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Well Posedness of the Nearest Points Problem for Two Sets in Asymmetric Seminormed Spaces

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Optimization and Applications (OPTIMA 2019)

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

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Abstract

In this paper we consider spaces with an asymmetric seminorm and continue to study weakly convex sets. If we consider the Minkowski functional of the epigraph of some convex function as a seminorm, then the results obtained for weakly convex sets can be applied to weakly convex functions whose epigraphs are weakly convex sets with respect to this seminorm. We consider two sets in an asymmetric seminormed space, one of which is weakly convex with respect to an asymmetric seminorm, and the other one is strongly convex with respect to the asymmetric seminorm. We study the nearest points (in the sense of seminorm) problem and prove that this problem is well posed. Well posedness is an important property in the optimization theory. If a minimization problem is well posed, then one can build stable numerical algorithms, used for finding the solution of the problem.

Supported by the Russian Foundation for Basic Research, grant 18-01-00209.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation

    Mariana S. Lopushanski

Authors
  1. Mariana S. Lopushanski
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Correspondence to Mariana S. Lopushanski .

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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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Lopushanski, M.S. (2020). Well Posedness of the Nearest Points Problem for Two Sets in Asymmetric Seminormed Spaces. 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_34

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  • DOI: https://doi.org/10.1007/978-3-030-38603-0_34

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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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