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Continuous knapsack sets with divisible capacities

Author

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  • WOLSEY, Laurence

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

  • YAMAN , Hand

    (Department of Industrial Engineering, Bilkent University, Turkey)

  • ,

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.

Suggested Citation

  • WOLSEY, Laurence & YAMAN , Hand & ,, 2013. "Continuous knapsack sets with divisible capacities," CORE Discussion Papers 2013063, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2013063
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    File URL: http://uclouvain.be/cps/ucl/doc/core/documents/coredp2013_63web.pdf
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    1. Förster, Manuel & Mauleon, Ana & Vannetelbosch, Vincent J., 2016. "Trust and manipulation in social networks," Network Science, Cambridge University Press, vol. 4(02), pages 216-243, June.
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    More about this item

    Keywords

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

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C26 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Instrumental Variables (IV) Estimation

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