Gauge Optimization and Duality

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by [R. M. Freund, Math. Programming , 38 (1987), pp. 47--67], seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a...

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Bibliographic Details
Published in:SIAM journal on optimization 2014-01, Vol.24 (4), p.1999-2022
Main Authors: Friedlander, Michael P., Macêdo, Ives, Pong, Ting Kei
Format: Article
Language:English
Subjects:
Arrays
Gages
Gauges
Mathematical analysis
Mathematical models
Norms
Optimization
Programming
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Summary:Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by [R. M. Freund, Math. Programming , 38 (1987), pp. 47--67], seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, and related problems that arise in machine learning and signal processing. The gauge structure of these problems allows for a special kind of duality framework. This paper explores the duality framework proposed by Freund, and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework (such as the variational properties of its value function), and yet maintains most of the generality of the abstract form of gauge optimization.
ISSN:1052-6234
1095-7189
DOI:10.1137/130940785