AI 中文总结
该研究为割平面方法的遗传性质建立两个充分条件，证明split等多种闭包满足该性质，同时指出Gomory分数割等四类经典方法不满足该性质。
AI 中文摘要
设$K'$表示凸集$K$在给定割平面方法下的闭包。若对$K$的每个面$F$都有$F' = K' \bigcap F$，则该方法满足“遗传性质”。该性质是有限秩和闭包多面体性的归纳证明基础，也是割平面方法抽象框架中的一个可容许性要求，但此前尚未在知名割平面方法中被系统研究。我们为该性质建立了两个充分条件：第一个适用于可表示为一族无格凸集的交集割闭包的方法，当单一族实现$K$及其每个面的闭包时，该方法对一般凸集应用的split、lift-and-project、Lovász--Schrijver、Sherali--Adams和Lasserre闭包，以及非暴露面均满足遗传性质；第二个适用于割为Gomory角多面体有效不等式的方法，该方法对所有基导出的Dantzig割闭包满足遗传性质。我们补充了四个不满足该遗传性质的经典方法：Gomory分数割闭包（来自所有基或仅可行基）、混合整数Chvátal闭包、$+$-割闭包，以及来自可行基的Dantzig割闭包。
英文摘要
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.
Comments29 pages, 1 figure
