A stochastic radial basis function method for the global optimization of expensive functions We introduce a new framework for the global optimization of computationally expensive multimodal functions when derivatives are unavailable. The proposed Stochastic Response Surface (SRS) Method iteratively utilizes a response surface model to approximate the expensive function and identifies a promising point for function evaluation from a set of randomly generated points, called candidate points. Assuming some mild technical conditions, SRS converges to the global minimum in a probabilistic sense. We also propose Metric SRS (MSRS), which is a special case of SRS where the function evaluation point in each iteration is chosen to be the best candidate point according to two criteria: the estimated function value obtained from the response surface model, and the minimum distance from previously evaluated points. We develop a global optimization version and a multistart local optimization version of MSRS. In the numerical experiments, we used a radial basis function (RBF) model for MSRS and the resulting algorithms, Global MSRBF and Multistart Local MSRBF, were compared to 6 alternative global optimization methods, including a multistart derivative-based local optimization method. Multiple trials of all algorithms were compared on 17 multimodal test problems and on a 12-dimensional groundwater bioremediation application involving partial differential equations. The results indicate that Multistart Local MSRBF is the best on most of the higher dimensional problems, including the groundwater problem. It is also at least as good as the other algorithms on most of the lower dimensional problems. Global MSRBF is competitive with the other alternatives on most of the lower dimensional test problems and also on the groundwater problem. These results suggest that MSRBF is a promising approach for the global optimization of expensive functions.

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  1. Chen, Leilei; Cheng, Ruhui; Li, Shengze; Lian, Haojie; Zheng, Changjun; Bordas, Stéphane P. A.: A sample-efficient deep learning method for multivariate uncertainty qualification of acoustic-vibration interaction problems (2022)
  2. Wang, Wenyu; Akhtar, Taimoor; Shoemaker, Christine A.: Integrating (\varepsilon)-dominance and RBF surrogate optimization for solving computationally expensive many-objective optimization problems (2022)
  3. Dong, Huachao; Wang, Peng; Fu, Chongbo; Song, Baowei: Kriging-assisted teaching-learning-based optimization (KTLBO) to solve computationally expensive constrained problems (2021)
  4. Müller, Juliane; Park, Jangho; Sahu, Reetik; Varadharajan, Charuleka; Arora, Bhavna; Faybishenko, Boris; Agarwal, Deborah: Surrogate optimization of deep neural networks for groundwater predictions (2021)
  5. Sampaio, Phillipe R.: DEFT-FUNNEL: an open-source global optimization solver for constrained grey-box and black-box problems (2021)
  6. Wang, Nanzhe; Chang, Haibin; Zhang, Dongxiao: Efficient uncertainty quantification for dynamic subsurface flow with surrogate by theory-guided neural network (2021)
  7. Xia, Wei; Shoemaker, Christine: GOPS: efficient RBF surrogate global optimization algorithm with high dimensions and many parallel processors including application to multimodal water quality PDE model calibration (2021)
  8. Chen, Ray-Bing; Wang, Yuan; Wu, C. F. Jeff: Finding optimal points for expensive functions using adaptive RBF-based surrogate model via uncertainty quantification (2020)
  9. Gao, Han; Zhu, Xueyu; Wang, Jian-Xun: A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations (2020)
  10. García-García, José Carlos; García-Ródenas, Ricardo; Codina, Esteve: A surrogate-based cooperative optimization framework for computationally expensive black-box problems (2020)
  11. He, Xinyu; Reyes, Kristofer G.; Powell, Warren B.: Optimal learning with local nonlinear parametric models over continuous designs (2020)
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  14. Ahmadvand, Mohammad; Esmaeilbeigi, Mohsen; Kamandi, Ahmad; Yaghoobi, Farajollah Mohammadi: An improved hybrid-ORBIT algorithm based on point sorting and MLE technique (2019)
  15. Cortesi, Andrea F.; Jannoun, Ghina; Congedo, Pietro M.: Kriging-sparse polynomial dimensional decomposition surrogate model with adaptive refinement (2019)
  16. He, Jiachuan; Mattis, Steven A.; Butler, Troy D.; Dawson, Clint N.: Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines (2019)
  17. Larson, Jeffrey; Menickelly, Matt; Wild, Stefan M.: Derivative-free optimization methods (2019)
  18. Müller, Juliane; Day, Marcus: Surrogate optimization of computationally expensive black-box problems with hidden constraints (2019)
  19. Costa, Alberto; Nannicini, Giacomo: RBFOpt: an open-source library for black-box optimization with costly function evaluations (2018)
  20. Zhou, Zhe; Bai, Fusheng: An adaptive framework for costly black-box global optimization based on radial basis function interpolation (2018)

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