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Stochastic mesh adaptive direct search for blackbox optimization using probabilistic estimates

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Abstract

We present a stochastic extension of the mesh adaptive direct search (MADS) algorithm originally developed for deterministic blackbox optimization. The algorithm, called StoMADS, considers the unconstrained optimization of an objective function f whose values can be computed only through a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based on an algorithmic framework similar to that of MADS and uses random estimates of function values obtained from stochastic observations since the exact deterministic computable version of f is not available. Such estimates are required to be accurate with a sufficiently large but fixed probability and to satisfy a variance condition. The ability of the proposed algorithm to generate an asymptotically dense set of search directions is then exploited using martingale theory to prove convergence to a Clarke stationary point of f with probability one.

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Notes

  1. The notations \(\tilde{f}(x,\xi )\) [20], \(\tilde{f}(x;\xi )\) [38] and \(f(x;\varepsilon )\) [23] are often used for the noisy computable versions of f, where \(\xi \) and \(\varepsilon \) are random variables. We use the more compact notation \(f_{\Theta }(x)\).

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Acknowledgements

The authors are grateful to Erick Delage from HEC Montréal and Richard Labib from Polytechnique Montréal for valuable discussions and constructive suggestions. They would also like to thank an anonymous referee for his/her careful reading and helpful remarks that contributed to improve this work. This research is supported by the NSERC CRD RDCPJ 490744-15 Grant and by an InnovÉÉ grant, both in collaboration with Hydro-Québec and Rio Tinto, and by a FRQNT fellowship.

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Authors and Affiliations

  1. GERAD and Département de Mathématiques et de Génie Industriel, Polytechnique Montréal, Montreal, Canada

    Charles Audet, Kwassi Joseph Dzahini & Sébastien Le Digabel

  2. GERAD and Department of Mechanical Engineering, McGill University, Montreal, Canada

    Michael Kokkolaras

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  1. Charles Audet
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  2. Kwassi Joseph Dzahini
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  4. Sébastien Le Digabel
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Correspondence to Kwassi Joseph Dzahini.

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Audet, C., Dzahini, K.J., Kokkolaras, M. et al. Stochastic mesh adaptive direct search for blackbox optimization using probabilistic estimates. Comput Optim Appl 79, 1–34 (2021). https://doi.org/10.1007/s10589-020-00249-0

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