Consider a semidefinite program (SDP) involving an n×n positive
semidefinite matrix X. The Burer-Monteiro method uses the substitution X=YYT to obtain a nonconvex optimization problem in terms of an n×p
matrix Y. Boumal et al. showed that this nonconvex method provably solves
equality-constrained SDPs with a generic cost matrix when p≳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.Comment: 10 page