In bilevel optimization there are two decision makers, commonly denoted as the leader and the follower, and decisions are made in a hierarchical manner: the leader makes the first move, and then the follower reacts optimally to the leader's action. It is assumed that the leader can anticipate the decisions of the follower, hence the leader optimization task is a nested optimization problem that takes into consideration the follower's response.In this talk we focus on new branch-and-cut (B&C) algorithms for dealing with mixed-integer bilevel linear programs (MIBLPs). We first address a general case in which intersection cuts are used to cut off infeasible solutions. We then focus on a subfamily of MIBLPs in which the leader and the follower share a set of items, and the leader can select some of the items to inhibit their usage by the follower. Interdiction Problems, Blocker Problems, Critical Node/Edge Detection Problems are some examples of optimization problems that satisfy the later condition. We show that, in case the follower subproblem satisfies monotonicity property, a family of "interdiction-cuts" can be derived resulting in a more efficient B&C scheme.These new B&C algorithms consistently outperform (often by a large margin) alternative state-of-the-art methods from the literature, including methods that exploit problem specific information for special instance classes.

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