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International Conference on Integer Programming and Combinatorial Optimization

IPCO 2010: Integer Programming and Combinatorial Optimization pp 327–340Cite as

The Chvátal-Gomory Closure of an Ellipsoid Is a Polyhedron

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6080))

Abstract

It is well-know that the Chvátal-Gomory (CG) closure of a rational polyhedron is a rational polyhedron. In this paper, we show that the CG closure of a bounded full-dimensional ellipsoid, described by rational data, is a rational polytope. To the best of our knowledge, this is the first extension of the polyhedrality of the CG closure to a non-polyhedral set. A key feature of the proof is to verify that all non-integral points on the boundary of ellipsoids can be separated by some CG cut. Given a point u on the boundary of an ellipsoid that cannot be trivially separated using the CG cut parallel to its supporting hyperplane, the proof constructs a sequence of CG cuts that eventually separates u. The polyhedrality of the CG closure is established using this separation result and a compactness argument. The proof also establishes some sufficient conditions for the polyhedrality result for general compact convex sets.

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Author information

Authors and Affiliations

  1. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, USA

    Santanu S. Dey

  2. Business Analytics and Mathematical Sciences Department, IBM T. J. Watson Research Center, Yorktown Heights, NY, USA

    Juan Pablo Vielma

  3. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Juan Pablo Vielma

Authors
  1. Santanu S. Dey
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  2. Juan Pablo Vielma
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, École Polytechnique Féderale de Lausanne, 1015, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. McGill University, 805 Sherbrooke West, H3A 2K6, Montreal, Quebec, Canada

    F. Bruce Shepherd

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Dey, S.S., Vielma, J.P. (2010). The Chvátal-Gomory Closure of an Ellipsoid Is a Polyhedron. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_25

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  • DOI: https://doi.org/10.1007/978-3-642-13036-6_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13035-9

  • Online ISBN: 978-3-642-13036-6

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