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

TL;DR

This paper demonstrates that ADMM, when combined with Krylov acceleration, is highly insensitive to parameter choices and can efficiently solve saddle-point problems with theoretical and empirical guarantees.

Contribution

It proves that ADMM-GMRES converges independently of parameter selection and provides improved iteration bounds for solving saddle-point problems.

Findings

ADMM-GMRES converges regardless of parameter choice.

Empirical results confirm parameter insensitivity.

Iteration bounds are improved to O(√κ log ε^{-1}).

Abstract

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