Advertisement

Journal of Mathematical Imaging and Vision

, Volume 49, Issue 3, pp 551–568 | Cite as

Convergence Analysis for a Primal-Dual Monotone + Skew Splitting Algorithm with Applications to Total Variation Minimization

  • Radu Ioan BoţEmail author
  • Christopher Hendrich
Article

Abstract

In this paper we investigate the convergence behavior of a primal-dual splitting method for solving monotone inclusions involving mixtures of composite, Lipschitzian and parallel sum type operators proposed by Combettes and Pesquet (in Set-Valued Var. Anal. 20(2):307–330, 2012). Firstly, in the particular case of convex minimization problems, we derive convergence rates for the partial primal-dual gap function associated to a primal-dual pair of optimization problems by making use of conjugate duality techniques. Secondly, we propose for the general monotone inclusion problem two new schemes which accelerate the sequences of primal and/or dual iterates, provided strong monotonicity assumptions for some of the involved operators are fulfilled. Finally, we apply the theoretical achievements in the context of different types of image restoration problems solved via total variation regularization.

Keywords

Splitting method Fenchel duality Convergence statements Image processing 

Notes

Acknowledgements

The authors are thankful to the anonymous reviewers for pertinent comments and suggestions which improved the quality of the paper.

References

  1. 1.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011) CrossRefzbMATHGoogle Scholar
  2. 2.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010) zbMATHGoogle Scholar
  4. 4.
    Boţ, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators. SIAM J. Optim. 23(4), 2011–2036 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boţ, R.I., Csetnek, E.R., Heinrich, A.: On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems (2013). arXiv:1303.2875
  6. 6.
    Boţ, R.I., Hendrich, C.: A Douglas–Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23(4), 2541–2565 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230–1250 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  9. 9.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460–479 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103(1), 127–152 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Optim. 29(1), 119–138 (1991) CrossRefzbMATHGoogle Scholar
  15. 15.
    Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667–681 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002) CrossRefzbMATHGoogle Scholar
  17. 17.
    Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. CAM Reports 08-34, UCLA, Center for Applied Mathematics (2008) Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Department of MathematicsChemnitz University of TechnologyChemnitzGermany

Personalised recommendations