Analysis of the bfgs method with errors

The classical convergence analysis of quasi-Newton methods assumes that the func- tion 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 mod- ification 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 [4], 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.