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Design optimization under uncertainty is notoriously difficult when theobjective function is expensive to evaluate. State-of-the-art techniques, e.g,stochastic optimization or sampling average approximation, fail to learnexploitable patterns from collected data and require an excessive number ofobjective function evaluations. There is a need for techniques that alleviatethe high cost of information acquisition and select sequential simulationsoptimally. In the field of deterministic single-objective unconstrained globaloptimization, the Bayesian global optimization (BGO) approach has beenrelatively successful in addressing the information acquisition problem. BGObuilds a probabilistic surrogate of the expensive objective function and usesit to define an information acquisition function (IAF) whose role is toquantify the merit of making new objective evaluations. Specifically, BGOiterates between making the observations with the largest expected IAF andrebuilding the probabilistic surrogate, until a convergence criterion is met.In this work, we extend the expected improvement (EI) IAF to the case of designoptimization under uncertainty wherein the EI policy is reformulated to filterout parametric and measurement uncertainties. To increase the robustness of ourapproach in the low sample regime, we employ a fully Bayesian interpretation ofGaussian processes by constructing a particle approximation of the posterior ofits hyperparameters using adaptive Markov chain Monte Carlo. We verify andvalidate our approach by solving two synthetic optimization problems underuncertainty and demonstrate it by solving the oil-well-placement problem withuncertainties in the permeability field and the oil price time series.

Extending Expected Improvement for High-dimensional Stochastic Optimization of Expensive Black-Box Functions