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On compositions of special cases of Lipschitz continuous operators

Giselsson, Pontus LU orcid and Moursi, Walaa M. (2021) In Fixed Point Theory and Algorithms for Sciences and Engineering 2021(1).
Abstract

Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.

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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Compositions of operators, Conically nonexpansive operators, Douglas–Rachford algorithm, Forward-backward algorithm, Hypoconvex function, Maximally monotone operator, Proximal operator, Resolvent
in
Fixed Point Theory and Algorithms for Sciences and Engineering
volume
2021
issue
1
article number
25
publisher
Springer
external identifiers
  • pmid:34993526
  • scopus:85121491628
ISSN
1687-1820
DOI
10.1186/s13663-021-00709-0
language
English
LU publication?
yes
additional info
Publisher Copyright: © 2021, The Author(s).
id
2d3256b0-afb8-450e-a575-030219e524f5
date added to LUP
2022-02-21 09:19:02
date last changed
2024-06-23 09:46:58
@article{2d3256b0-afb8-450e-a575-030219e524f5,
  abstract     = {{<p>Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.</p>}},
  author       = {{Giselsson, Pontus and Moursi, Walaa M.}},
  issn         = {{1687-1820}},
  keywords     = {{Compositions of operators; Conically nonexpansive operators; Douglas–Rachford algorithm; Forward-backward algorithm; Hypoconvex function; Maximally monotone operator; Proximal operator; Resolvent}},
  language     = {{eng}},
  number       = {{1}},
  publisher    = {{Springer}},
  series       = {{Fixed Point Theory and Algorithms for Sciences and Engineering}},
  title        = {{On compositions of special cases of Lipschitz continuous operators}},
  url          = {{http://dx.doi.org/10.1186/s13663-021-00709-0}},
  doi          = {{10.1186/s13663-021-00709-0}},
  volume       = {{2021}},
  year         = {{2021}},
}