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

Abstract

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.

How Do Designers Choose Among Multiple Noisy Information Sources in Engineering Design Optimization? An Experimental Study

“…For example in Figure 4, compare the mean value of z µ n (equal to 0.8) at interval n ∈ (6,7) for total budget 20, and that of (equal to 0.55) at the same interval for total budget 40. In the Save-remaining Budget treatments, similar trends are observed except for between the fixed budgets of 40 and 60 (compare at n ∈ (10,11)). This may be because subjects stop early in Saveremaining Budget (60) treatment in expectation of saving larger budget.…”

“…For instance, in the Use-or-lose Budget treatments, the mean of z σ 2 n at n ∈ (6,7) increases as the fixed budget increases. This trend is observed for intervals (10,11) and (14,15) of n between total budgets of 40 and 60. In the Save-remaining Budget treatments, the mean of z σ 2 n increases for interval (6,7) between budgets 20 and 40.…”

“…Numerous optimization methods for optimal design have been developed in literature [6,7,8]. In recent studies in design literature, optimal strategies based on meta-modeling of performance observations have been proposed [4,9,10,5,11]. The optimal decisions are defined as picking alternatives in design space that maximize the expectation of payoff.…”

“…The optimal strategies specify decisions that maximize the expectation of payoff. For the decision to choose next x, the optimal strategy is to select x where the expected improvement (EI) in the maximum of predictive mean is the highest [25,10]. This strategy is however independent of fixed budget and encodes next x only based on previous observations.…”

“…Ignore intervals(10,11) and(14,15) in the treatments of total budget 20 and 40 respectively. Subjects who sampled this many points could do so because they did not sample any physical observations.…”