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Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications

Author

Listed:
  • Wenli Huang

    (School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China)

  • Yuchao Tang

    (Department of Mathematics, Nanchang University, Nanchang 330031, China)

  • Meng Wen

    (School of Science, Xi’an Polytechnic University, Xi’an 710048, China)

  • Haiyang Li

    (School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China)

Abstract

In this paper, a relaxed variable metric primal-dual fixed-point algorithm is proposed for solving the convex optimization problem involving the sum of two convex functions where one is differentiable with the Lipschitz continuous gradient while the other is composed of a linear operator. Based on the preconditioned forward–backward splitting algorithm, the convergence of the proposed algorithm is proved. At the same time, we show that some existing algorithms are special cases of the proposed algorithm. Furthermore, the ergodic convergence and linear convergence rates of the proposed algorithm are established under relaxed parameters. Numerical experiments on the image deblurring problems demonstrate that the proposed algorithm outperforms some existing algorithms in terms of the number of iterations.

Suggested Citation

  • Wenli Huang & Yuchao Tang & Meng Wen & Haiyang Li, 2022. "Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications," Mathematics, MDPI, vol. 10(22), pages 1-16, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4372-:d:978601
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    References listed on IDEAS

    as
    1. Franck Iutzeler & Jérôme Malick, 2018. "On the Proximal Gradient Algorithm with Alternated Inertia," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 688-710, March.
    2. Laurent Condat, 2013. "A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 460-479, August.
    3. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    4. Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    5. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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