A real-binary coded genetic algorithm for solving nonlinear bilevel programming with nonconvex objective functions

This work deals with a class of nonlinear bilevel programming problems with nonconvex objective functions, in which the follower objective is a function of linear expression of all variables and the follower constraints are linear. For the leader functions, there are no any restrictions on the convexity as well as the differentiability. The distinguished feature of the problem is the nonconvexity of the follower objective function, which breaks through the barrier that the follower must be convex or concave in literature. First, for any fixed leader variable x, two linear programming are got from the follower and used to obtain the follower optimal solution y. In addition, in order to avoid solving directly the follower problem for each x, a real-binary encoding scheme is given which consists of x and the bases of two linear programming. Finally, a new crossover operator is designed based on the characteristics of the mixed encoding, and a novel genetic algorithm is proposed. The numerical results on 15 examples illustrate that the proposed algorithm is effective and stable.