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Pré-Publication, Document De Travail Année : 2023

Submodular maximization and its generalization through an intersection cut lens

Résumé

We study a mixed-integer set $S:={(x,t)∈{0,1}^n \times \mathbf{R}:f(x)≥t}$ 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 $\mathbf{R}^n$. We show that the epigraph of $F$ is an $S$-free set, and characterize maximal $S$-free sets including the epigraph. 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, 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.
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Dates et versions

hal-04304919 , version 1 (24-11-2023)

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Liding Xu, Leo Liberti. Submodular maximization and its generalization through an intersection cut lens. 2023. ⟨hal-04304919⟩
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