On the search of the shape parameter in radial basis functions using univariate global optimization methods

In this paper we consider the problem of finding an optimal value of the shape parameter in radial basis function interpolation. In particular, we propose the use of a leave-one-out cross validation (LOOCV) technique combined with univariate global optimization methods, which involve strategies of global optimization with pessimistic improvement (GOPI) and global optimization with optimistic improvement (GOOI). This choice is carried out to overcome serious issues of commonly used optimization routines that sometimes result in shape parameter values are not globally optimal. New locally-biased versions of geometric and information Lipschitz global optimization algorithms are presented. Numerical experiments and applications to real-world problems show a promising performance and efficacy of the new algorithms, called LOOCV-GOPI and LOOCV-GOOI, in comparison with their direct competitors.