A common approach for minimizing a smooth nonlinear function is to employ finitedifference 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 near-optimal estimates of the finite-difference interval at low cost without knowledge of the 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 synthetically noisy unconstrained CUTEst problems.

The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. It consists of computing gradients of a smoothed approximation of the objective function (and constraints), and employing them within established codes. These gradient approximations are calculated by finite differences, with a differencing interval determined by the noise level in the functions and a bound on the second or third derivatives. It is assumed that noise level is known or can be estimated by means of difference tables or sampling. The use of finite differences has been largely dismissed in the derivative-free optimization literature as too expensive in terms of function evaluations and/or as impractical when the objective function contains noise. The test results presented in this paper suggest that such views should be re-examined and that the finite-difference approach has much to be recommended. The tests compared newuoa, dfo-ls and cobyla against the finite-difference approach on three classes of problems: general unconstrained problems, nonlinear least squares, and general nonlinear programs with equality constraints.

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