  Design optimization under uncertainty is notoriously difficult when the
objective function is expensive to evaluate. State-of-the-art techniques, e.g,
stochastic optimization or sampling average approximation, fail to learn
exploitable patterns from collected data and require an excessive number of
objective function evaluations. There is a need for techniques that alleviate
the high cost of information acquisition and select sequential simulations
optimally. In the field of deterministic single-objective unconstrained global
optimization, the Bayesian global optimization (BGO) approach has been
relatively successful in addressing the information acquisition problem. BGO
builds a probabilistic surrogate of the expensive objective function and uses
it to define an information acquisition function (IAF) whose role is to
quantify the merit of making new objective evaluations. Specifically, BGO
iterates between making the observations with the largest expected IAF and
rebuilding the probabilistic surrogate, until a convergence criterion is met.
In this work, we extend the expected improvement (EI) IAF to the case of design
optimization under uncertainty wherein the EI policy is reformulated to filter
out parametric and measurement uncertainties. To increase the robustness of our
approach in the low sample regime, we employ a fully Bayesian interpretation of
Gaussian processes by constructing a particle approximation of the posterior of
its hyperparameters using adaptive Markov chain Monte Carlo. We verify and
validate our approach by solving two synthetic optimization problems under
uncertainty and demonstrate it by solving the oil-well-placement problem with
uncertainties in the permeability field and the oil price time series.
