Parameter Insensitivity in ADMM-Preconditioned Solution of Saddle-Point Problems

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 $O(\sqrt{\kappa}\log\epsilon^{-1})$.