(710c) Constrained Black-Box Optimization for Stochastic Simulations with Polynomial-Chaos-Based Stochastic Kriging

Surrogate modeling techniques have gained popularity in different engineering fields when an accurate closed-form formulation is not available or when the computational cost for the problem of interest is prohibitively high¹. When the surrogate model is selected, sampling strategies are used to calibrate the model, and different optimization algorithms can be applied to find the optimal or near-optimal solution given a certain computational budget². Among various surrogate models and optimization algorithms, the Kriging-based model with the Bayesian optimization framework has become increasingly common³. This approach assumes the behavior of a computational model to be a realization of a Gaussian random process⁴ and uses some infill criterion to propose new sample points while maintaining the balance between exploration and exploitation³.

The Kriging-based models consist of two parts: the mean (also known as the trend) and the variance of the Gaussian process⁴. Based on the choice of the trend, Kriging models can be divided into three categories: simple Kriging, ordinary Kriging, and universal Kriging^4-7. Simple Kriging assumes a known constant value for the trend, ordinary Kriging assumes an unknown constant value, and universal Kriging uses some deterministic model for the trend function to capture the global behavior. In the work of Schobi et al., polynomial chaos expansion was used as the trend of a universal Kriging⁴. It was found that the performance of this polynomial-chaos-based Kriging was at least as good as ordinary Kriging for a deterministic simulator⁴. In the work of Ankenman et al., stochastic Kriging was proposed to extend the deterministic case to stochastic simulations by explicitly accounting for the uncertainty inherent to the simulator⁸. Recently, the work of García-Merino et al. proposed the polynomial-chaos-expansion-based stochastic Kriging and reported a superior performance compared to the ordinary stochastic Kriging⁹. Despite these promising findings, the use of polynomial-chaos-expansion-based stochastic Kriging in Bayesian optimization remains unexplored. This work aims to propose a tailored framework that capitalizes on the strength of this novel surrogate model in constrained black-box optimization for stochastic simulations.

In this work, we use the polynomial-chaos-based stochastic Kriging as the surrogate model given its advantages in accommodating the intrinsic uncertainty of the simulation and its capability to capture both the global and local behavior of the simulator. A tailored framework that combines the Bayesian optimization with polynomial-chaos-based stochastic Kriging is applied to search for the near-optimal solution. Different infill criteria are applied for adaptive sampling, and the computational performances are compared with the case in which the ordinary stochastic Kriging is used as the surrogate model. Different benchmark problems are used to evaluate the ability of the proposed method to find a feasible solution, locate the global optimum, and make accurate predictions around the optimum. A case study in the chemical recycling of plastic waste is included to illustrate the application of the proposed method in real-world chemical engineering problems.

References

(1) Sudret, B.; Marelli, S.; Wiart, J. Surrogate models for uncertainty quantification: An overview. 2017 11th European Conference on Antennas and Propagation (EUCAP) 2017.

(2) Bhosekar, A.; Ierapetritou, M. Advances in surrogate based modeling, feasibility analysis, and optimization: A review. Computers & Chemical Engineering 2018, 108, 250-267.

(3) Wang, Z.; Ierapetritou, M. Constrained Optimization of black-box stochastic systems using a novel feasibility enhanced kriging-based method. Computers & Chemical Engineering 2018, 210-223.

(4) Schobi, R.; Sudret, B.; Wiart, J. Polynomial-Chaos-based kriging. International Journal for Uncertainty Quantification 2015, 5 (2), 171-193.

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(8) Ankenman, B.; Nelson, B. L.; Staum, J. Stochastic Kriging for simulation metamodeling. Operations Research 2010, 58 (2), 371-382.

(9) García-Merino, J. C.; Calvo-Jurado, C.; García-Macías, E. Sparse polynomial chaos expansion for Universal Stochastic Kriging. Journal of Computational and Applied Mathematics 2024.