A radial basis function method for noisy global optimisation

Abstract

We present a novel response surface method for global optimisation of an expensive and noisy (black-box) objective function, where error bounds on the deviation of the observed noisy function values from their true counterparts are available. The method is based on Gutmann's well-established RBF method for minimising an expensive and deterministic objective function, which has become popular both from a theoretical and practical perspective. To construct suitable radial basis function approximants to the objective function and to determine new sample points for successive evaluation of the expensive noisy objective, the method uses a regularised least-squares criterion. In particular, new points are defined by means of a target value, analogous to the original RBF method. We provide essential convergence results, and provide a numerical illustration of the method by means of a simple test problem.

Keywords: Approximation; Controlled noise; Expensive noisy objective function; Global optimisation; Radial basis functions; Response surface methods.

Fig. 1

Objective function f and global minimum (red point) (colour figure online)

Fig. 2

The sample points $x_i$, their observed function values $\widehat{f}(x_i)$ and ${\widehat{f}}^{(n)}(x_i)$ with associated error bounds ${ϵ}_i$ and ${ϵ}^{(n)}$, respectively, as well as the response surfaces $s_n^n$ of the RBF method for noisy objective functions after $n=10$, $n=20$, and $n=40$ iterations