Theta Bodies for Polynomial Ideals
Name
Gouveia-2010-Theta Bodies for Polynomial Ideals.pdf
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352.4 KB
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Adobe PDF
Checksum (MD5)
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Author(s)
Parrilo, Pablo A.
Thomas, Rekha R.
Gouveia, João
Alternative Title
THETA BODIES FOR POLYNOMIAL IDEALS
Date Issued
March 2010
Journal
SIAM Journal on Optimization
Publisher
Society for Industrial and Applied Mathematics
Citation
Gouveia, João, Pablo A. Parrilo, and Rekha R. Thomas. "Theta Bodies for Polynomial Ideals." SIAM J. Optim. Volume 20, Issue 4, pp. 2097-2118 (2010) ©2010 Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal. These relaxations generalize Lovász's construction of the theta body of a graph. We establish a relationship between theta bodies and Lasserre's relaxations for real varieties which allows, in many cases, for theta bodies to be expressed as feasible regions of semidefinite programs. Examples from combinatorial optimization are given. Lovász asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety. We answer this question for vanishing ideals of finite point sets via several equivalent characterizations. We also give a geometric description of the first theta body for all ideals.
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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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/090746525
