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On unions and dominants of polytopes

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  • BALAS, Egon
  • BOCKMAYR, Alexander
  • PISARUK, Nicolai
  • WOLSEY, Laurence

Abstract

A well-known result on unions of polyhedra in the same space gives an extended formulation whose projection is the convex hull of the union. Here in contrast we study the unions of polytopes in different spaces, giving a complete description of the convex hull without additional variables. When the underlying polytopes are monotone, the facets are described explicitly, generalizing results of Hong and Hooker on cardinality rules, and an efficient separation algorithm is proposed. These results are based on an explicit representation of the dominant of a polytope, and an algorithm for the separation problem for the dominant. For non-monotone polytopes, both the dominant and the union are characterized. We also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of Conforti and Laurent.
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Suggested Citation

  • BALAS, Egon & BOCKMAYR, Alexander & PISARUK, Nicolai & WOLSEY, Laurence, 2004. "On unions and dominants of polytopes," LIDAM Reprints CORE 1684, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:1684
    DOI: 10.1007/s10107-003-0432-4
    Note: In : Mathematical Programming, Serie A, 99, 223-239, 2004
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    Cited by:

    1. L. Foulds & B. Toklu & J. Wilson, 2009. "Modelling either-or relations in integer programming," Annals of Operations Research, Springer, vol. 166(1), pages 203-222, February.
    2. Dan A. Iancu & Marek Petrik & Dharmashankar Subramanian, 2015. "Tight Approximations of Dynamic Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 655-682, March.
    3. Ioannis Mourtos, 2016. "Cardinality constraints and systems of restricted representatives," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1061-1089, April.

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