A matheuristic for the discrete bilevel problem with multiple objectives at the lower level

In this paper, we solve a discrete bilevel problem with multiple objectives at the lower level and constraints at the upper level coupling variables of both levels. In the case of a multiobjective lower level, we deal with a set of Pareto-efficient solutions rather than a single optimal lower level solution. To calculate the upper level objective function value, we need to select one solution out of a potentially large set of efficient lower level solutions. To avoid the enumeration of the whole set of Pareto solutions, we formulate an auxiliary mixed integer linear programming problem with a large number of constraints. We propose an iterative exact method to solve it. To find a near-optimal upper level solution, we apply a metaheuristic. The method is tested on the discrete (r vertical bar p)-centroid problem with multiple objectives at the lower level.