The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-04-22 DOI:10.1134/s0001434624010152

M. V. Balashov, K. Z. Biglov

引用次数: 0

Abstract

We prove that the strong convexity supporting condition is typical for an arbitrary convex compact set in \(\mathbb R^n\). It is shown that, in a certain sense for almost all points, the metric projection onto a convex compact set satisfies the Lipschitz condition with Lipschitz constant strictly less than 1. This condition characterizes the strong convexity supporting condition. The linear convergence of the alternating projection method for a convex compact set with the strong convexity supporting condition and for a proximally smooth set is proved under a certain relation between the constant in the strong convexity supporting condition and the proximal smoothness constant.

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公设投影的强凸支持条件和 Lipschitz 条件

摘要我们证明，对于 \(\mathbb R^n\) 中的任意凸紧凑集，强凸性支持条件是典型的。证明了在一定意义上对于几乎所有点，凸紧凑集上的度量投影满足 Lipschitz 条件，且 Lipschitz 常量严格小于 1。根据强凸性支撑条件中的常数与近似光滑常数之间的某种关系，证明了交替投影法对具有强凸性支撑条件的凸紧凑集和近似光滑集的线性收敛性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.