  We consider the solution of linear saddle-point problems, using the
alternating direction method-of-multipliers (ADMM) as a preconditioner for the
generalized minimum residual method (GMRES). We show, using theoretical bounds
and empirical results, that ADMM is made remarkably insensitive to the
parameter choice with Krylov subspace acceleration. We prove that ADMM-GMRES
can consistently converge, irrespective of the exact parameter choice, to an
$\epsilon$-accurate solution of a $\kappa$-conditioned problem in
$O(\kappa^{2/3}\log\epsilon^{-1})$ iterations. The accelerated method is
applied to randomly generated problems, as well as the Newton direction
computation for the interior-point solution of semidefinite programs in the
SDPLIB test suite. The empirical results confirm this parameter insensitivity,
and suggest a slightly improved iteration bound of
$O(\sqrt{\kappa}\log\epsilon^{-1})$.
