A fresh CP look at mixed-binary QPs: new formulations and relaxations

Triggered by Burer’s seminal characterization from 2009, many copositive reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely)positive cones are replaced by tractable approximations. While the widely used approx...

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Bibliographic Details
Published in:Mathematical programming Vol. 166; no. 1-2; pp. 159 - 184
Main Authors: Bomze, Immanuel M., Cheng, Jianqiang, Dickinson, Peter J. C., Lisser, Abdel
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017
Springer Nature B.V
Springer Verlag
Subjects:
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computer Science
Cones
Formulations
Full Length Paper
Hierarchies
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Matrix methods
Numerical Analysis
Operations Research
Optimization and Control
Theoretical
90C20
Nonconvex optimization
90C30
Quadratic optimization
Copositivity
Completely positive
Reformulations
Nonlinear optimization
90C26
Conic optimization
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:Triggered by Burer’s seminal characterization from 2009, many copositive reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely)positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders which rapidly increase with the level of approximation, alternatives focus on the problem of keeping psd matrix orders small, with the aim to avoid memory problems in the interior point algorithms. This work continues this approach, proposing new reformulations and relaxations. Moreover, we provide a thorough comparison of the respective duals and establish a monotonicity relation among their duality gaps. We also identify sufficient conditions for strong duality/zero duality gap in some of these formulations and generalize some of our observations to general conic problems.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1109-8