  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.

Optimization and Control