There is no variational characterization of the cycles in the method of periodic projections
type: Item
instance of: scholarly article
title: There is no variational characterization of the cycles in the method of periodic projections
zbMATH Open document ID: 1241.46015
author: Patrick L. Combettes
author: J. B. Baillon
author: Roberto Cominetti
published in: Journal of Functional Analysis
publication date: 2011-12-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.1378
review text: The title of this paper meets two old questions, namely first, given a force, is there a potential, belonging to this force, or, given a differential equation, can it be seen as Euler's equation of a variational problem? And secondly, what about the behaviour of alternating projections on more than two nonempty closed convex (ordered) possibly nonintersecting subsets of a Hilbert space? As for the second problem remember Schwarz's alternating method, there are also (for two sets) old results of J. von Neumann, mentioned in this paper. The authors' main result is that for more than two sets in Hilbert spaces with more than two dimensions there is (under conditions) no function \(F\), such that the solution points of the cyclic projections on that sets are the unique solution of a constrained variational problem with \(F\). The proof is given by contradiction using Froda's theorem. This theorem answers a long standing question from the sixties in the negative. By the way, the authors mention that for two sets there are (under assumptions) even explicit results; there are variational problems characterizing the alternating projections. Some remarks about motivations and related projection algorithms are added.
reviewed by: Alfred Göpfert
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 47J99
zbMATH DE Number: 5988401
zbMATH Keywords: alternating projections
zbMATH Keywords: best approximation
zbMATH Keywords: limit cycle
zbMATH Keywords: Froda's theorem
zbMATH Keywords: von Neumann algorithm
MaRDI profile type: Publication
OpenAlex ID: W1976095572
arXiv ID: 1102.1378
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DOI: 10.1016/J.JFA.2011.09.002
description: scientific article