Partial gradient optimal thresholding algorithms for a class of sparse optimization problems

Nan Meng, Yun-Bin Zhao, Michal Kocvara, Zhongfeng Sun

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Abstract

The optimization problems with a sparsity constraint is a class of important global optimization problems. A typical type of thresholding algorithms for solving such a problem adopts the traditional full steepest descent direction or Newton-like direction as a search direction to generate an iterate on which a certain thresholding is performed. Traditional hard thresholding discards a large part of a vector, and thus some important information contained in a dense vector has been lost in such a thresholding process. Recent study (Zhao in SIAM J Optim 30(1): 31–55, 2020) shows that the hard thresholding should be applied to a compressible vector instead of a dense vector to avoid a big loss of information. On the other hand, the optimal k-thresholding as a novel thresholding technique may overcome the intrinsic drawback of hard thresholding, and performs thresholding and objective function minimization simultaneously. This motivates us to propose the so-called partial gradient optimal thresholding (PGOT) method and its relaxed versions in this paper. The PGOT is an integration of the partial gradient and the optimal k-thresholding technique. The solution error bound and convergence for the proposed algorithms have been established in this paper under suitable conditions. Application of our results to the sparse optimization problems arising from signal recovery is also discussed. Experiment results from synthetic data indicate that the proposed algorithm is efficient and comparable to several existing algorithms.
Original languageEnglish
Number of pages21
JournalJournal of Global Optimization
Early online date23 Feb 2022
DOIs
Publication statusE-pub ahead of print - 23 Feb 2022

Bibliographical note

Funding Information:
This work was founded by the National Natural Science Foundation of China (NSFC) under the grant 12071307.

Publisher Copyright:
© 2022, The Author(s).

Keywords

  • Sparse optimization
  • Signal recovery
  • Optimal k-thresholding
  • Partial gradient method
  • Error bound
  • Convergence

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