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The Generalized Ellipsoid Method and Its Implementation

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Book cover Optimization and Applications (OPTIMA 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1145))

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Abstract

We consider an algorithm with space dilation. For a certain choice of the dilation coefficient, this is a method of outer approximation of semi-ellipsoids by ellipsoids with monotonous decrease in their volume. It is shown that the Yudin-Nemirovski-Shor ellipsoid method is a specific case. Two forms of the algorithm are dealt with: the B-form, where the inverse space transformation matrix B is updated, and the H-form, where the symmetric matrix \(H=BB^{\top }\) is updated. Our test results show that the B-form of the algorithm is computationally more robust to error accumulation than the H-form. The application of the algorithm for finding a minimizer of a convex function, for solving convex programming problems, and for determining a saddle point of a convex-concave function is described. Possible ways of accelerating the algorithm by deeper ellipsoid approximations are discussed as well.

Supported by Volkswagen Foundation (grant No 90 306).

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References

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Acknowledgement

The authors would like to thank Oleksii Lykhovyd and Volodymyr Zhydkov for their help in preparing this paper. Moreover, the authors are grateful to an anonymous referee for comments that helped to improve the paper.

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Authors and Affiliations

  1. V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, 40 Glushkov Avenue, Kyiv, 03187, Ukraine

    Petro Stetsyuk & Olga Khomyak

  2. Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    Andreas Fischer

Authors
  1. Petro Stetsyuk
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  2. Andreas Fischer
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  3. Olga Khomyak
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Corresponding author

Correspondence to Petro Stetsyuk .

Editor information

Editors and Affiliations

  1. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  2. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

  4. FRC CSC RAS, Moscow, Russia

    Mikhail Posypkin

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Stetsyuk, P., Fischer, A., Khomyak, O. (2020). The Generalized Ellipsoid Method and Its Implementation. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2019. Communications in Computer and Information Science, vol 1145. Springer, Cham. https://doi.org/10.1007/978-3-030-38603-0_26

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  • DOI: https://doi.org/10.1007/978-3-030-38603-0_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-38602-3

  • Online ISBN: 978-3-030-38603-0

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