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On the Burer-Monteiro method for general semidefinite programs

Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method uses the substitution $X=Y Y^T$ to obtain a nonconvex optimization problem in terms of an $n\times p$ matrix $Y$. Boumal et al. showed that this nonconvex method provably solves equality-constrained SDPs with a generic cost matrix when $p \gtrsim \sqrt{2m}$, where $m$ is the number of constraints. In this note we extend their result to arbitrary SDPs, 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.