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On the Burer–Monteiro method for general semidefinite programs

Author(s)
Cifuentes, Diego
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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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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.
Date issued
2021-01-28
URI
https://hdl.handle.net/1721.1/136891
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Springer Berlin Heidelberg

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