Branch-and-Cut Method for Solving the Integer Linear Multiplicative Bilevel Problem

Abstract

We study a class of integer bilevel problems, the so-called Integer Linear Multiplicative Bilevel Problem, ILMBP, where the constraints are linear and both the upper level problem and the lower level problem are integer linear multiplicative problems. We assume that the k-linear factors of the upper level problem and the l-linear factors of the lower level problem to be non-negative. In this paper, we propose an algorithm based on branch-and-cut method for solving the Integer Linear Multiplicative Bilevel Problem. First, the continuous upper level linear multiplicative problem is solved, then the optimal integer solution obtained is tested for the optimality of the main problem by solving the lower level linear multiplicative problem. If this solution is not optimal for the problem ILMBP, a cut is added to the upper level problem and a new best integer solution is determined. The integer solution is generated by the branching process well known in the branch and bound procedure. Following, the presentation and validation of the algorithm, an example is provided to better visualize the algorithm.