IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v83y2022i2d10.1007_s10589-022-00396-6.html
   My bibliography  Save this article

Some modified fast iterative shrinkage thresholding algorithms with a new adaptive non-monotone stepsize strategy for nonsmooth and convex minimization problems

Author

Listed:
  • Hongwei Liu

    (Xidian University)

  • Ting Wang

    (Xidian University)

  • Zexian Liu

    (Guizhou University)

Abstract

The “ fast iterative shrinkage-thresholding algorithm " (FISTA) is one of the most famous first order optimization schemes, and the stepsize, which plays an important role in theoretical analysis and numerical experiment, is always determined by a constant relating to the Lipschitz constant or by a backtracking strategy. In this paper, we design a new adaptive non-monotone stepsize strategy (NMS), which allows the stepsize to increase monotonically after finite iterations. It is remarkable that NMS can be successfully implemented without knowing the Lipschitz constant or without backtracking. And the additional cost of NMS is less than the cost of some existing backtracking strategies. For using NMS to the original FISTA (FISTA_NMS) and the modified FISTA (MFISTA_NMS), we show that the convergence results stay the same. Moreover, under the error bound condition, we show that FISTA_NMS achieves the rate of convergence to $$o\left( {\frac{1}{{{k^6}}}} \right) $$ o 1 k 6 and MFISTA_NMS enjoys the convergence rate related to the value of parameter of $$t_k$$ t k , that is $$o\left( {\frac{1}{{{k^{2\left( {a + 1} \right) }}}}} \right) ;$$ o 1 k 2 a + 1 ; and the iterates generated by the above two algorithms are convergent. In addition, by taking advantage of the restart technique to accelerate the above two methods, we establish the linear convergences of the function values and iterates under the error bound condition. We conduct some numerical experiments to examine the effectiveness of the proposed algorithms.

Suggested Citation

  • Hongwei Liu & Ting Wang & Zexian Liu, 2022. "Some modified fast iterative shrinkage thresholding algorithms with a new adaptive non-monotone stepsize strategy for nonsmooth and convex minimization problems," Computational Optimization and Applications, Springer, vol. 83(2), pages 651-691, November.
    • Handle: RePEc:spr:coopap:v:83:y:2022:i:2:d:10.1007_s10589-022-00396-6
    DOI: 10.1007/s10589-022-00396-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-022-00396-6
    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/s10589-022-00396-6?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. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    2. Cesare Molinari & Jingwei Liang & Jalal Fadili, 2019. "Convergence Rates of Forward–Douglas–Rachford Splitting Method," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 606-639, August.
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. A. Chambolle & Ch. Dossal, 2015. "On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 968-982, September.
    5. Amir Beck & Marc Teboulle, 2006. "A Linearly Convergent Dual-Based Gradient Projection Algorithm for Quadratically Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 398-417, May.
    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. Donghwan Kim & Jeffrey A. Fessler, 2018. "Adaptive Restart of the Optimized Gradient Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 240-263, July.
    2. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    3. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Sun, Shilin & Wang, Tianyang & Yang, Hongxing & Chu, Fulei, 2022. "Damage identification of wind turbine blades using an adaptive method for compressive beamforming based on the generalized minimax-concave penalty function," Renewable Energy, Elsevier, vol. 181(C), pages 59-70.
    5. S. Bonettini & M. Prato & S. Rebegoldi, 2018. "A block coordinate variable metric linesearch based proximal gradient method," Computational Optimization and Applications, Springer, vol. 71(1), pages 5-52, September.
    6. Patrick R. Johnstone & Pierre Moulin, 2017. "Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions," Computational Optimization and Applications, Springer, vol. 67(2), pages 259-292, June.
    7. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    8. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
    9. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    10. Masoud Ahookhosh & Arnold Neumaier, 2018. "Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 )," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 110-145, April.
    11. Oumaima Benchettou & Abdeslem Hafid Bentbib & Abderrahman Bouhamidi, 2023. "An Accelerated Tensorial Double Proximal Gradient Method for Total Variation Regularization Problem," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 111-134, July.
    12. Kaiwen Ma & Nikolaos V. Sahinidis & Sreekanth Rajagopalan & Satyajith Amaran & Scott J Bury, 2021. "Decomposition in derivative-free optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 269-292, October.
    13. Hao Wang & Hao Zeng & Jiashan Wang, 2022. "An extrapolated iteratively reweighted $$\ell _1$$ ℓ 1 method with complexity analysis," Computational Optimization and Applications, Springer, vol. 83(3), pages 967-997, December.
    14. Guillaume Sagnol & Edouard Pauwels, 2019. "An unexpected connection between Bayes A-optimal designs and the group lasso," Statistical Papers, Springer, vol. 60(2), pages 565-584, April.
    15. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    16. Junhong Lin & Lorenzo Rosasco & Silvia Villa & Ding-Xuan Zhou, 2018. "Modified Fejér sequences and applications," Computational Optimization and Applications, Springer, vol. 71(1), pages 95-113, September.
    17. A. Scagliotti & P. Colli Franzone, 2022. "A piecewise conservative method for unconstrained convex optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 251-288, January.
    18. Ren Jiang & Zhifeng Ji & Wuling Mo & Suhua Wang & Mingjun Zhang & Wei Yin & Zhen Wang & Yaping Lin & Xueke Wang & Umar Ashraf, 2022. "A Novel Method of Deep Learning for Shear Velocity Prediction in a Tight Sandstone Reservoir," Energies, MDPI, vol. 15(19), pages 1-20, September.
    19. Silvia Bonettini & Peter Ochs & Marco Prato & Simone Rebegoldi, 2023. "An abstract convergence framework with application to inertial inexact forward–backward methods," Computational Optimization and Applications, Springer, vol. 84(2), pages 319-362, March.
    20. Puya Latafat & Panagiotis Patrinos, 2017. "Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators," Computational Optimization and Applications, Springer, vol. 68(1), pages 57-93, September.

    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:coopap:v:83:y:2022:i:2:d:10.1007_s10589-022-00396-6. 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.