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Some Properties of Convex Hulls of Integer Points Contained in General Convex Sets

Santanu S. Dey (santanu.dey***at***isye.gatech.edu)
Diego A. Moran (dmoran***at***isye.gatech.edu)

Abstract: In this paper, we study properties of general closed convex sets that determine the closed-ness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special class of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present sufficient conditions for the convex hull of integer points in general convex sets to be polyhedron. These sufficient conditions generalize the sufficient conditions given in Meyer (1974). Under a simple technical condition, we show that these sufficient conditions are also necessary conditions for the convex hull of integer points contained in general convex sets to be polyhedra.

Keywords: Convex Integer Programming, Closed-ness, Polyhedrality

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Citation:

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Entry Submitted: 01/18/2011
Entry Accepted: 01/19/2011
Entry Last Modified: 01/23/2011

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