  Tenfold improvements in computation speed can be brought to the alternating
direction method of multipliers (ADMM) for Semidefinite Programming with
virtually no decrease in robustness and provable convergence simply by
projecting approximately to the Semidefinite cone. Instead of computing the
projections via "exact" eigendecompositions that scale cubically with the
matrix size and cannot be warm-started, we suggest using state-of-the-art
factorization-free, approximate eigensolvers, thus achieving almost quadratic
scaling and the crucial ability of warm-starting. Using a recent result from
[Goulart et al., 2020], we are able to circumvent the numerical instability of
the eigendecomposition and thus maintain tight control on the projection
accuracy. This in turn guarantees convergence, either to a solution or a
certificate of infeasibility, of the ADMM algorithm. To achieve this, we extend
recent results from [Banjac et al., 2019] to prove that reliable infeasibility
detection can be performed with ADMM even in the presence of approximation
errors. In all of the considered problems of SDPLIB that "exact" ADMM can solve
in a few thousand iterations, our approach brings a significant, up to 20x,
speedup without a noticeable increase on ADMM's iterations.
