  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.
