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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 378568, 10 pages
http://dx.doi.org/10.1155/2013/378568
Research Article

A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems

1School of Computer Science and Technology, Xidian University, Xi’an 710071, China
2School of Science, Xidian University, Xi’an 710071, China

Received 26 March 2013; Accepted 14 July 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Aihong Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. F. Bard, “Coordination of a multidivisional organization through two levels of management,” Omega, vol. 11, no. 5, pp. 457–468, 1983. View at Google Scholar
  2. B. Colson, P. Marcotte, and G. Savard, “An overview of bilevel optimization,” Annals of Operations Research, vol. 153, pp. 235–256, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Dempe, “Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints,” Optimization, vol. 52, no. 3, pp. 333–359, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. N. Vicente and P. H. Calamai, “Bilevel and multilevel programming: a bibliography review,” Journal of Global Optimization, vol. 5, no. 3, pp. 291–306, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, vol. 30 of Nonconvex Optimization and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1998. View at MathSciNet
  6. S. Dempe, Foundations of Bilevel Programming, vol. 61 of Nonconvex Optimization and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2002. View at MathSciNet
  7. R. Mathieu, L. Pittard, and G. Anandalingam, “Genetic algorithm based approach to bi-level linear programming,” RAIRO Recherche Opérationnelle, vol. 28, no. 1, pp. 1–21, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. K.-M. Lan, U.-P. Wen, H.-S. Shih, and E. S. Lee, “A hybrid neural network approach to bilevel programming problems,” Applied Mathematics Letters, vol. 20, no. 8, pp. 880–884, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. I. Calvete, C. Galé, and P. M. Mateo, “A new approach for solving linear bilevel problems using genetic algorithms,” European Journal of Operational Research, vol. 188, no. 1, pp. 14–28, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. J. Kuo and C. C. Huang, “Application of particle swarm optimization algorithm for solving bi-level linear programming problem,” Computers & Mathematics with Applications, vol. 58, no. 4, pp. 678–685, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. J. Kuo and Y. S. Han, “A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem—a case study on supply chain model,” Applied Mathematical Modelling, vol. 35, no. 8, pp. 3905–3917, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y. P. Wang, Y. C. Jiao, and H. Li, “An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handing scheme,” IEEE Transactions on Systems, Man, and Cybernetics, Part C, vol. 35, no. 2, pp. 221–232, 2005. View at Google Scholar
  13. K. Deb and S. Sinha, “An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm,” Evolutionary Computation, vol. 18, no. 3, pp. 403–449, 2010. View at Google Scholar
  14. Y. Jiang, X. Li, C. Huang, and X. Wu, “Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4332–4339, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  15. H. Li and L. Fang, “An evolutionary algorithm for solving bilevel programming problems using duality conditions,” Mathematical Problems in Engineering, vol. 2012, Article ID 471952, 14 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. P. Wan, G. M. Wang, and B. Sun, “A hybrid intelligent algorithm by combining particle swarm optimization with chaos searching technique for solving nonlinear bilevel programming problems,” Swarm and Evolutionary Computation, vol. 8, pp. 26–32, 2013. View at Google Scholar
  17. P. Larranaga and J. A. Lozano, Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation, Kluwer Academic, 2002.
  18. P. Pelikan, Hierachical Bayesian Optimization Algorithm: Toward a New Generation of Evolutionary Algorithms, Springer, 2005.
  19. H. Tuy, A. Migdalas, and N. T. Hoai-Phuong, “A novel approach to bilevel nonlinear programming,” Journal of Global Optimization, vol. 38, no. 4, pp. 527–554, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Wang, X. Wang, Z. Wan, and Y. Lv, “A globally convergent algorithm for a class of bilevel nonlinear programming problem,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 166–172, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. I. Calvete and C. Galé, “Bilevel multiplicative problems: a penalty approach to optimality and a cutting plane based algorithm,” Journal of Computational and Applied Mathematics, vol. 218, no. 2, pp. 259–269, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Z. Wan, G. Wang, and Y. Lv, “A dual-relax penalty function approach for solving nonlinear bilevel programming with linear lower level problem,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 652–660, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. Li and Y. P. Wang, “A hybrid genetic algorithm for nonlinear bilevel programming,” Journal of XiDian University, vol. 29, no. 6, pp. 840–843, 2002. View at Google Scholar
  24. H. C. Li and Y. P. Wang, “A mixed-encoding genetic algorithm for nonlinear bilevel programming problems,” in Proceedings of the International Joint Conference on INC, IMS and IDC, 2009.
  25. G. S. Qiu and J. F. Wang, “Penalty function of the nonlinear bilevel programming problem,” Journal of NanChang HangKong University, vol. 23, no. 3, pp. 61–64, 2009. View at Google Scholar
  26. H. I. Calvete and C. Galé, “A penalty method for solving bilevel linear fractional/linear programming problems,” Asia-Pacific Journal of Operational Research, vol. 21, no. 2, pp. 207–224, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. H. Mühlenbein and G. Paaß, “From recombination of genes to the estimation of distributions I. binary parameter,” Lecture Notes in Computer Science, vol. 1141, pp. 178–187, 1996. View at Google Scholar
  28. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Computer Journal, vol. 7, no. 4, pp. 308–313, 1965. View at Google Scholar
  29. K. T. Fang, “Uniform design: an application of number-theoretic methods to experimental designs,” Acta Mathematicae Applicatae Sinica, vol. 3, no. 4, pp. 363–372, 1980. View at Google Scholar · View at MathSciNet
  30. Y. Wang and K. T. Fang, “A note on uniform distribution and experimental design,” Kexue Tongbao, vol. 26, no. 6, pp. 485–489, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. Y. W. Leung and Y. P. Wang, “Multiobjective programming using uniform design and genetic algorithm,” IEEE Transactions on Systems, Man and Cybernetics C, vol. 30, no. 3, pp. 293–304, 2000. View at Google Scholar