of the First National Industrial Engineering Conference (2001)Google Scholar
  15. 15.
    Wu, C.P.: Hybrid technique for global optimization of hierarchical systems. In: Proceedings of the IEEE International Conference on Systems, Man and Cybernatics, vol. 3, pp. 1706–1711 (1996)Google Scholar
  16. 16.
    Yin, Y.: Genetic algorithms based approach for bilevel programming models. Journal of Transportion Engineering 126(2), 115–120 (2000)CrossRefGoogle Scholar
  17. 17.
    Shih, H.S., Wen, U.P.: A neural network approach for multi-objective and multilevel programming problems. Journal of Computer and Mathematics with Applications 48(1-2), 95–108 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lan, K.M., Wen, U.P.: A hybrid neural network approach to bilevel programming problems. Applied Mathematics Letters 20(8), 880–884 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Markowitz, H.: Mean-variance analysis in portfolio choice and capital markets. Blackwell, Malden (1987)MATHGoogle Scholar
  20. 20.
    Ackley, D.H., Hinton, G.E., Sejnowski, T.J.: A learning algorithm for Bolzmann machine. Cognitive Science 9(1), 147–169 (1985)CrossRefGoogle Scholar
  21. 21.
    Watada, J., Oda, K.: Formulation of a two-layered Boltzmann machine for portfolio selection. International Journal Fuzzy Systems 2(1), 39–44 (2000)Google Scholar
  22. 22.
    Muu, L.D., Quy, N.V.: A global optimization method for solving convex quadratic bilevel programming problems. Journal of Global Optimization 26(2), 199–219 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shamshul Bahar Yaakob
    • 1
    • 2
  • Junzo Watada
    • 1
  1. 1.Graduate School of IPSWaseda UniversityFukuokaJapan
  2. 2.School of Electrical Systems EngineeringUniversiti Malaysia PerlisJejawiMalaysia

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