A radial basis function method for noisy global optimisation

• Published in: Mathematical programming Vol. 211; no. 1; pp. 49 - 92
• Main Authors: Banholzer, Dirk, Fliege, Jörg, Werner, Ralf
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2025; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Global optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Monte Carlo simulation; Numerical Analysis; Radial basis function; Response surface methodology; Stochastic models; Theoretical; Response surface methods; Radial basis functions; Approximation; 90C30; Controlled noise; Global optimisation; Expensive noisy objective function; 90C26
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-024-02125-9

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.