# Scalable Preconditioning of Block-Structured Linear Algebra Systems using ADMM

## TL;DR

This paper introduces a scalable iterative method combining ADMM and GMRES to efficiently solve large, block-structured linear systems common in optimization, outperforming traditional Schur complement methods in speed and robustness.

## Contribution

The paper presents a novel ADMM-preconditioned GMRES approach that overcomes scalability issues in solving large, coupled linear systems in optimization problems.

## Key findings

- ADMM-GMRES is nearly ten times faster than Schur decomposition.
- The method handles systems with up to 2 million variables efficiently.
- It is robust to the choice of the augmented Lagrangian penalty parameter.

## Abstract

We study the solution of block-structured linear algebra systems arising in optimization by using iterative solution techniques. These systems are the core computational bottleneck of many problems of interest such as parameter estimation, optimal control, network optimization, and stochastic programming. Our approach uses a Krylov solver (GMRES) that is preconditioned with an alternating method of multipliers (ADMM). We show that this ADMM-GMRES approach overcomes well-known scalability issues of Schur complement decomposition in problems that exhibit a high degree of coupling. The effectiveness of the approach is demonstrated using linear systems that arise in stochastic optimal power flow problems and that contain up to 2 million total variables and 4,000 coupling variables. We find that ADMM-GMRES is nearly an order of magnitude faster than Schur complement decomposition. Moreover, we demonstrate that the approach is robust to the selection of the augmented Lagrangian penalty parameter, which is a key advantage over the direct use of ADMM.