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On Hölder global optimization method using piecewise affine bounding functions

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Abstract

This paper is devoted to solving the one-dimensional global optimization problems where the objective function f satisfies a Hölder condition over a real interval [a, b]. We suggest two algorithms as an extended version of the Piyavskii’s method. The first one is based on the use of the tangent at the midpoint of each subinterval of [a, b] of the parabolic lower bounding functions of f. The second one is a combination of two successive phases. In phase 1, we obtain with a few iterations, a subinterval which often contains the point of the global minimum using a \(K_{\alpha }\)-Lipschitz lower bounding functions with a tolerance \(\delta \) much larger than the given accuracy \(\varepsilon \). In phase 2, we apply the first algorithm on the interval obtained in phase 1. A convergence result is proved and compared with the existing methods. The numerical experiments of the two algorithms on some test functions are encouraging.

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Funding

This work is funded by the Directorate-General of Scientific Research and Technological Development (DGRSDT) of Algeria.

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  1. Laboratory of Fundamental And Numerical Mathematics (LMFN), Department of Mathematics, Ferhat Abbas University Setif 1, 19000, Setif, Algeria

    Chahinaz Chenouf & Mohamed Rahal

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  1. Chahinaz Chenouf
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  2. Mohamed Rahal
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Correspondence to Chahinaz Chenouf.

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Chenouf, C., Rahal, M. On Hölder global optimization method using piecewise affine bounding functions. Numer Algor 94, 905–935 (2023). https://doi.org/10.1007/s11075-023-01524-x

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