  This paper introduces a new storage-optimal first-order method (FOM),
CertSDP, for solving a special class of semidefinite programs (SDPs) to high
accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is
characterized by low-rank solutions, a priori knowledge of the restriction of
the SDP solution to a small subspace, and standard regularity assumptions such
as strict complementarity. Crucially, we show how to use a certificate of
strict complementarity to construct a low-dimensional strongly convex minimax
problem whose optimizer coincides with a factorization of the SDP optimizer.
From an algorithmic standpoint, we show how to construct the necessary
certificate and how to solve the minimax problem efficiently. We accompany our
theoretical results with preliminary numerical experiments suggesting that
CertSDP significantly outperforms current state-of-the-art methods on large
sparse exact QMP-like SDPs.
