Submodular maximization and its generalization through an intersection cut lens

We study a mixed-integer set 𝒮:={(x,t) ∈{0,1}^n ×ℝ: f(x) ≥ t} arising in the submodular maximization problem, where f is a submodular function defined over {0,1}^n . We use intersection cuts to tighten a polyhedral outer approximation of 𝒮 . We construct a continuous extension _f of f, which is convex and defined over the entire space ℝ^n . We show that the epigraph epi (_f) of _f is an 𝒮 -free set, and characterize maximal 𝒮 -free sets containing epi (_f) . We propose a hybrid discrete Newton algorithm to compute an intersection cut efficiently and exactly. Our results are generalized to the hypograph or the superlevel set of a submodular-supermodular function over the Boolean hypercube, which is a model for discrete nonconvexity. A consequence of these results is intersection cuts for Boolean multilinear constraints. We evaluate our techniques on max cut, pseudo Boolean maximization, and Bayesian D-optimal design problems within a MIP solver.