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The strength of multi-row models

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Abstract

We develop a method for computing facet-defining valid inequalities for any mixed-integer set \(P_J\). While our practical implementation does not return only facet-defining inequalities, it is able to find a separating cut whenever one exists. The separator is not comparable in speed with the specific cutting-plane generators used in branch-and-cut solvers, but it is general-purpose. We can thus use it to compute cuts derived from any reasonably small relaxation \(P_J\) of a general mixed-integer problem, even when there exists no specific implementation for computing cuts with \(P_J\). Exploiting this, we evaluate, from a computational perspective, the usefulness of cuts derived from several types of multi-row relaxations. In particular, we present results with four different strengthenings of the two-row intersection cut model, and multi-row models with up to fifteen rows. We conclude that only fully-strengthened two-row cuts seem to offer a significant advantage over two-row intersection cuts. Our results also indicate that the improvement obtained by going from models with very few rows to models with up to fifteen rows may not be worth the increased computing cost.

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Acknowledgments

We are grateful to three anonymous referees for their valuable comments.

Author information

Authors and Affiliations

  1. Montefiore Institute, University of Liège, Liège, Belgium

    Quentin Louveaux

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Laurent Poirrier

  3. Department of Information Engineering, University of Padova, Padua, Italy

    Domenico Salvagnin

Authors
  1. Quentin Louveaux
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  2. Laurent Poirrier
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  3. Domenico Salvagnin
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Corresponding author

Correspondence to Laurent Poirrier.

Additional information

D. Salvagnin was supported by the University of Padova (Progetto di Ateneo 325 “Exploiting randomness in Mixed Integer Linear Programming”), and by MiUR, Italy (PRIN project “Mixed-Integer Nonlinear Optimization: Approaches and Applications”).

This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Laurent Poirrier was supported by Progetto di Eccellenza 2008–2009 of the Fondazione Cassa Risparmio di Padova e Rovigo, Italy, by NSERC Discovery Council Grant Number RGPIN-37 1937–2009, and by Early Researcher Award number ER11-08-174.

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Louveaux, Q., Poirrier, L. & Salvagnin, D. The strength of multi-row models. Math. Prog. Comp. 7, 113–148 (2015). https://doi.org/10.1007/s12532-014-0076-9

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