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We study a mixed-integer set $S:=\{(x,t) \in \{0,1\}^n \times \mathbb{R}:f(x) \ge t\}$ arising in the submodular maximization problem, where $f$ is asubmodular function defined over $\{0,1\}^n$. We use intersection cuts totighten a polyhedral outer approximation of $S$. We construct a continuousextension $F$ of $f$, which is convex and defined over the entire space$\mathbb{R}^n$. We show that the epigraph of $F$ is an $S$-free set, andcharacterize maximal $S$-free sets including the epigraph. We propose a hybriddiscrete Newton algorithm to compute an intersection cut efficiently andexactly. Our results are generalized to the hypograph or the superlevel set ofa submodular-supermodular function, which is a model for discrete nonconvexity.A consequence of these results is intersection cuts for Boolean multilinearconstraints. We evaluate our techniques on max cut, pseudo Booleanmaximization, and Bayesian D-optimal design problems within a MIP solver.

Submodular maximization and its generalization through an intersection cut lens