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Research Papers: Design Automation

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

[+] Author and Article Information
Piyush Pandita

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: ppandit@purdue.edu

Ilias Bilionis

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: ibilion@purdue.edu

Jitesh Panchal

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: panchal@purdue.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 16, 2016; final manuscript received June 26, 2016; published online September 12, 2016. Assoc. Editor: Zissimos P. Mourelatos.

J. Mech. Des 138(11), 111412 (Sep 12, 2016) (8 pages) Paper No: MD-16-1215; doi: 10.1115/1.4034104 History: Received March 16, 2016; Revised June 26, 2016

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 a lot 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) that quantifies the merit of making new objective evaluations. In this work, we reformulate the expected improvement (EI) IAF to filter out parametric and measurement uncertainties. We bypass the curse of dimensionality, since the method does not require learning the response surface as a function of the stochastic parameters, and we employ a fully Bayesian interpretation of Gaussian processes (GPs) by constructing a particle approximation of the posterior of its hyperparameters using adaptive Markov chain Monte Carlo (MCMC) to increase the methods robustness. Also, our approach quantifies 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 and demonstrate it by solving the oil-well placement problem (OWPP) with uncertainties in the permeability field and the oil price time series.

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Figures

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Fig. 1

One-dimensional synthetic example (s(x)=0.1,n=5). Subfigure (a) depicts our initial state of knowledge about the true expected objective (dotted line) conditioned on n = 5 noisy observations (crosses). Subfigure (b) shows a histogram of the predictive distribution of the optimal design x*.

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Fig. 2

One-dimensional synthetic example (s(x)=0.1,n=5). The dashed red line in subfigure (b) marks the real optimal value.

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Fig. 3

One-dimensional synthetic example (n = 10)

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Fig. 4

One-dimensional synthetic example (n = 10)

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Fig. 5

Two-dimensional synthetic example (n = 20)

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Fig. 6

Two-dimensional synthetic example (n = 20)

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Fig. 7

OWPP: Samples from the stochastic permeability model (in logarithmic scale) defined in Eq. (28)

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Fig. 8

OWPP: Samples from the stochastic oil price model

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