On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima
(2019) In Journal of Optimization Theory and Applications 183(1). p.66-84- Abstract
We provide theory for the computation of convex envelopes of non-convex functionals including an ℓ2-term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the ℓ2-term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch... (More)
We provide theory for the computation of convex envelopes of non-convex functionals including an ℓ2-term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the ℓ2-term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional, we necessarily have a direction in which the convex envelope is affine.
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- author
- Carlsson, Marcus LU
- organization
- publishing date
- 2019-05-29
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Convex envelope, Fenchel conjugate, Non-convex/non-smooth optimization, Proximal hull, Regularization
- in
- Journal of Optimization Theory and Applications
- volume
- 183
- issue
- 1
- pages
- 66 - 84
- publisher
- Springer
- external identifiers
-
- scopus:85066786522
- ISSN
- 0022-3239
- DOI
- 10.1007/s10957-019-01541-8
- language
- English
- LU publication?
- yes
- id
- 54692c0d-b7e9-4614-8052-f0b3dc60b7f5
- date added to LUP
- 2019-06-19 14:28:17
- date last changed
- 2022-04-26 01:40:36
@article{54692c0d-b7e9-4614-8052-f0b3dc60b7f5,
abstract = {{<p>We provide theory for the computation of convex envelopes of non-convex functionals including an ℓ<sup>2</sup>-term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the ℓ<sup>2</sup>-term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional, we necessarily have a direction in which the convex envelope is affine.</p>}},
author = {{Carlsson, Marcus}},
issn = {{0022-3239}},
keywords = {{Convex envelope; Fenchel conjugate; Non-convex/non-smooth optimization; Proximal hull; Regularization}},
language = {{eng}},
month = {{05}},
number = {{1}},
pages = {{66--84}},
publisher = {{Springer}},
series = {{Journal of Optimization Theory and Applications}},
title = {{On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima}},
url = {{http://dx.doi.org/10.1007/s10957-019-01541-8}},
doi = {{10.1007/s10957-019-01541-8}},
volume = {{183}},
year = {{2019}},
}