Solving Bilevel Combinatorial Optimization as Bilinear Min-max Optimization via a Branch-and-cut Algorithm

In this paper, we propose a generic branch-and-cut algorithm for a special class of bi-level combinatorial optimization problems. Namely, we study such problems that can be reformulated as bilinear min-max combinatorial optimization problems. We show that the reformulation can be efficiently solved by a branch-and-cut algorithm whose cuts represent the inner maximization feasibility set. The algorithm generalizes a method developed recently by Roboredo and Pessoa for two particular bi-level problems. In addition, we apply the algorithm on the r-Interdiction Median Problem with Fortification (RIMF). The RIMF considers sets of facilities and customers where each customer is served by the nearest facility unless the facility is interdicted and not fortified. The objective is to minimize the total weighted distance by fortifying q facilities knowing that r facilities will be interdicted. Our numerical results show that our method is more suitable on large instances than the best exact method found in literature.