Journal of Applied Mathematics

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

Aihong Ren, Yuping Wang, and Fei Jia

Full-text: Open access

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.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 378568, 10 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808089

Digital Object Identifier
doi:10.1155/2013/378568

Mathematical Reviews number (MathSciNet)
MR3090625

Citation

Ren, Aihong; Wang, Yuping; Jia, Fei. A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems. J. Appl. Math. 2013 (2013), Article ID 378568, 10 pages. doi:10.1155/2013/378568. https://projecteuclid.org/euclid.jam/1394808089


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