# Preconditioning ideas for the Augmented Lagrangian method

## TL;DR

This paper introduces a modular preconditioning strategy for the Augmented Lagrangian method that leverages problem structure, improves convergence, and is adaptable to various preconditioning techniques, with promising initial results.

## Contribution

A novel, structure-exploiting preconditioning scheme for ALM that is flexible, efficient, and applicable to both linear and nonlinear problems, enhancing convergence without frequent updates.

## Key findings

- Preconditioned matrices show improved spectral properties.
- Numerical experiments demonstrate potential benefits on benchmark problems.
- Scheme is effective for problems with few constraints relative to search space.

## Abstract

A preconditioning strategy for the Powell-Hestenes-Rockafellar Augmented Lagrangian method (ALM) is presented. The scheme exploits the structure of the Augmented Lagrangian Hessian. It is a modular preconditioner consisting of two blocks. The first one is associated with the Lagrangian of the objective while the second administers the Jacobian of the constraints and possible low-rank corrections to the Hessian. The proposed updating strategies take advantage of ALM convergence results and avoid frequent refreshing. Constraint administration takes into account complementarity over the Lagrange multipliers and admits relaxation. The preconditioner is designed for problems where constraint quantity is small compared to the search space. A virtue of the scheme is that it is agnostic to the preconditioning technique used for the Hessian of the Lagrangian function. The strategy described can be used for linear and nonlinear preconditioning. Numerical experiments report on spectral properties of preconditioned matrices from Matrix Market while some optimization problems where taken from the CUTEst collection. Preliminary results indicate that the proposed scheme could be attractive and further experimentation is encouraged.