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Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding

Author

Listed:
  • Kristian Bredies

    (University of Graz)

  • Dirk A. Lorenz

    (TU Braunschweig)

  • Stefan Reiterer

    (University of Graz)

Abstract

Convergence analysis is carried out for a forward-backward splitting/generalized gradient projection method for the minimization of a special class of non-smooth and genuinely non-convex minimization problems in infinite-dimensional Hilbert spaces. The functionals under consideration are the sum of a smooth, possibly non-convex and non-smooth, necessarily non-convex functional. For separable constraints in the sequence space, we show that the generalized gradient projection method amounts to a discontinuous iterative thresholding procedure, which can easily be implemented. In this case we prove strong subsequential convergence and moreover show that the limit satisfies strengthened necessary conditions for a global minimizer, i.e., it avoids a certain set of non-global minimizers. Eventually, the method is applied to problems arising in the recovery of sparse data, where strong convergence of the whole sequence is shown, and numerical tests are presented.

Suggested Citation

  • Kristian Bredies & Dirk A. Lorenz & Stefan Reiterer, 2015. "Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 78-112, April.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:1:d:10.1007_s10957-014-0614-7
    DOI: 10.1007/s10957-014-0614-7
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    Cited by:

    1. Hao Jiang & Daniel P. Robinson & René Vidal & Chong You, 2018. "A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis," Computational Optimization and Applications, Springer, vol. 70(2), pages 395-418, June.
    2. 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.
    3. Daria Ghilli & Karl Kunisch, 2019. "On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 609-641, November.
    4. 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.
    5. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.
    6. Carolin Natemeyer & Daniel Wachsmuth, 2021. "A proximal gradient method for control problems with non-smooth and non-convex control cost," Computational Optimization and Applications, Springer, vol. 80(2), pages 639-677, November.
    7. Victor A. Kovtunenko & Karl Kunisch, 2022. "Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 597-635, August.
    8. Peng, Bo & Xu, Hong-Kun, 2020. "Proximal methods for reweighted lQ-regularization of sparse signal recovery," Applied Mathematics and Computation, Elsevier, vol. 386(C).

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