A Hybrid Genetic Algorithm for Solving Nonlinear Bilevel Programming Problems Based on the Simplex Method

In this paper, a hybrid genetic algorithm is proposed for solving nonlinear bilevel programming problems (BLPPs). In order to improve the feasibility of the individuals, for each fixed leader-level variable x, the follower's problem is solved to get its optimal solution y. Then, based on the simplex method, a new crossover operator is designed, in which the best individuals generated so far are employed to yield a good direction of evolvement. Furthermore, a penalty method is developed to deal with the leader-level constraints, in which the penalty parameter can be adjusted by considering the status of the individuals in the population. At last, when the follower's problem has more than one optimal solutions for a fixed x, a selection scheme is given by solving a constructed single-level programming problem. The simulation on 20 benchmark problems demonstrates the effectiveness of the proposed algorithm.