Solving constrained nonsmooth group sparse optimization via group Capped-ℓ1 relaxation and group smoothing proximal gradient algorithm

In this paper, we study the constrained group sparse regularization optimization problem, where the loss function is convex but nonsmooth, and the penalty term is the group sparsity which is then proposed to be relaxed by the group Capped- ℓ 1 for the convenience of computation. Firstly, we introduc...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 83; no. 3; pp. 801 - 844
Main Authors: Zhang, Xian, Peng, Dingtao
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2022
Springer Nature B.V
Subjects:
Algorithms
Convex and Discrete Geometry
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Regularization
Smoothing
Statistics
Constrained group sparse optimization
Exact continuous relaxation
90C30
65K05
90C46
Lifted stationary points
relaxation
Group Capped
Group smoothing proximal gradient algorithm
90C26
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:In this paper, we study the constrained group sparse regularization optimization problem, where the loss function is convex but nonsmooth, and the penalty term is the group sparsity which is then proposed to be relaxed by the group Capped- ℓ 1 for the convenience of computation. Firstly, we introduce three kinds of stationary points for the continuous relaxation problem and describe the relationship of the three kinds of stationary points. We give the optimality conditions for the group Capped- ℓ 1 problem and the original group sparse regularization problem, and investigate the link between the relaxation problem and the original problem in terms of global and local optimal solutions. The results reveal the equivalence of the original problem and the relaxation problem in some sense. Secondly, we propose a group smoothing proximal gradient (GSPG) algorithm for the constrained group sparse optimization, and prove that the proposed GSPG algorithm globally converges to a lifted stationary point of the relaxation problem. Finally, we present some numerical results on recovery of the simulated group sparse signals and the real group sparse images to illustrate the efficiency of the continuous relaxation model and the proposed algorithm.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00419-2