# A radial basis function method for noisy global optimisation

## TL;DR

This paper introduces a new method for optimizing noisy and expensive functions using radial basis functions and regularization.

## Contribution

The novel approach extends Gutmann’s RBF method to handle noisy objective functions with error bounds.

## Key findings

- The method uses a regularised least-squares criterion to construct radial basis function approximants.
- New sample points are determined using a target value similar to the original RBF method.
- The paper provides convergence results and demonstrates the method on a test problem.

## 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.