A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems
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 opti...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/378568 |
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| author | Aihong Ren Yuping Wang Fei Jia |
| author_facet | Aihong Ren Yuping Wang Fei Jia |
| author_sort | Aihong Ren |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ef2883e5a2fa4fb5bc9a7a050b295924 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-ef2883e5a2fa4fb5bc9a7a050b2959242025-02-03T01:11:59ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/378568378568A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming ProblemsAihong Ren0Yuping Wang1Fei Jia2School of Computer Science and Technology, Xidian University, Xi’an 710071, ChinaSchool of Computer Science and Technology, Xidian University, Xi’an 710071, ChinaSchool of Science, Xidian University, Xi’an 710071, ChinaWe 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.http://dx.doi.org/10.1155/2013/378568 |
| spellingShingle | Aihong Ren Yuping Wang Fei Jia A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems Journal of Applied Mathematics |
| title | A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems |
| title_full | A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems |
| title_fullStr | A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems |
| title_full_unstemmed | A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems |
| title_short | A Hybrid Estimation of Distribution Algorithm and Nelder-Mead Simplex Method for Solving a Class of Nonlinear Bilevel Programming Problems |
| title_sort | hybrid estimation of distribution algorithm and nelder mead simplex method for solving a class of nonlinear bilevel programming problems |
| url | http://dx.doi.org/10.1155/2013/378568 |
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