Solving constrained nonsmooth group sparse optimization via group Capped- $$\ell _1$$ ℓ 1 relaxation and group smoothing proximal gradient algorithm
Xian Zhang () and
Dingtao Peng ()
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Xian Zhang: Guizhou University
Dingtao Peng: Guizhou University
Computational Optimization and Applications, 2022, vol. 83, issue 3, No 4, 844 pages
Abstract 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- $$\ell _1$$ 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- $$\ell _1$$ 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.
Keywords: Constrained group sparse optimization; Exact continuous relaxation; Group Capped- $$\ell _1$$ ℓ 1 relaxation; Lifted stationary points; Group smoothing proximal gradient algorithm; 90C26; 90C30; 90C46; 65K05 (search for similar items in EconPapers)
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