On the Burer–Monteiro method for general semidefinite programs

Abstract

Abstract Consider a semidefinite program involving an $$n\times n$$ n × n positive semidefinite matrix X. The Burer–Monteiro method uses the substitution $$X=Y Y^T$$ X = Y Y T to obtain a nonconvex optimization problem in terms of an $$n\times p$$ n × p matrix Y. Boumal et al. showed that this nonconvex method provably solves equality-constrained semidefinite programs with a generic cost matrix when $$p > rsim \sqrt{2m}$$ p ≳ 2 m , 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