Title: An efficient memetic algorithm using approximation scheme for solving nonlinear integer bilevel programming problems

Authors: Yuhui Liu; Hecheng Li; Huafei Chen; Jing Huang

Addresses: School of Computer Science and Technology, Qinghai Normal University, Xining 810016, China; Department of Fundamental Courses, Qinghai University, Xining 810016, China ' School of Mathematics and Statistics, Qinghai Normal University, Xining 810016, China; Academy of Plateau Science and Sustainability, Xining 810016, China ' School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China; School of Mathematics and Statistics, Qinghai Normal University, Xining 810016, China ' School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China; School of Mathematics and Statistics, Qinghai Normal University, Xining 810016, China

Abstract: Nonlinear integer bilevel programming problems (NIBLPPs) are mathematical models with hierarchical structure, which are known as strongly NP-hard problems. In general, it is extremely hard to solve this kind of problem because they are always non-convex and non-differentiable, especially when integer constraints are involved. In this manuscript, based on a simplified branch and bound method as well as interpolation technique, a memetic algorithm is developed to solve NIBLPPs. Firstly, the leader's variable values are taken as individuals in populations, for each individual in the initial population, a simplified branch and bound method is adopted to obtain the follower's optimal solutions. Then, in order to reduce the computation cost in frequently solving the follower's problems for lots of offspring generated in evolution, the interpolation method is applied to approximate the solutions to the follower's problem for each individuals in populations. In addition, among these approximated points, only potential better points can be chosen to endure further optimisation procedure, so as to obtain precise optimal solutions to the follower's problems. The simulation results show that the proposed memetic algorithm is efficient in dealing with NIBLPPs.

Keywords: nonlinear integer bilevel programming problem; NIBLPP; memetic algorithm; branch and bound method; interpolation function; optimal solutions.

DOI: 10.1504/IJCSE.2020.110194

International Journal of Computational Science and Engineering, 2020 Vol.23 No.1, pp.63 - 73

Received: 25 Jul 2019
Accepted: 24 Feb 2020
Published online: 08 Oct 2020 *

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