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. Zhou, Zhe; Bai, Fusheng: An adaptive framework for costly black-box global optimization based on radial basis function interpolation (2018)
  2. Beiranvand, Vahid; Hare, Warren; Lucet, Yves: Best practices for comparing optimization algorithms (2017)
  3. Boukouvala, Fani; Faruque Hasan, M. M.; Floudas, Christodoulos A.: Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption (2017)
  4. Corveleyn, Samuel; Vandewalle, Stefan: Computation of the output of a function with fuzzy inputs based on a low-rank tensor approximation (2017)
  5. Martinez, Nadia; Anahideh, Hadis; Rosenberger, Jay M.; Martinez, Diana; Chen, Victoria C. P.; Wang, Bo Ping: Global optimization of non-convex piecewise linear regression splines (2017)
  6. Müller, Juliane; Woodbury, Joshua D.: GOSAC: global optimization with surrogate approximation of constraints (2017)
  7. Vu, Ky Khac; D’Ambrosio, Claudia; Hamadi, Youssef; Liberti, Leo: Surrogate-based methods for black-box optimization (2017)
  8. Akhtar, Taimoor; Shoemaker, Christine A.: Multi objective optimization of computationally expensive multi-modal functions with RBF surrogates and multi-rule selection (2016)
  9. Boukouvala, Fani; Misener, Ruth; Floudas, Christodoulos A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO (2016)
  10. Jamshidi, Arta A.; Powell, Warren B.: A recursive local polynomial approximation method using Dirichlet clouds and radial basis functions (2016)
  11. Krityakierne, Tipaluck; Akhtar, Taimoor; Shoemaker, Christine A.: SOP: parallel surrogate global optimization with Pareto center selection for computationally expensive single objective problems (2016)
  12. Müller, Juliane: MISO: mixed-integer surrogate optimization framework (2016)
  13. Chow, Joseph Y. J.; Regan, Amelia C.: A surrogate-based multiobjective metaheuristic and network degradation simulation model for robust toll pricing (2014)
  14. Müller, Juliane; Shoemaker, Christine A.: Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems (2014)
  15. Müller, Juliane; Shoemaker, Christine A.; Piché, Robert: SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications (2014)
  16. Regis, Rommel G.; Shoemaker, Christine A.: A quasi-multistart framework for global optimization of expensive functions using response surface models (2013)
  17. Le Thi, H. A.; Vaz, A. I. F.; Vicente, L. N.: Optimizing radial basis functions by d.c. programming and its use in direct search for global derivative-free optimization (2012)
  18. Regis, Rommel G.: Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions (2011) ioport
  19. Jakobsson, Stefan; Patriksson, Michael; Rudholm, Johan; Wojciechowski, Adam: A method for simulation based optimization using radial basis functions (2010)
  20. Laguna, M.; Molina, J.; Pérez, F.; Caballero, R.; Hernández-Díaz, A. G.: The challenge of optimizing expensive black boxes: a scatter search/rough set theory approach (2010)

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