A polynomial-time algorithm to check closedness of simple second order mixed-integer sets

  • Authors:

  • Diego Alejandro Morán Ramírez;Santanu S. Dey

  • Affiliations:

  • Industrial and Systems Engineering, Georgia Institute of Technology;Industrial and Systems Engineering, Georgia Institute of Technology

  • Venue:
  • IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
  • Year:
  • 2013

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Abstract

Let $\textbf{L}^{m}$ be the Lorentz cone in ℝm. Given $A \in {\mathbb{Q}}^{m \times n_1}$, $B \in {\mathbb{Q}}^{m \times n_2}$ and b∈ℚm, 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.