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Envelope Functions : Unifications and Further Properties

Giselsson, Pontus LU and Fält, Mattias LU (2018) In Journal of Optimization Theory and Applications 178(3). p.673-698
Abstract

Forward–backward and Douglas–Rachford splitting are methods for structured nonsmooth optimization. With the aim to use smooth optimization techniques for nonsmooth problems, the forward–backward and Douglas–Rachford envelopes where recently proposed. Under specific problem assumptions, these envelope functions have favorable smoothness and convexity properties and their stationary points coincide with the fixed-points of the underlying algorithm operators. This allows for solving such nonsmooth optimization problems by minimizing the corresponding smooth convex envelope function. In this paper, we present a general envelope function that unifies and generalizes existing ones. We provide properties of the general envelope function that... (More)

Forward–backward and Douglas–Rachford splitting are methods for structured nonsmooth optimization. With the aim to use smooth optimization techniques for nonsmooth problems, the forward–backward and Douglas–Rachford envelopes where recently proposed. Under specific problem assumptions, these envelope functions have favorable smoothness and convexity properties and their stationary points coincide with the fixed-points of the underlying algorithm operators. This allows for solving such nonsmooth optimization problems by minimizing the corresponding smooth convex envelope function. In this paper, we present a general envelope function that unifies and generalizes existing ones. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present a new interpretation of the underlying methods as being majorization–minimization algorithms applied to their respective envelope functions.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Envelope functions, First-order methods, Large-scale optimization, Nonsmooth optimization, Smooth reformulations
in
Journal of Optimization Theory and Applications
volume
178
issue
3
pages
673 - 698
publisher
Springer New York
external identifiers
  • scopus:85048362045
ISSN
0022-3239
DOI
10.1007/s10957-018-1328-z
language
English
LU publication?
yes
id
621a87ae-86a0-4620-b09d-982d7ccb858e
date added to LUP
2018-06-25 14:51:45
date last changed
2019-01-14 16:47:28
@article{621a87ae-86a0-4620-b09d-982d7ccb858e,
  abstract     = {<p>Forward–backward and Douglas–Rachford splitting are methods for structured nonsmooth optimization. With the aim to use smooth optimization techniques for nonsmooth problems, the forward–backward and Douglas–Rachford envelopes where recently proposed. Under specific problem assumptions, these envelope functions have favorable smoothness and convexity properties and their stationary points coincide with the fixed-points of the underlying algorithm operators. This allows for solving such nonsmooth optimization problems by minimizing the corresponding smooth convex envelope function. In this paper, we present a general envelope function that unifies and generalizes existing ones. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present a new interpretation of the underlying methods as being majorization–minimization algorithms applied to their respective envelope functions.</p>},
  author       = {Giselsson, Pontus and Fält, Mattias},
  issn         = {0022-3239},
  keyword      = {Envelope functions,First-order methods,Large-scale optimization,Nonsmooth optimization,Smooth reformulations},
  language     = {eng},
  month        = {06},
  number       = {3},
  pages        = {673--698},
  publisher    = {Springer New York},
  series       = {Journal of Optimization Theory and Applications},
  title        = {Envelope Functions : Unifications and Further Properties},
  url          = {http://dx.doi.org/10.1007/s10957-018-1328-z},
  volume       = {178},
  year         = {2018},
}