On the Convergence of GMRES-Accelerated ADMM in $O(\kappa^{1/4}\log\epsilon^{-1})$ Iterations for Quadratic Objectives

This paper investigates the generalized minimum residual method (GMRES) in its ability to accelerate the convergence of the alternating direction method-of-multipliers (ADMM). We provide evidence that ADMM-GMRES can consistently converge to an $epsilon$-accurate solution for a $kappa$-conditioned problem in $O(kappa^{1/4}logepsilon^{-1})$ iterations, and characterize two broad classes of problems for which the enhanced convergence is guaranteed. At the same time, we construct a class of problems that forces ADMM-GMRES to converge at the same asymptotic rate as ADMM. To demonstrate the enhanced convergence rate in practice, the accelerated method is applied to the Newton direction computation for the interior-point solution of semidefinite programs in the SDPLIB test suite.