Abstract:
We study the metric introduced by Pliś on the set of convex closed bounded subsets of a Banach space. For a real Hilbert space it is proved that metric projection and (under certain conditions) metric antiprojection from a point onto a set satisfy a Lipschitz condition with respect to the set in the Pliś metric. It is proved that solutions of a broad class of minimization problems are also Lipschitz stable with respect to the set. Several examples are discussed.
Bibliography: 18 titles.
\Bibitem{Bal19}
\by M.~V.~Balashov
\paper The Pli\'s metric and Lipschitz stability of minimization problems
\jour Sb. Math.
\yr 2019
\vol 210
\issue 7
\pages 911--927
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Linking options:
https://www.mathnet.ru/eng/sm9128
https://doi.org/10.1070/SM9128
https://www.mathnet.ru/eng/sm/v210/i7/p3
This publication is cited in the following 3 articles:
A. R. Alimov, K. S. Ryutin, I. G. Tsar'kov, “Existence, uniqueness, and stability of best and near-best approximations”, Russian Math. Surveys, 78:3 (2023), 399–442
M. V. Balashov, “The Lipschitz condition of the metric projection in the plis metric”, J. Convex Anal., 27:3 (2020), 923–934
M. V. Balashov, “Lipschitz stability of extremal problems with a strongly convex set”, J. Convex Anal., 27:1 (2020), 103–116
Citation in format AMSBIB
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