Abstract
Let C be a closed convex subset of a real Hilbert space containing the origin, and assume that K is the homogenization cone of C, i.e., the smallest closed convex cone containing \(C\times \{1\}\). Homogenization cones play an important role in optimization for the construction of examples and counterexamples. A famous examples is the second-order/Lorentz/“ice cream” cone which is the homogenization cone of the unit ball. In this paper, we discuss the polar cone of K as well as an algorithm for finding the projection onto K provided that the projection onto C is available. Various examples illustrate our results.
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Acknowledgements
The authors thank Professors Simeon Reich and Tadeusz Kuczumow for comments concerning linearly bounded sets and providing us with a copy of [6]; Professor Daniel Klain for comments on polar sets and pointing us to [10] as a reference; Professor Levent Tunçel for comments on the homogenization cone and providing us with [9]. Last, but not least, we thank the anonymous reviewers for their remarkably careful reading and for their very constructive and encouraging comments.
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HHB is supported by the Natural Sciences and Engineering Research Council of Canada.
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Communicated by Maicon Marques Alves.
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Theo Bendit and Hansen Wang contributed equally to this work
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Bauschke, H.H., Bendit, T. & Wang, H. The Homogenization Cone: Polar Cone and Projection. Set-Valued Var. Anal 31, 29 (2023). https://doi.org/10.1007/s11228-023-00687-y
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DOI: https://doi.org/10.1007/s11228-023-00687-y
Keywords
- Conification
- Convex cone
- Convex set
- Hilbert space
- Homogenization cone
- Ice cream cone
- Lorentz cone
- Polar cone
- Projection
- Recession cone
- Second order cone