Closedness of Integer Hulls of Simple Conic Sets

Abstract

Let ${{\textbf{C}}}$ be a full-dimensional pointed closed convex cone in ${\mathbb{R}}^m$ obtained by taking the conic hull of a strictly convex set. Given $A \in {\mathbb{Q}} ^{m \times n_1}$, $B \in \mathbb{Q}^{m \times n_2}$, and $b \in {\mathbb{Q}}^m$, a simple conic mixed-integer set (SCMIS) is a set of the form $\{(x,y)\in {\mathbb{Z}}^{n_1} \times {\mathbb{R}}^{n_2}\,|\, Ax +By -b \in {{\textbf{C}}}\}$. In this paper, we give a complete characterization of the closedness of convex hulls of SCMISs. Under certain technical conditions on the cone ${{\textbf{C}}}$, we show that the closedness characterization can be used to construct a polynomial-time algorithm to check the closedness of convex hulls of SCMISs. Moreover, we also show that the Lorentz cone satisfies these technical conditions. In the special case of pure integer problems, we present sufficient conditions, which can be checked in polynomial time, to verify the closedness of intersection of SCMISs.