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On the Quantile Cut Closure of Chance-Constrained Problems

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Integer Programming and Combinatorial Optimization (IPCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9682))

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Abstract

A chance constrained problem involves a set of scenario constraints from which a small subset can be violated. Existing works typically consider a mixed integer programming (MIP) formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated. A variety of cutting plane approaches for this MIP formulation have been developed. In this paper we consider a family of cuts for chance constrained problems in the original space rather than those in the extended space of the MIP reformulation. These cuts, known as quantile cuts, can be viewed as a projection of the well known family of mixing inequalities for the MIP reformulation, onto the original problem space. We show the following results regarding quantile cuts: (i) the closure of all quantile cuts is a polyhedral set; (ii) separation of quantile cuts is in general NP-hard; (iii) successive application of quantile cut closures achieves the convex hull of the chance constrained problem in the limit; and (iv) in the pure integer setting this convergence is finite.

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Weijun Xie & Shabbir Ahmed

Authors
  1. Weijun Xie
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  2. Shabbir Ahmed
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Correspondence to Shabbir Ahmed .

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Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Quentin Louveaux

  2. Technische Universität Berlin, Berlin, Germany

    Martin Skutella

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Xie, W., Ahmed, S. (2016). On the Quantile Cut Closure of Chance-Constrained Problems. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_33

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  • DOI: https://doi.org/10.1007/978-3-319-33461-5_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33460-8

  • Online ISBN: 978-3-319-33461-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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