An adaptive radial basis function method using weighted improvement

This paper introduces an adaptive Radial Basis Function (RBF) method using weighted improvement for the global optimization of black-box problems subject to box constraints. The proposed method applies rank-one update to efficiently build RBF models and derives a closed form for the leave-one-out cross validation (LOOCV) error of RBF models, allowing an adaptive choice of radial basis functions. In addition, we develop an estimated error bound, which share several desired properties with the kriging variance. This error estimate motivates us to design a novel sampling criterion called weighted improvement, capable of balancing between global search and local search with a tunable parameter. Computational results on 45 popular test problems indicate that the proposed algorithm outperforms several benchmark algorithms. Results also suggest that multiquadrics introduces lowest LOOCV error for small sample size while thin plate splines and inverse multiquadrics shows lower LOOCV error for large sample size.