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Facial Structure and Representation of Integer Hulls of Convex Sets

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7801))

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Abstract

For a convex set S, we study the facial structure of its integer hull, S ℤ. Crucial to our study is the decomposition of the integer hull into the convex hull of its extreme points, conv(ext(S ℤ)), and its recession cone. Although conv(ext(S ℤ)) might not be a polyhedron, or might not even be closed, we show that it shares several interesting properties with polyhedra: all faces are exposed, perfect, and isolated, and maximal faces are facets. We show that S ℤ has an infinite number of extreme points if and only if conv(ext(S ℤ)) has an infinite number of facets. Using these results, we provide a necessary and sufficient condition for semidefinite representability of conv(ext(S ℤ)).

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Authors and Affiliations

  1. Industrial Engineering and Operations Research, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Vishnu Narayanan

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  1. Vishnu Narayanan
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Editors and Affiliations

  1. Massachusetts Institute of Technology, 77 Massachusetts Ave., 02139, Cambridge, MA, USA

    Michel Goemans

  2. Universidad de Chile, Santiago, Chile

    José Correa

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Narayanan, V. (2013). Facial Structure and Representation of Integer Hulls of Convex Sets. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_26

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  • DOI: https://doi.org/10.1007/978-3-642-36694-9_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36693-2

  • Online ISBN: 978-3-642-36694-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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