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.