On the Burer–Monteiro method for general semidefinite programs

Abstract

Consider a semidefinite program 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 semidefinite programs with a generic cost matrix when \(p > rsim \sqrt{2m}\), where m is the number of constraints. In this note we extend their result to arbitrary semidefinite programs, 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.