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Solving constrained nonsmooth group sparse optimization via group Capped- $$\ell _1$$ ℓ 1 relaxation and group smoothing proximal gradient algorithm

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  • Xian Zhang

    (Guizhou University)

  • Dingtao Peng

    (Guizhou University)

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.

Suggested Citation

  • Xian Zhang & Dingtao Peng, 2022. "Solving constrained nonsmooth group sparse optimization via group Capped- $$\ell _1$$ ℓ 1 relaxation and group smoothing proximal gradient algorithm," Computational Optimization and Applications, Springer, vol. 83(3), pages 801-844, December.
  • Handle: RePEc:spr:coopap:v:83:y:2022:i:3:d:10.1007_s10589-022-00419-2
    DOI: 10.1007/s10589-022-00419-2
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    References listed on IDEAS

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    1. Lukas Meier & Sara Van De Geer & Peter Bühlmann, 2008. "The group lasso for logistic regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 53-71, February.
    2. Wei Bian & Xiaojun Chen, 2017. "Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1063-1084, November.
    3. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
    4. Wei, Fengrong & Zhu, Hongxiao, 2012. "Group coordinate descent algorithms for nonconvex penalized regression," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 316-326.
    5. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    6. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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