Optimization by "expected improvement" and exponential interpolation: rigorous results for analytic functions

Optimization by "expected improvement" is a popular type of optimization of "black boxes" in engineering applications. I will present some rigorous results in this area. The emphasis will be on analytic kernels and univariate functions, where the results are rather complete. In particular, it turns out that the Gaussian-kernel-based optimization converges exponentially fast to the global optimum if the objective function is analytic, but may diverge if the objective function is only infinitely differentiable.
The proofs rely heavily on some new results in interpolation, which are interesting on their own. In particular, I will present integral formulas for the errors of 1D interpolation by Gaussians and exponential functions. These formulas are based on the Harish-Chandra-Itzykson-Zuber integral and generalize the classical Hermite-Genocchi formula for the error of polynomial interpolation.
The talk is based on preprints arXiv:1109.1320 and arXiv:1205.5961.