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MIR closures of polyhedral sets

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An Erratum to this article was published on 19 November 2009

Abstract

We study the mixed-integer rounding (MIR) closures of polyhedral sets. The MIR closure of a polyhedral set is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a given mixed-integer set. We linearize the objective using additional variables to produce a linear MIP model that solves the separation problem exactly. Using a subset of these additional variables yields an MIP model which solves the separation problem approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields an alternative proof of the result of Cook et al. (1990) that the split closure of a polyhedral set is again a polyhedron. We also discuss a heuristic to obtain MIR cuts based on our approximate separation model, and present some computational results.

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Authors and Affiliations

  1. IBM, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    Sanjeeb Dash & Oktay Günlük

  2. DEIS, University of Bologna, viale Risorgimento 2, 40136, Bologna, Italy

    Andrea Lodi

Authors
  1. Sanjeeb Dash
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  2. Oktay Günlük
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  3. Andrea Lodi
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Corresponding author

Correspondence to Sanjeeb Dash.

Additional information

Andrea Lodi was supported in part by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract n. FP6-021235-2).

An erratum to this article can be found at http://dx.doi.org/10.1007/s10107-009-0328-z

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Dash, S., Günlük, O. & Lodi, A. MIR closures of polyhedral sets. Math. Program. 121, 33–60 (2010). https://doi.org/10.1007/s10107-008-0225-x

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