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Steklov regularization and trajectory methods for univariate global optimization

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Abstract

We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.

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Acknowledgements

The authors offer their warm thanks to an anonymous referee whose comments and suggestions improved the paper.

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Authors and Affiliations

  1. Electrical and Electronics Engineering Department, Bilkent University, 06800, Bilkent, Ankara, Turkey

    Orhan Arıkan

  2. School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, SA, 5095, Australia

    Regina S. Burachik & C. Yalçın Kaya

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  1. Orhan Arıkan
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  2. Regina S. Burachik
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  3. C. Yalçın Kaya
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Correspondence to C. Yalçın Kaya.

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Arıkan, O., Burachik, R.S. & Kaya, C.Y. Steklov regularization and trajectory methods for univariate global optimization. J Glob Optim 76, 91–120 (2020). https://doi.org/10.1007/s10898-019-00837-3

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  • DOI: https://doi.org/10.1007/s10898-019-00837-3

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