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GOSH: derivative-free global optimization using multi-dimensional space-filling curves

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Abstract

Global optimization is a field of mathematical programming dealing with finding global (absolute) minima of multi-dimensional multiextremal functions. Problems of this kind where the objective function is non-differentiable, satisfies the Lipschitz condition with an unknown Lipschitz constant, and is given as a “black-box” are very often encountered in engineering optimization applications. Due to the presence of multiple local minima and the absence of differentiability, traditional optimization techniques using gradients and working with problems having only one minimum cannot be applied in this case. These real-life applied problems are attacked here by employing one of the mostly abstract mathematical objects—space-filling curves. A practical derivative-free deterministic method reducing the dimensionality of the problem by using space-filling curves and working simultaneously with all possible estimates of Lipschitz and Hölder constants is proposed. A smart adaptive balancing of local and global information collected during the search is performed at each iteration. Conditions ensuring convergence of the new method to the global minima are established. Results of numerical experiments on 1000 randomly generated test functions show a clear superiority of the new method w.r.t. the popular method DIRECT and other competitors.

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Acknowledgements

The authors thank the unknown reviewers for their very useful comments that have allowed the authors to improve the manuscript. The research of Ya. D. Sergeyev was supported by the Russian Science Foundation, project No 15-11-30022 “Global optimization, supercomputing computations, and applications”.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Cagliari, Cagliari, Italy

    Daniela Lera

  2. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Università della Calabria and the Institute of High Performance Computing and Networking of the National Research Council of Italy, Via Pietro Bucci 42C, 87036, Rende, CS, Italy

    Yaroslav D. Sergeyev

  3. Department of Software and Supercomputing, Lobachevskiy University of Nizhni Novgorod, Gagarin Av. 23, Nizhni Novgorod, Russia

    Yaroslav D. Sergeyev

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  1. Daniela Lera
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Correspondence to Yaroslav D. Sergeyev.

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Lera, D., Sergeyev, Y.D. GOSH: derivative-free global optimization using multi-dimensional space-filling curves. J Glob Optim 71, 193–211 (2018). https://doi.org/10.1007/s10898-017-0589-7

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