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Continuous global optimization on fractals through \(\alpha \)-dense curves

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In the present paper we propose a method to approximate, under suitable conditions, the global minimum of a continuous function which is defined in a compact metric space and the its global minimum is attained on a fractal subset of such compact metric space. For this goal, we construct a new class of \(\alpha \)-dense curves, which allow us to approximate, with arbitrarily small error, the solution of the given optimization problem by the solutions of certain single variable optimization problems. Also, by using the \(\alpha \)-dense curves and an algorithm due to J. Calvin, we introduce a new method to approximate the global minimum of a several variables continuous functions on a densifiable set. We illustrate our results with some numerical experiments.

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The author declares that the data of Tables 1 and 2 of this paper are available within such tables and were generated by a software. No external repositories are available.

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Acknowledgements

The author is grateful to the anonymous referee for his/her careful reading of the paper and proposed comments and corrections to improve the quality of the paper.

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  1. UNED, Departamento de Matemáticas, 03202, Elche Alicante, Spain

    G. García

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  1. G. García
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García, G. Continuous global optimization on fractals through \(\alpha \)-dense curves. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 165 (2023). https://doi.org/10.1007/s13398-023-01493-9

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