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Mixed-integer nonlinear programs featuring “on/off” constraints

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Abstract

In this paper, we study MINLPs featuring “on/off” constraints. An “on/off” constraint is a constraint f(x)≤0 that is activated whenever a corresponding 0–1 variable is equal to 1. Our main result is an explicit characterization of the convex hull of the feasible region when the MINLP consists of simple bounds on the variables and one “on/off” constraint defined by an isotone function f. When extended to general convex MINLPs, we show that this result yields tight lower bounds compared to classical formulations. This allows us to introduce new models for the delay-constrained routing problem in telecommunications. Numerical results show gains in computing time of up to one order of magnitude compared to state-of-the-art approaches.

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Authors and Affiliations

  1. LIX, École Polytechnique, 91128, Palaiseau, France

    Hassan Hijazi

  2. LIF, Parc Scientifique et Technologique de Luminy, CNRS-Aix Marseille Universite, Marseille, France

    Pierre Bonami

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

    Gérard Cornuéjols

  4. Orange Labs R&D/CORE-TPN, 38-40 rue du Général Leclerc, 92794, Issy-Les-Moulineaux cedex 9, France

    Adam Ouorou

Authors
  1. Hassan Hijazi
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  2. Pierre Bonami
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  3. Gérard Cornuéjols
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  4. Adam Ouorou
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Corresponding author

Correspondence to Pierre Bonami.

Additional information

First author is supported by Digiteo Emergence PASO, Digiteo Chair 2009-14D RMNCCO and Digiteo Emergence 2009-55D ARM. A major part of this work was accomplished while first author was working at Orange Labs R&D.

Second author is supported by ANR grant ANR06-BLAN-0375 and by a Google Research Award.

Third author is supported by NSF grant CMMI1024554 and ONR grant N00014-03-1-0133.

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Hijazi, H., Bonami, P., Cornuéjols, G. et al. Mixed-integer nonlinear programs featuring “on/off” constraints. Comput Optim Appl 52, 537–558 (2012). https://doi.org/10.1007/s10589-011-9424-0

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