An Evolutionary Algorithm for Nonlinear Bilevel Programming Problems Based on a New Penalty Method

Abstract

A class of nonlinear bilevel programming problems (BLPPs) are discussed in this paper, in which the follower's problem is linear. Based on the duality theory and a new penalty method, an evolutionary algorithm is proposed for solving this class of nonlinear bilevel programming problems. At first, for the leader's variables x, the solution functions y(x) of the follower's problem are gotten by using the primal-dual relationship of the follower's programming, and a penalty approach is given totransform the BLPP into a single-level unconstrained problem. Then, a new crossover operator is designed, in which some better individuals generated so far are employed to yield a good direction of evolvement. At last,in order to improve the efficiency of the proposed algorithm, a mutation operator is given based on an exponential distribution, which can make the mutation offspring generated in the neighborhoods of the better points. The simulation on 15 benchmark problems demonstrates the effectiveness and efficiency of the proposed algorithm.