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Evolving Smoothing Kernels for Global Optimization

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Applications of Evolutionary Computation (EvoApplications 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9598))

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Abstract

The Diffusion-Equation Method (DEM) – sometimes synonymously called the Continuation Method – is a well-known natural computation approach in optimization. The DEM continuously transforms the objective function by a (Gaussian) kernel technique to reduce barriers separating local and global minima. Now, the DEM can successfully solve problems of small sizes. Here, we present a generalization of the DEM to use convex combinations of smoothing kernels in Fourier space. We use a genetic algorithm to incrementally optimize the (meta-)combinatorial problem of finding better performing kernels for later optimization of an objective function. For two test applications we derive and show their transferability to larger problems. Most strikingly, the original DEM failed on a number of the test instances to find the global optimum while our transferable kernels – obtained via evolutionary computations – were able to find the global optimum.

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Notes

  1. 1.

    For example, the DEM fails to converge to the global optimum of \(g(x) = \cos (x-3.4) + 6\cos (2x + 1) + 3\cos (3x+2.5)\) for all starting points.

  2. 2.

    For \(\alpha \)’s the width of the normal distribution was set to 0.2 and for the \(\nu \)’s to 1.0.

  3. 3.

    Here degrees of freedom = \(3(N-2)\) due to a free choice of the point of origin and orientation, thus translational and rotational invariance of the potential V. The prefactor 3 is the relation between no. of particles and no. of Eucledian coordinates.

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Acknowledgements

KH gratefully acknowledges funding by the LOEWE project compuGene of the Hessen State Ministry of Higher Education, Research and the Arts.

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Authors and Affiliations

  1. Department of Computer Science, TU Darmstadt, Darmstadt, Germany

    Paul Manns & Kay Hamacher

  2. Department of Biology, TU Darmstadt, Darmstadt, Germany

    Kay Hamacher

  3. Department of Physics, TU Darmstadt, Darmstadt, Germany

    Kay Hamacher

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  1. Paul Manns
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  2. Kay Hamacher
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Correspondence to Kay Hamacher .

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Editors and Affiliations

  1. Politecnico di Torino, Turin, Italy

    Giovanni Squillero

  2. Aalborg University, Copenhagen, Denmark

    Paolo Burelli

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Manns, P., Hamacher, K. (2016). Evolving Smoothing Kernels for Global Optimization. In: Squillero, G., Burelli, P. (eds) Applications of Evolutionary Computation. EvoApplications 2016. Lecture Notes in Computer Science(), vol 9598. Springer, Cham. https://doi.org/10.1007/978-3-319-31153-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-31153-1_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-31152-4

  • Online ISBN: 978-3-319-31153-1

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