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Journal of Applied Mathematics
Volume 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.

Abstract

We propose a hybrid algorithm based on estimation of distribution algorithm (EDA) and Nelder-Mead simplex method (NM) to solve a class of nonlinear bilevel programming problems where the follower’s problem is linear with respect to the lower level variable. The bilevel programming is an NP-hard optimization problem, for which EDA-NM is applied as a new tool aiming at obtaining global optimal solutions of such a problem. In fact, EDA-NM is very easy to be implementedsince it does not require gradients information. Moreover, the hybrid algorithm intends to produce faster and more accurate convergence. In the proposed approach, for fixed upper level variable, we make use of the optimality conditions of linear programming to deal with the follower’s problem and obtain its optimal solution. Further, the leader’s objective function is taken as the fitness function. Based on these schemes, the hybrid algorithm is designed by combining EDA with NM. To verify the performance of EDA-NM, simulations on some test problems are made, and the results demonstrate that the proposed algorithm has a better performance than the compared algorithms. Finally, the proposed approach is used to solve a practical example about pollution charges problem.