Sampling Algebraic Varieties for Sum of Squares Programs
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15m1052548.pdf
Description
Published version
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304.48 KB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Cifuentes, Diego Fernando
Parrilo, Pablo A.
Date Issued
November 2017
Journal
SIAM journal on optimization
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Cifuentes, Diego and Pablo A. Parrilo. "Sampling Algebraic Varieties for Sum of Squares Programs." SIAM journal on optimization 27, no. 4 (2017): pp. 2381-2404.
Version
Final published version
Abstract
We study sum of squares (SOS) relaxations to optimize polynomial functions over a set V ∩ Rn, where V is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents V with a generic set of complex samples. This approach depends only on the geometry of V, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties, we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Terms of Use
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/10.1137/15M1052548
