A Hereditary Property of Cutting Plane Procedures

Abstract

Let $K'$ denote the closure of a convex set $K$ under a given cutting-plane procedure. The procedure satisfies the \emph{hereditary property} if $F' = K' \cap F$ for every face $F$ of $K$. The property underlies inductive proofs of finite-rank and the polyhedrality of closures, and it is an admissibility requirement in abstract frameworks for cutting-plane procedures. Yet, it has not been studied systematically across well-known cutting-plane procedures. We establish two sufficient conditions for the property. The first applies to procedures that can be expressed as closures under intersection cuts from a family of lattice-free convex sets, when a single family realizes the closure of $K$ and of each of its faces. It yields the hereditary property for the split, lift-and-project, Lovász--Schrijver, Sherali--Adams, and Lasserre closures, applied over general convex sets and for faces that need not be exposed. The second applies to procedures whose cuts are valid inequalities for Gomory's corner polyhedron, and it yields the hereditary property for the closure of Dantzig cuts derived from all bases. We complement these results with four classical procedures for which the hereditary property fails, namely the closure of Gomory's fractional cuts (derived from either all bases or feasible bases only), the mixed-integer Chvátal closure, the $+$-cut closure, and the closure of Dantzig cuts derived from feasible bases.