# Analysis of the BFGS Method with Errors

## TL;DR

This paper analyzes the convergence of the BFGS quasi-Newton method when errors are present in function and gradient evaluations, extending classical results and demonstrating robustness through numerical experiments.

## Contribution

It extends BFGS convergence analysis to cases with bounded errors and introduces a modified algorithm that converges near the solution despite noise.

## Key findings

- BFGS converges to a neighborhood of the solution with errors.
- A modified BFGS algorithm maintains convergence properties.
- Numerical results show robustness of the method under noise.

## Abstract

The classical convergence analysis of quasi-Newton methods assumes that the function and gradients employed at each iteration are exact. In this paper, we consider the case when there are (bounded) errors in both computations and establish conditions under which a slight modification of the BFGS algorithm with an Armijo-Wolfe line search converges to a neighborhood of the solution that is determined by the size of the errors. One of our results is an extension of the analysis presented in Byrd, R. H., & Nocedal, J. (1989), which establishes that, for strongly convex functions, a fraction of the BFGS iterates are good iterates. We present numerical results illustrating the performance of the new BFGS method in the presence of noise.