Convergence properties of the expected improvement algorithm

This paper 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. The first result is that under some mild hypotheses on the covariance function $k$ of the Gaussian process, the expected improvement algorithm produces a dense sequence of evaluation points in the search domain, when the function to be optimized is in the reproducing kernel Hilbert space (RKHS) generated by $k$. The second result states that the density property also holds for $\P$-almost all continuous functions, where $\P$ is the (prior) probability distribution induced by the Gaussian process.