Advertisement

A Neurodynamic Optimization Approach to Bilevel Linear Programming

  • Sitian Qin
  • Xinyi Le
  • Jun Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9377)

Abstract

This paper presents new results on neurodynamic optimization approach to solve bilevel linear programming problems (BLPPs) with linear inequality constraints. A sub-gradient recurrent neural network is proposed for solving the BLPPs. It is proved that the state convergence time period is finite and can be quantitatively estimated. Compared with existing recurrent neural networks for BLPPs, the proposed neural network does not have any design parameter and can solve the BLPPs in finite time. Some numerical examples are introduced to show the effectiveness of the proposed neural network.

Keywords

Bilevel linear programming problem sub-gradient recurrent neural network convergence in finite time 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, J.P., Cellina, A.: Differential inclusions. Springer, Berlin (1984)CrossRefGoogle Scholar
  2. 2.
    Calvete, H.I., Gale, C.: On linear bilevel problems with multiple objectives at the lower level. Omega 39(1), 33–40 (2011)CrossRefGoogle Scholar
  3. 3.
    Cheng, L., Hou, Z.G., Lin, Y., Tan, M., Zhang, W.C., Wu, F.: Recurrent neural network for non-smooth convex optimization problems with application to the identification of genetic regulatory networks. IEEE Trans. Neural Networks 22(5), 714–726 (2011)CrossRefGoogle Scholar
  4. 4.
    Cichocki, A., Unbehauen, R.: Neural networks for optimization and signal processing. Wiley (1993)Google Scholar
  5. 5.
    Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Annals of Operations Research 153(1), 235–256 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    He, X., Li, C., Huang, T., Li, C.: Neural network for solving convex quadratic bilevel programming problems. Neural Networks 51, 17–25 (2014)CrossRefGoogle Scholar
  7. 7.
    He, X., Li, C., Huang, T., Li, C., Huang, J.: A recurrent neural network for solving bilevel linear programming problem. IEEE Transactions on Neural Networks and Learning Systems 25(4), 824–830 (2014)CrossRefGoogle Scholar
  8. 8.
    Jiang, Y., Li, X., Huang, C., Wu, X.: An augmented lagrangian multiplier method based on a chks smoothing function for solving nonlinear bilevel programming problems. Knowledge-Based Systems 55, 9–14 (2014)CrossRefGoogle Scholar
  9. 9.
    Kennedy, M.P., Chua, L.O.: Neural networks for nonlinear programming. IEEE Trans. Circuits and Systems I: Regular Papers 35(5), 554–562 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kuo, R., Huang, C.: Application of particle swarm optimization algorithm for solving bi-level linear programming problem. Computers & Mathematics with Applications 58(4), 678–685 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lv, Y., Chen, Z., Wan, Z.: A neural network for solving a convex quadratic bilevel programming problem. Journal of Computational and Applied Mathematics 234(2), 505–511 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mersha, A.G., Dempe, S.: Feasible direction method for bilevel programming problem. Optimization 61(5), 597–616 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Muu, L.D., Quy, N.V.: A global optimization method for solving convex quadratic bilevel programming problems. J. Global Optimization 26(2), 199–219 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Qin, S., Xue, X.: Global exponential stability and global convergence in finite time of neural networks with discontinuous activations. Neural Processing Letters 29(3), 189–204 (2009)CrossRefGoogle Scholar
  15. 15.
    Qin, S., Xue, X.: Dynamical analysis of neural networks of subgradient system. IEEE Trans. Automat. Contr. 55(10), 2347–2352 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM Journal on Optimization 11(4), 918–936 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shih, H.S., Wen, U.P., Lee, S., Lan, K.M., Hsiao, H.C.: A neural network approach to multiobjective and multilevel programming problems. Computers & Mathematics with Applications 48(1-2), 95–108 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wan, Z., Mao, L., Wang, G.: Estimation of distribution algorithm for a class of nonlinear bilevel programming problems. Information Sciences 256, 184–196 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wen, U.P., Hsu, S.T.: Linear bi-level programming problems–a review. Journal of the Operational Research Society, 125–133 (1991)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

<SimplePara><Emphasis Type="Bold">Open Access</Emphasis> This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. </SimplePara> <SimplePara>The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.</SimplePara>

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyWeihaiChina
  2. 2.Department of Mechanical and Automation EngineeringThe Chinese University of Hong KongShatinHong Kong
  3. 3.School of Control Science and EngineeringDalian University of TechnologyDalianChina

Personalised recommendations