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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2019 Volume 210, Number 7, Pages 3–20 (Mi sm9128)

This article is cited in 3 papers

The Pliś metric and Lipschitz stability of minimization problems

M. V. Balashov

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

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.

Keywords: Pliś metric, Hausdorff metric, support function, strong convexity, Lipschitz continuous gradient.

UDC: 517.98

MSC: Primary 49J53, 52A20; Secondary 90C26

Received: 23.04.2018

DOI: 10.4213/sm9128


 English version:
Sbornik: Mathematics, 2019, 210:7, 911–927

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© Steklov Math. Inst. of RAS, 2025