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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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
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