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Foundations of Computational Mathematics

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19-02-2020

On the Minimal Displacement Vector of Compositions and Convex Combinations of Nonexpansive Mappings

Authors: Heinz H. Bauschke, Walaa M. Moursi

Published in: Foundations of Computational Mathematics | Issue 6/2020

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Abstract

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or “almost have” fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis–Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.

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We shall write \({\text {dom}}A = \big \{{x\in X}~\big |~{Ax\ne \varnothing }\big \}\) for the domain of A, \({\text {ran}}A = A(X) = \bigcup _{x\in X}Ax\) for the range of A, and \({\text {gra}}A=\big \{{(x,u)\in X\times X}~\big |~{u\in Ax}\big \}\) for the graph of A.

Here and elsewhere, \({\text {Id}}\) denotes the identity operator on X.

Given a nonempty closed convex subset C of X, we denote its projection mapping or projector by \({{\text {P}}}_ {C}\).

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Title: On the Minimal Displacement Vector of Compositions and Convex Combinations of Nonexpansive Mappings
Authors: Heinz H. Bauschke
Walaa M. Moursi
Publication date: 19-02-2020
Publisher: Springer US
Published in: Foundations of Computational Mathematics / Issue 6/2020
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-020-09449-w

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