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Unique lifting of integer variables in minimal inequalities

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Abstract

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting (This result first appeared in the proceedings of IPCO 2010). The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Amitabh Basu

  2. Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Fortaleza, Brazil

    Manoel Campêlo

  3. Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121, Padua, Italy

    Michele Conforti

  4. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  5. London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK

    Giacomo Zambelli

Authors
  1. Amitabh Basu
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  2. Manoel Campêlo
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  3. Michele Conforti
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  4. Gérard Cornuéjols
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  5. Giacomo Zambelli
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Correspondence to Giacomo Zambelli.

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Basu, A., Campêlo, M., Conforti, M. et al. Unique lifting of integer variables in minimal inequalities. Math. Program. 141, 561–576 (2013). https://doi.org/10.1007/s10107-012-0560-9

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  • DOI: https://doi.org/10.1007/s10107-012-0560-9

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