On piecewise linear approximations of bilinear terms: structural comparison of univariate and bivariate mixed-integer programming formulations
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Published:2022-11-03
Issue:4
Volume:85
Page:789-819
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ISSN:0925-5001
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Container-title:Journal of Global Optimization
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language:en
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Short-container-title:J Glob Optim
Author:
Bärmann Andreas,Burlacu Robert,Hager Lukas,Kleinert Thomas
Abstract
AbstractBilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an $$ \varepsilon $$
ε
-approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error $$ \varepsilon $$
ε
, univariate $$ \varepsilon $$
ε
-approximations require fewer simplices than bivariate $$ \varepsilon $$
ε
-approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.
Funder
Bayerisches Staatsministerium für Wirtschaft, Infrastruktur, Verkehr und Technologie
Deutsche Forschungsgemeinschaft
Bayerische Staatsregierung
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Management Science and Operations Research,Control and Optimization,Computer Science Applications,Business, Management and Accounting (miscellaneous)
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