On the Burer–Monteiro method for general semidefinite programs
Name
11590_2021_1705_ReferencePDF.pdf
Size
321.74 KB
Format
Adobe PDF
Checksum (MD5)
85f92ab85832b223caf306c6ba40d07e
Author(s)
Cifuentes, Diego
Date Issued
January 28, 2021
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
Consider a semidefinite program involving an
$$n\times n$$
n
×
n
positive semidefinite matrix X. The Burer–Monteiro method uses the substitution
$$X=Y Y^T$$
X
=
Y
Y
T
to obtain a nonconvex optimization problem in terms of an
$$n\times p$$
n
×
p
matrix Y. Boumal et al. showed that this nonconvex method provably solves equality-constrained semidefinite programs with a generic cost matrix when
$$p > rsim \sqrt{2m}$$
p
≳
2
m
, where m is the number of constraints. In this note we extend their result to arbitrary semidefinite programs, possibly involving inequalities or multiple semidefinite constraints. We derive similar guarantees for a fixed cost matrix and generic constraints. We illustrate applications to matrix sensing and integer quadratic minimization.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s11590-021-01705-4
