# Burer-Monteiro guarantees for general semidefinite programs

Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method consists in solving a nonconvex program in $Y$, where $Y$ is an $n\times p$ matrix such that $X = Y Y^T$. Despite nonconvexity, Boumal et al. showed that the method provably solves generic equality-constrained SDP's when $p > \sqrt{2m}$, where $m$ is the number of constraints. We extend this result to arbitrary SDP's, possibly involving inequalities or multiple semidefinite constraints. We illustrate applications to sensor network localization and to matrix sensing.

arXiv:1904.07147