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

摘要

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 implemented since 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, wemake 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 aremade, and the results demonstrate that the proposed algorithmhas a better performance than the compared algorithms. Finally, the proposed approach is used to solve a practical example about pollution charges problem.