A quasi-multistart framework for global optimization of expensive functions using response surface models We present the AQUARS (a QUAsi-multistart response surface) framework for finding the global minimum of a computationally expensive black-box function subject to bound constraints. In a traditional multistart approach, the local search method is blind to the trajectories of the previous local searches. Hence, the algorithm might find the same local minima even if the searches are initiated from points that are far apart. In contrast, AQUARS is a novel approach that locates the promising local minima of the objective function by performing local searches near the local minima of a response surface (RS) model of the objective function. It ignores neighborhoods of fully explored local minima of the RS model and it bounces between the best partially explored local minimum and the least explored local minimum of the RS model. We implement two AQUARS algorithms that use a radial basis function model and compare them with alternative global optimization methods on an 8-dimensional watershed model calibration problem and on 18 test problems. The alternatives include EGO, GLOBALm, MLMSRBF [{it R. G. Regis} and {it C. A. Shoemaker}, INFORMS J. Comput. 19, No. 4, 497--509 (2007; Zbl 1241.90192)], CGRBF-Restart [{it R. G. Regis} and {it C. A. Shoemaker}, J. Glob. Optim. 37, No. 1, 113--135 (2007; Zbl 1149.90120)], and multilevel single linkage (MLSL) coupled with two types of local solvers: SQP and mesh adaptive direct search (MADS) combined with kriging. The results show that the AQUARS methods generally use fewer function evaluations to identify the global minimum or to reach a target value compared to the alternatives. In particular, they are much better than EGO and MLSL coupled to MADS with kriging on the watershed calibration problem and on 15 of the test problems.

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  1. Mathesen, Logan; Pedrielli, Giulia; Ng, Szu Hui; Zabinsky, Zelda B.: Stochastic optimization with adaptive restart: a framework for integrated local and global learning (2021)
  2. Costa, Alberto; Nannicini, Giacomo: RBFOpt: an open-source library for black-box optimization with costly function evaluations (2018)
  3. Kieslich, Chris A.; Boukouvala, Fani; Floudas, Christodoulos A.: Optimization of black-box problems using Smolyak grids and polynomial approximations (2018)
  4. 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)
  5. Müller, Juliane; Woodbury, Joshua D.: GOSAC: global optimization with surrogate approximation of constraints (2017)
  6. Tenne, Yoel: Machine-learning in optimization of expensive black-box functions (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. Custódio, A. L.; Madeira, J. F. A.: GLODS: global and local optimization using direct search (2015)
  11. 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)
  12. Regis, Rommel G.; Shoemaker, Christine A.: A quasi-multistart framework for global optimization of expensive functions using response surface models (2013)