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 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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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. |
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| 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 |
| Author_xml | – sequence: 1 givenname: Pietro surname: Belotti fullname: Belotti, Pietro organization: FICO – sequence: 2 givenname: Pierre surname: Bonami fullname: Bonami, Pierre organization: IBM – sequence: 3 givenname: Matteo surname: Fischetti fullname: Fischetti, Matteo organization: University of Padova – sequence: 4 givenname: Andrea surname: Lodi fullname: Lodi, Andrea email: andrea.lodi@polymtl.ca organization: University of Bologna, École Polytechnique de Montréal – sequence: 5 givenname: Michele surname: Monaci fullname: Monaci, Michele organization: University of Bologna – sequence: 6 givenname: Amaya surname: Nogales-Gómez fullname: Nogales-Gómez, Amaya organization: Mathematical and Algorithmic Sciences Lab, Huawei France R&D – sequence: 7 givenname: Domenico 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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| 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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