Lifting for Simplicity: Concise Descriptions of Convex Sets
Name
20m1324417.pdf
Description
Published version
Size
5.22 MB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Fawzi, Hamza
Gouveia, Joao
Parrilo, Pablo A
Saunderson, James
Thomas, Rekha R
Date Issued
November 2022
Journal
SIAM Review
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Fawzi, Hamza, Gouveia, Joao, Parrilo, Pablo A, Saunderson, James and Thomas, Rekha R. 2022. "Lifting for Simplicity: Concise Descriptions of Convex Sets." SIAM Review, 64 (4).
Version
Final published version
Abstract
This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for optimization problems. We consider both the classical case of polyhedral lifts, described by linear inequalities, as well as that of spectrahedral lifts, defined by linear matrix inequalities, with a focus on recent developments related to spectrahedral lifts. Given a convex set, ideally we would like to either find a (low-complexity) polyhedral or spectrahedral lift or find an obstruction proving that no such lift is possible. To this end, we explain the connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator. Based on this characterization, we describe a uniform approach, via sums of squares, to the construction of spectrahedral lifts of convex sets and illustrate the method on several families of examples. Finally, we discuss two flavors of obstruction to the existence of lifts: one related to facial structure, and the other related to algebraic properties of the set in question. Rather than being exhaustive, our aim is to illustrate the richness of the area. We touch on a range of different topics related to the existence of lifts and present many examples of lifts from different areas of mathematics and its applications.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1137/20M1324417
