On the Burer-Monteiro method for general semidefinite programs

Consider a semidefinite program involving an n x 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 x p matrix Y. Boumal et al. showed that this nonconvex method provably solves equality-constrained semidefinite programs with a generic cost matrix when p greater than or similar to root 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.

Automated summary

The Burer-Monteiro method can solve equality-constrained semidefinite programs with a generic cost matrix, as shown by Boumal et al. This study extends their result to arbitrary semidefinite programs, including inequalities or multiple constraints, deriving similar guarantees for fixed cost matrix and generic constraints. Applications to matrix sensing and integer quadratic minimization are also illustrated.