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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Published in:Computational optimization and applications Vol. 65; no. 3; pp. 545 - 566
Main Authors: Belotti, Pietro, Bonami, Pierre, Fischetti, Matteo, Lodi, Andrea, Monaci, Michele, Nogales-Gómez, Amaya, Salvagnin, Domenico
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2016
Springer Nature B.V
Springer Verlag
Subjects:
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
Indicator constraints
Mixed-integer quadratic programming
Mixed-integer linear programming
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary: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.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-016-9847-8