Computational Optimization and Applications

, Volume 65, Issue 3, pp 545–566

On handling indicator constraints in mixed integer programming

  • Pietro Belotti
  • Pierre Bonami
  • Matteo Fischetti
  • Andrea Lodi
  • Michele Monaci
  • Amaya Nogales-Gómez
  • Domenico Salvagnin
Article

DOI: 10.1007/s10589-016-9847-8

Cite this article as:
Belotti, P., Bonami, P., Fischetti, M. et al. Comput Optim Appl (2016) 65: 545. doi:10.1007/s10589-016-9847-8

Abstract

Mixed integer programming (MIP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice; this is what happens, for e.g., in the case of Classification problems with Ramp Loss functions that represent an important application in this context. In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MIP. One of these methods is currently part of the arsenal of IBM-Cplex  since version 12.6.1. More generally, we argue that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present.

Keywords

Mixed-integer linear programming Mixed-integer quadratic programming Indicator constraints 

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Pietro Belotti
    • 1
  • Pierre Bonami
    • 2
  • Matteo Fischetti
    • 3
  • Andrea Lodi
    • 4
    • 5
  • Michele Monaci
    • 4
  • Amaya Nogales-Gómez
    • 6
  • Domenico Salvagnin
    • 3
    • 7
  1. 1.FICOBirminghamUK
  2. 2.IBMMadridSpain
  3. 3.University of PadovaPaduaItaly
  4. 4.University of BolognaBolognaItaly
  5. 5.École Polytechnique de MontréalMontrealCanada
  6. 6.Mathematical and Algorithmic Sciences Lab, Huawei France R&DParisFrance
  7. 7.IBMMilanoItaly

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