Vietnam Journal of Mathematics

, Volume 43, Issue 2, pp 323–341

# A Splitting Algorithm for System of Composite Monotone Inclusions

• Dinh Dũng
• Bằng Công Vũ
Article

## Abstract

We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resulting algorithm is an extension of the algorithm in Becker and Combettes (J. Convex Nonlinear Anal. 15, 137–159, 2014). The weak convergence of the algorithm proposed is proved. Applications to minimization problems is demonstrated.

## Keywords

Coupled system Monotone inclusion Monotone operator Splitting method Lipschitzian operator Forward-backward-forward algorithm Composite operator Duality Primal-dual algorithm

## Mathematics Subject Classification (2010)

47H05 49M29 49M27 90C25

## Notes

### Acknowledgments

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2014.02. A part of the research work of Dinh Dũng was done when the author was working as a research professor at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for providing a fruitful research environment and working condition. We thank the referees for their suggestions and corrections which helped to improve the first version of the manuscript.

## References

1. 1.
Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim. 48, 3246–3270 (2010)
2. 2.
Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3, 1–24 (1996)
3. 3.
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
4. 4.
Becker, S., Combettes, P.L.: An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery. J. Nonlinear Convex Anal. 15, 137–159 (2014)
5. 5.
Boţ, R.I., Csetnek, E.R., Nagy, E.: Solving systems of monotone inclusions via primal-dual splitting techniques. Taiwan. J. Math. 17, 1983–2009 (2013)
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, 2541–2565 (2013)
7. 7.
Boţ, R.I., Hendrich, C.: Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators. arXiv:1306.3191(2013)
8. 8.
Briceño-Arias, L.M., Combettes, P.L.: Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery. Numer. Math. Theory Methods Appl. 2, 485–508 (2009)
9. 9.
Briceño-Arias, L.M., Combettes, P.L., Pesquet, J.-C., Pustelnik, N.: Proximal algorithms for multicomponent image recovery problems. J. Math. Imaging Vis. 41, 3–22 (2011)
10. 10.
Briceño-Arias, L.M., Combettes, P.L.: Monotone operator methods for Nash equilibria in non-potential games. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Thera, M., Vanderwerff, J.D., Wolkowicz, H. (eds.) Computational and Analytical Mathematics. Springer, New York (2013)Google Scholar
11. 11.
Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21, 1230–1250 (2011)
12. 12.
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)
13. 13.
Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270. Academic Press, New York (1996)Google Scholar
14. 14.
Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)
15. 15.
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model Simul. 4, 1168–1200 (2005)
16. 16.
Combettes, P.L.: Systems of structured monotone inclusions: duality, algorithms, and applications. SIAM J. Optim. 23, 2420–2447 (2013)
17. 17.
Combettes, P.L., Dũng, D, Vũ, B.C.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380, 680–688 (2011)
18. 18.
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, 307–330 (2012)
19. 19.
Jezierska, A., Chouzenoux, E., Pesquet, J.-C., Talbot, H.: A primal-dual proximal splitting approach for restoring data corrupted with Poisson–Gaussian noise. In: Proc. Int. Conf. Acoust., Speech Signal Process., pp. 1085–1088, Kyoto, Japan (2012)Google Scholar
20. 20.
Eckstein, J., Svaiter, B.F.: A family of projective splitting methods for the sum of two maximal monotone operators. Math. Program., Ser. B 111, 173–199 (2008)
21. 21.
Papafitsoros, K., Schönlieb, C.-B., Sengul, B.: Combined first and second order total variation inpainting using split Bregman. Image Process On Line 3, 112–136 (2013)
22. 22.
Svaiter, B.F.: On weak convergence on Douglas–Rachford method. SIAM J. Control Optim. 49, 280–287 (2011)
23. 23.
Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)
24. 24.
Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667–681 (2013)
25. 25.
Vũ, B.C.: A splitting algorithm for coupled system of primal-dual monotone inclusions. J. Optim. Theory Appl. (2013). doi: Google Scholar