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Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2 –ℓ p minimization

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  • Xiaojun Chen
  • Weijun Zhou

Abstract

The iteratively reweighted ℓ 1 minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ℓ 2 –ℓ p minimization problem with 0>p>1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Xiaojun Chen & Weijun Zhou, 2014. "Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2 –ℓ p minimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 47-61, October.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:47-61
    DOI: 10.1007/s10589-013-9553-8
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    Citations

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    Cited by:

    1. Lei Wu, 2020. "A residual-based algorithm for solving a class of structured nonsmooth optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 137-153, January.
    2. Xuerui Gao & Yanqin Bai & Qian Li, 2021. "A sparse optimization problem with hybrid $$L_2{\text {-}}L_p$$ L 2 - L p regularization for application of magnetic resonance brain images," Journal of Combinatorial Optimization, Springer, vol. 42(4), pages 760-784, November.
    3. Yun-Bin Zhao & Zhi-Quan Luo, 2017. "Constructing New Weighted ℓ 1 -Algorithms for the Sparsest Points of Polyhedral Sets," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 57-76, January.
    4. S. M. Mirhadi & S. A. MirHassani, 2022. "A solution approach for cardinality minimization problem based on fractional programming," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 583-602, August.
    5. Peiran Yu & Ting Kei Pong, 2019. "Iteratively reweighted $$\ell _1$$ ℓ 1 algorithms with extrapolation," Computational Optimization and Applications, Springer, vol. 73(2), pages 353-386, June.
    6. Zhili Ge & Zhongming Wu & Xin Zhang & Qin Ni, 2023. "An extrapolated proximal iteratively reweighted method for nonconvex composite optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 821-844, August.
    7. Daria Ghilli & Karl Kunisch, 2019. "On monotone and primal-dual active set schemes for $$\ell ^p$$ ℓ p -type problems, $$p \in (0,1]$$ p ∈ ( 0 , 1 ]," Computational Optimization and Applications, Springer, vol. 72(1), pages 45-85, January.
    8. Xuerui Gao & Yanqin Bai & Qian Li, 0. "A sparse optimization problem with hybrid $$L_2{\text {-}}L_p$$L2-Lp regularization for application of magnetic resonance brain images," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-25.
    9. Xianchao Xiu & Lingchen Kong & Yan Li & Houduo Qi, 2018. "Iterative reweighted methods for $$\ell _1-\ell _p$$ ℓ 1 - ℓ p minimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 201-219, May.
    10. Zhaosong Lu & Yong Zhang & Jian Lu, 2017. "$$\ell _p$$ ℓ p Regularized low-rank approximation via iterative reweighted singular value minimization," Computational Optimization and Applications, Springer, vol. 68(3), pages 619-642, December.
    11. Tao Sun & Hao Jiang & Lizhi Cheng, 2017. "Global convergence of proximal iteratively reweighted algorithm," Journal of Global Optimization, Springer, vol. 68(4), pages 815-826, August.

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