  Gaussian processes~(Kriging) are interpolating data-driven models that are
frequently applied in various disciplines. Often, Gaussian processes are
trained on datasets and are subsequently embedded as surrogate models in
optimization problems. These optimization problems are nonconvex and global
optimization is desired. However, previous literature observed computational
burdens limiting deterministic global optimization to Gaussian processes
trained on few data points. We propose a reduced-space formulation for
deterministic global optimization with trained Gaussian processes embedded. For
optimization, the branch-and-bound solver branches only on the degrees of
freedom and McCormick relaxations are propagated through explicit Gaussian
process models. The approach also leads to significantly smaller and
computationally cheaper subproblems for lower and upper bounding. To further
accelerate convergence, we derive envelopes of common covariance functions for
GPs and tight relaxations of acquisition functions used in Bayesian
optimization including expected improvement, probability of improvement, and
lower confidence bound. In total, we reduce computational time by orders of
magnitude compared to state-of-the-art methods, thus overcoming previous
computational burdens. We demonstrate the performance and scaling of the
proposed method and apply it to Bayesian optimization with global optimization
of the acquisition function and chance-constrained programming. The Gaussian
process models, acquisition functions, and training scripts are available
open-source within the "MeLOn - Machine Learning Models for Optimization"
toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn).
