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“Facet” separation with one linear program

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Abstract

Given polyhedron P and and a point \(x^*\), the separation problem for polyhedra asks to certify that \(x^*\in P\) and if not, to determine an inequality that is satisfied by P and violated by \(x^*\). This problem is repeatedly solved in cutting plane methods for Integer Programming and the quality of the violated inequality is an essential feature in the performance of such methods. In this paper we address the problem of finding efficiently an inequality that is violated by \(x^*\) and either defines an improper face or a facet of P. We show that, by solving a single linear program, one almost surely obtains such an improper face or facet.

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Acknowledgements

The authors gratefully acknowledge that this work started at the University of Bonn when they were participants in the Hausdorff Trimester Program on “Combinatorial Optimization”. They also thank three referees for their constructive comments.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universitá di Padova, Padua, Italy

    Michele Conforti

  2. CORE, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Laurence A. Wolsey

Authors
  1. Michele Conforti
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  2. Laurence A. Wolsey
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Correspondence to Laurence A. Wolsey.

Additional information

M. Conforti: Supported by Grants SID 2016 and PRIN 2016.

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Conforti, M., Wolsey, L.A. “Facet” separation with one linear program. Math. Program. 178, 361–380 (2019). https://doi.org/10.1007/s10107-018-1299-8

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  • DOI: https://doi.org/10.1007/s10107-018-1299-8

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