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

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, as a result, they tend to 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 in an optimal way. 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. This involves a reformulation of the EI policy that is able to filter out parametric and measurement uncertainties. We by-pass the curse of dimensionality, since the method does not require learning the response surface as a function of the stochastic parameters. 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. An addendum of our approach is that it can quantify the epistemic uncertainty on the location of the optimum and the optimal value as induced by the limited number of objective evaluations used in obtaining it. We verify and validate our approach by solving two synthetic optimization problems under uncertainty. We demonstrate our approach by solving a challenging engineering problem: the oil-well-placement problem with uncertainties in the permeability field and the oil price time series.