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The ChvátalGomory Closure of a Strictly Convex Body
"... In this paper, we prove that the Ch´vatalGomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver[29] which shows that the Ch´vatalGomory closure of a rational polyhedron is a polyhedron. Key ..."
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In this paper, we prove that the Ch´vatalGomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver[29] which shows that the Ch´vatalGomory closure of a rational polyhedron is a polyhedron. Key words: nonlinear integer programming; cutting planes; Ch´vatalGomory closure
The ChvátalGomory Closure of an Ellipsoid is a Polyhedron
"... It is wellknow that the ChvátalGomory (CG) closure of a rational polyhedron is a rational polyhedron. In this paper, we show that the CG closure of a bounded fulldimensional ellipsoid, described by rational data, is a rational polytope. To the best of our knowledge, this is the first extension ..."
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Cited by 6 (3 self)
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It is wellknow that the ChvátalGomory (CG) closure of a rational polyhedron is a rational polyhedron. In this paper, we show that the CG closure of a bounded fulldimensional 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 nonpolyhedral set. A key feature of the proof is to verify that all nonintegral 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.