IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v76y2020i1d10.1007_s10898-019-00776-z.html
   My bibliography  Save this article

A residual-based algorithm for solving a class of structured nonsmooth optimization problems

Author

Listed:
  • Lei Wu

    (Jiangxi Normal University)

Abstract

In this paper, we consider a class of structured nonsmooth optimization problem in which the first component of the objective is a smooth function while the second component is the sum of one-dimensional nonsmooth functions. We first verify that every minimizer of this problem is a solution of an equation $$h(x)=0$$h(x)=0, where h is continuous but not differentiable, and moreover $$-h(x)$$-h(x) is a descent direction of the objective at $$x\in \mathbb {R}^n$$x∈Rn if $$h(x)\ne 0$$h(x)≠0. Then by using $$-h(x)$$-h(x) as a search direction, we propose a residual-based algorithm for solving this problem. Under proper conditions, we verify that any accumulation point of the sequence of iterates generated by our algorithm is a first-order stationary point of the problem. Additionally, we prove that the worst-case iteration-complexity for finding an $$\epsilon $$ϵ first-order stationary point is $$O(\epsilon ^{-2})$$O(ϵ-2). Numerical results have shown the efficiency of this algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00776-z
    DOI: 10.1007/s10898-019-00776-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00776-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-019-00776-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zhaosong Lu & Xiaorui Li, 2018. "Sparse Recovery via Partial Regularization: Models, Theory, and Algorithms," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1290-1316, November.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    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. Kai Tu & Haibin Zhang & Huan Gao & Junkai Feng, 2020. "A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems," Journal of Global Optimization, Springer, vol. 76(4), pages 665-693, April.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Lei Yang, 2024. "Proximal Gradient Method with Extrapolation and Line Search for a Class of Non-convex and Non-smooth Problems," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 68-103, January.
    10. 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.
    11. 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.
    12. Hideaki Iiduka, 2021. "Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning," Journal of Global Optimization, Springer, vol. 80(2), pages 479-505, June.
    13. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00776-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.