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Continuous knapsack sets with divisible capacities

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  • WOLSEY, Laurence

    ()
    (Université catholique de Louvain, CORE, Belgium)

  • YAMAN , Hand

    (Department of Industrial Engineering, Bilkent University, Turkey)

  • ,
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    Abstract

    We study two continuous knapsack sets Y≥ and Y≤ with n integer, one unbounded continuous and m bounded continuous variables in either ≥ or ≤ form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and 2m polyhedra arising from a continuous knapsack set with a single unbounded continuous variable. The latter polyhedra are in turn completely described by an exponential family of partition inequalities. A polynomial size extended formulation is known in the ≥ case. We provide an extended formulation for the ≤ case. It follows that, given a specific objective function, optimization over both Y≥ and Y≤ can be carried out by solving a polynomial size linear program. A consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality.

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    Bibliographic Info

    Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2013063.

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    Date of creation: 11 Dec 2013
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    Handle: RePEc:cor:louvco:2013063

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    Keywords: continuous knapsack set; splittable flow arec set; divisible capacities; partition inequalities; convex hull;

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    1. Nigar Hashimzade & Jean Hindriks & Gareth D. Myles, 2006. "Solutions Manual to Accompany Intermediate Public Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582694, December.
    2. Manuel Förster & Ana Mauleon & Vincent Vannetelbosch, 2013. "Trust and Manipulation in Social Networks," Documents de travail du Centre d'Economie de la Sorbonne 13065, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. GAUTIER, Axel & POUDOU, Jean-Christophe, 2013. "Reforming the postal universal service," CORE Discussion Papers 2013024, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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