Intersection Cuts for Bilevel Optimization

We address a generic Mixed-Integer Bilevel Linear Program (MIBLP), i.e., a bilevel optimization problem where all objective functions and constraints are linear, and some/all variables are required to take integer values. Rather than proposing an ad-hoc method applicable only to specific cases, we describe a general-purpose MIBLP approach. We first propose necessary modifications needed to turn a standard branch-and-bound MILP solver into an exact and convergent MIBLP solver, also addressing MIBLP unboundedness and infeasibility. Contrarily to other approaches, in our convergent framework both leader and follower problems can be of mixed-integer type—provided that the leader variables influencing follower’s decisions are integer and bounded. We then introduce new classes of linear inequalities to be embedded in this branch-and-cut framework, some of which are intersection cuts based on feasible-free convex sets. We present a computational study on various classes of benchmark instances available from the literature, in which we demonstrate that our approach outperforms alternative state-of-the-art MIBLP methods.