A Polynomial-Time Algorithm to Check Closedness of Simple Second Order Mixed-Integer Sets

Abstract

Let $\textbf{L}^{m}$ be the Lorentz cone in ℝ^m. Given $A \in {\mathbb{Q}}^{m \times n_1}$, $B \in {\mathbb{Q}}^{m \times n_2}$ and b ∈ ℚ^m, a simple second order conic mixed-integer set (SOCMIS) is a set of the form $\{(x,y)\in {\mathbb{Z}}^{n_1} \times {\mathbb{R}}^{n_2}\,|\,\ Ax +By -b \in \textbf{L}^{m}\}$. We show that there exists a polynomial-time algorithm to check the closedness of the convex hull of simple SOCMISs. Moreover, in the special case of pure integer problems, we present sufficient conditions, that can be checked in polynomial-time, to verify the closedness of intersection of simple SOCMISs.