Lifts of Convex Sets and Cone Factorizations
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449.82 KB
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Author(s)
Parrilo, Pablo A.
Thomas, Rekha R.
Gouveia, João
Date Issued
May 2013
Journal
Mathematics of Operations Research
Publisher
Institute for Operations Research and the Management Sciences (INFORMS)
Citation
Gouveia, Joao, Pablo A. Parrilo, and Rekha R. Thomas. “Lifts of Convex Sets and Cone Factorizations.” Mathematics of Operations Research 38, no. 2 (May 2013): 248–264.
Version
Original manuscript
Abstract
In this paper, we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1287/moor.1120.0575
