The follower optimality cuts for mixed integer linear bilevel programming problems

We study linear bilevel programming problems, where (some of) the leader and the follower variables are restricted to be integer. A discussion on the relationships between the optimistic and the pessimistic setting is presented, providing necessary and sufficient conditions for them to be equivalent. A new class of inequalities, the follower optimality cuts, is introduced. They are used to derive a single-level non-compact reformulation of a bilevel problem, both for the optimistic and for the pessimistic case. The same is done for a family of known inequalities, the no-good cuts, and a polyhedral comparison of the related formulations is carried out. Finally, for both the optimistic and the pessimistic approach, we present a branch-and-cut algorithm and discuss computational results.