On handling indicator constraints in mixed integer programming

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 w...

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Vydáno v:Computational optimization and applications Ročník 65; číslo 3; s. 545 - 566
Hlavní autoři: Belotti, Pietro, Bonami, Pierre, Fischetti, Matteo, Lodi, Andrea, Monaci, Michele, Nogales-Gómez, Amaya, Salvagnin, Domenico
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.12.2016
Springer Nature B.V
Springer Verlag
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ISSN:0926-6003, 1573-2894
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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.
AbstractList 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.
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.
Author Monaci, Michele
Belotti, Pietro
Nogales-Gómez, Amaya
Lodi, Andrea
Bonami, Pierre
Fischetti, Matteo
Salvagnin, Domenico
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  surname: Fischetti
  fullname: Fischetti, Matteo
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  givenname: Andrea
  surname: Lodi
  fullname: Lodi, Andrea
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  surname: Salvagnin
  fullname: Salvagnin, Domenico
  organization: University of Padova, IBM
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Keywords Indicator constraints
Mixed-integer quadratic programming
Mixed-integer linear programming
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Snippet 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...
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SubjectTerms Algorithms
Classification
Computation
Computer Science
Constraint modelling
Convex and Discrete Geometry
Indicators
Integer programming
Linear programming
Management Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Quadratic programming
Statistics
Studies
Support vector machines
Tightening
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