  A common approach for minimizing a smooth nonlinear function is to employ
finite-difference approximations to the gradient. While this can be easily
performed when no error is present within the function evaluations, when the
function is noisy, the optimal choice requires information about the noise
level and higher-order derivatives of the function, which is often unavailable.
Given the noise level of the function, we propose a bisection search for
finding a finite-difference interval for any finite-difference scheme that
balances the truncation error, which arises from the error in the Taylor series
approximation, and the measurement error, which results from noise in the
function evaluation. Our procedure produces reliable estimates of the
finite-difference interval at low cost without explicitly approximating
higher-order derivatives. We show its numerical reliability and accuracy on a
set of test problems. When combined with L-BFGS, we obtain a robust method for
minimizing noisy black-box functions, as illustrated on a subset of
unconstrained CUTEst problems with synthetically added noise.
