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.
Mots clés
MINLP submodular maximization submodular-supermodular functions intersection cuts Boolean multilinear functions D-optimal design
MINLP
submodular maximization
submodular-supermodular functions
intersection cuts
Boolean multilinear functions
D-optimal design
Optimization and Control (math.OC)
FOS: Mathematics
Origine : Fichiers produits par l'(les) auteur(s)