On the Burer-Monteiro method for general semidefinite programs

Understand

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.