A bilevel programming problem with maximization of a supermodular function in the lower level

In bilevel programming there are two decision makers, the leader and the follower, who act in a hierarchy. In this paper we deal with a bilevel problem where the follower maximizes a supermodular function. The payoff for the leader is given by the weighted set that is chosen by the follower. To increase his payoff the leader can increase the supermodular function of the follower by a modular one, thus influencing the follower’s decision, but he has to pay a penalty for this. We want to find an optimum strategy for the leader. This is a bilevel programming problem with continuous variables in the upper level and a parametric supermodular maximization problem in the lower level. We analyze the structure of the bilevel problem. This we use to provide an equivalent one-level combinatorial problem. Finally, we investigate the properties of the new problem.