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Iterative reweighted methods for ℓ1−ℓp minimization

Iterative reweighted methods for ℓ1−ℓp minimization
Iterative reweighted methods for ℓ1−ℓp minimization
In this paper, we focus on the ℓ1−ℓpℓ1−ℓp minimization problem with 0<p<10<p<1, which is challenging due to the ℓpℓp norm being non-Lipschizian. In theory, we derive computable lower bounds for nonzero entries of the generalized first-order stationary points of ℓ1−ℓpℓ1−ℓpminimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous ϵϵ-approximation to ℓpℓp norm, we design several iterative reweighted ℓ1ℓ1 and ℓ2ℓ2methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of ℓ1−ℓpℓ1−ℓp minimization. This result, in particular, applies to the iterative reweighted ℓ1ℓ1methods based on the new Lipschitz continuous ϵϵ-approximation introduced by Lu (Math Program 147(1–2):277–307, 2014), provided that the approximation parameter ϵϵ is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.
generalized first-order stationary point, lower bound
0926-6003
201-219
Xiu, Xianchao
eeff9a9e-0e06-4853-89c6-2d0185e9a754
Kong, Lingchen
b3fb5253-440b-436b-977c-5a3140b572b8
Li, Yan
50de08b1-90ed-4557-8205-e799da43a47b
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Xiu, Xianchao
eeff9a9e-0e06-4853-89c6-2d0185e9a754
Kong, Lingchen
b3fb5253-440b-436b-977c-5a3140b572b8
Li, Yan
50de08b1-90ed-4557-8205-e799da43a47b
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85

Xiu, Xianchao, Kong, Lingchen, Li, Yan and Qi, Hou-Duo (2018) Iterative reweighted methods for ℓ1−ℓp minimization. Computational Optimization and Applications, 70 (1), 201-219. (doi:10.1007/s10589-017-9977-7).

Record type: Article

Abstract

In this paper, we focus on the ℓ1−ℓpℓ1−ℓp minimization problem with 0<p<10<p<1, which is challenging due to the ℓpℓp norm being non-Lipschizian. In theory, we derive computable lower bounds for nonzero entries of the generalized first-order stationary points of ℓ1−ℓpℓ1−ℓpminimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous ϵϵ-approximation to ℓpℓp norm, we design several iterative reweighted ℓ1ℓ1 and ℓ2ℓ2methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of ℓ1−ℓpℓ1−ℓp minimization. This result, in particular, applies to the iterative reweighted ℓ1ℓ1methods based on the new Lipschitz continuous ϵϵ-approximation introduced by Lu (Math Program 147(1–2):277–307, 2014), provided that the approximation parameter ϵϵ is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.

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Accepted/In Press date: 12 November 2017
e-pub ahead of print date: 5 January 2018
Published date: May 2018
Keywords: generalized first-order stationary point, lower bound

Identifiers

Local EPrints ID: 416895
URI: http://eprints.soton.ac.uk/id/eprint/416895
ISSN: 0926-6003
PURE UUID: 9f71b2b8-84ef-486c-80c3-006207070fb4
ORCID for Hou-Duo Qi: ORCID iD orcid.org/0000-0003-3481-4814

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Date deposited: 12 Jan 2018 17:30
Last modified: 17 Dec 2019 05:41

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