An Interpolation Based Genetic Algorithm for Solving Nonlinear Bilevel Programming Problems

Nonlinear bilevel programming problems are hierarchical optimization problems.For each fixed value of leader′s variables the existing algorithms are required to solve the follower′s optimization problem to obtain a feasible point for the whole problem,which results in a large amount of computation.Note that in the existing research works,the condition that the follower′soptimal solution is unique for each leader′s variable value is usually adopted.This condition means that each follower′s variable can be seen as a function of the leader′s variables although this function is unknown.Based on this observation and to avoid solving the follower′s problem frequently,a different skill from that of the existing works is used to tackle this difficulty,i.e.,the interpolation functions are adopted to approximate these unknown functions.First,the values of the interpolation function are gotten by solving the follower′s problem for some given leader′s variable values(i.e.,the interpolation points),and the interpolation polynomials(functions) are calculated by using these interpolation points.Then,the follower′s variables can be replaced by the corresponding interpolation polynomials in the leader′s problem.As a result,the original nonlinear bilevel programming can be approximated by a single-level programming.Finally,a specifically designed genetic algorithm is proposed for the single-level programming,and the interpolation points and the corresponding interpolation functions are adaptively modified and updated during the evolution so that the optimal solutions of the single-level programming can well approach to those of original nonlinear bilevel programming.Moreover,the computation amount will be decreased.The simulations on 25 test problems indicate the proposed algorithm can find the best solutions with a relatively small amount of computation for these test problems.