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Nonlinear Forward-Backward Splitting with Momentum Correction

Morin, Martin LU ; Banert, Sebastian LU and Giselsson, Pontus LU orcid (2023) In Set-Valued and Variational Analysis 31(4).
Abstract

The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new... (More)

The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Forward-backward splitting, Forward-reflected-backward splitting, Four-operator splitting, Momentum, Monotone inclusions, Nonlinear resolvent, Primal-dual splitting
in
Set-Valued and Variational Analysis
volume
31
issue
4
article number
37
publisher
Springer
external identifiers
  • scopus:85175694370
ISSN
1877-0533
DOI
10.1007/s11228-023-00700-4
language
English
LU publication?
yes
id
e08470c0-be3e-4457-a819-6adbf146dd19
date added to LUP
2023-11-24 13:57:32
date last changed
2023-12-05 12:39:05
@article{e08470c0-be3e-4457-a819-6adbf146dd19,
  abstract     = {{<p>The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.</p>}},
  author       = {{Morin, Martin and Banert, Sebastian and Giselsson, Pontus}},
  issn         = {{1877-0533}},
  keywords     = {{Forward-backward splitting; Forward-reflected-backward splitting; Four-operator splitting; Momentum; Monotone inclusions; Nonlinear resolvent; Primal-dual splitting}},
  language     = {{eng}},
  number       = {{4}},
  publisher    = {{Springer}},
  series       = {{Set-Valued and Variational Analysis}},
  title        = {{Nonlinear Forward-Backward Splitting with Momentum Correction}},
  url          = {{http://dx.doi.org/10.1007/s11228-023-00700-4}},
  doi          = {{10.1007/s11228-023-00700-4}},
  volume       = {{31}},
  year         = {{2023}},
}