On finite generation and infinite convergence of generalized closures from the theory of cutting planes

Abstract

For convex sets $K$ and $L$ in ${\mathbb{R}}^d$ we define $R_L(K)$ to be the convex hull of all points belonging to $K$ but not to the interior of $L$. Cutting-plane methods from integer and mixed-integer optimization can be expressed in geometric terms using functionals $R_L$ with appropriately chosen sets $L$. We describe the geometric properties of $R_L(K)$ and characterize those $L$ for which $R_L$ maps polyhedra to polyhedra. For certain natural classes ${\mathcal{L}}$ of convex sets in ${\mathbb{R}}^d$ we consider the functional $R_{\mathcal{L}}$ given by $R_{\mathcal{L}}(K):= \bigcap_{L \in {\mathcal{L}}}R_L(K)$. The functional $R_{\mathcal{L}}$ can be used to define various types of closure operations considered in the theory of cutting planes (such as the Chv\'atal closure, the split closure as well as generalized split closures recently introduced by Andersen, Louveaux and Weismantel). We study conditions on ${\mathcal{L}}$ under which $R_{\mathcal{L}}$ maps rational polyhedra to rational polyhedra. We also describe the limit of the sequence of sets obtained by iterative application of $R_{\mathcal{L}}$ to $K$. A part of the presented material gives generalized formulations and unified proofs of several recent results obtained by various authors.