A Genetic Algorithm Based on Duality for Linear Bilevel Programming Problems

Linear bilevel programming problem, as a NP-hard problem, is the linear version of bilevel programming, in this paper we design an efficient algorithm for solving this kind of problems by combining genetic algorithm with enumeration procedure of extreme points. First, based on the duality principle, the follower problem is replaced by the prime-dual conditions, and the original problem is transformed into an equivalent single-level programming in which all functions are linear except for one constraint. Then, the bases of the duality problem are considered as individuals. For each selected individual (base), the nonlinear constraint can be simplified to linear one, and some constraints can also be removed from the transformed single-level problem. Hence, the single-level problem is converted into a linear programming. At last, the linear programming is solved and the objective value is taken as the fitness of the individual. The distinguished feature of the algorithm is that the bases of follower´s duality problem are searched instead of taking all feasible points into account, which makes the search space smaller. In order to illustrate the efficiency of the algorithm, 4 problems selected from literature are solved, and the results show that the proposed algorithm is efficient and robust.