How Averaged is the Composition of Two Linear Projections?

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-29 DOI:10.1080/01630563.2023.2270308

Heinz H. Bauschke, Theo Bendit, Walaa M. Moursi

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引用次数: 0

Abstract

AbstractProjection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this note, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura–Yamada bound for the modulus of the composition is sharp.KEYWORDS: Averaged mappingFriedrichs anglemodulus of averagednessnonexpansive mappingOgura–Yamada boundprojectionMATHEMATICS SUBJECT CLASSIFICATION: Primary: 47H09Secondary: 65K0590C25 AcknowledgmentsThe authors thank the reviewers and the editors for careful reading and constructive comments. We also thank Dr. Andrzej Cegielski for making us aware of his recent work [Citation3] which contains complementary results.Notes1 Usually, one excludes the cases κ = 0 and κ = 1 in the study of averaged operators, but it is very convenient in this paper to allow for this case.2 We assume for convenience throughout the paper that the operators have full domain which is the case in all algorithmic applications we are aware of. One could obviously generalize this notion to allow for operators whose domains are proper subsets of X.Additional informationFundingThe research of the authors was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada.

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两个线性投影的合成有多平均?

摘要投影算子是分析与优化中的基本算法算子。众所周知，这些算符是绝对非膨胀的;然而，它们的组成通常只是平均的，而不再是稳定的非膨胀性的。在本文中，我们引入了平均模，并给出了两个线性投影算子复合的一个精确结果。因此，我们推导出复合模的Ogura-Yamada界是尖锐的。关键词:平均映射friedrichs角平均模非膨胀映射gogura - yamada边界投影数学学科分类:初级:47h09次级:65K0590C25致谢作者感谢审稿人和编辑的认真阅读和建设性的意见。我们还要感谢Andrzej Cegielski博士使我们了解到他最近的工作，其中包含了互补的结果。注1在研究平均算子时，通常会排除κ = 0和κ = 1的情况，但在本文中考虑到这种情况是非常方便的为了方便起见，我们在本文中假设算子具有完整的域，这在我们所知道的所有算法应用中都是如此。显然，我们可以将这一概念推广到允许算子的域是x的适当子集。附加信息资金本文作者的研究部分得到了加拿大自然科学与工程研究委员会的发现补助金的支持。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.