On the convergence of the expected improvement algorithm

This note deals with the convergence of the expected improvement algorithm, a popular global optimization algorithm based on a Gaussian process model of the function to be optimized. Under some mild hypotheses on the covariance function k of the Gaussian process, we prove that the expected improvement algorithm produces a dense sequence of evaluation points in the search domain when the function to be optimized belongs to the reproducing kernel Hilbert space (RKHS) generated by k. The extension of this convergence result to bigger spaces is an open problem.