# American Institute of Mathematical Sciences

August  2016, 10(3): 617-640. doi: 10.3934/ipi.2016014

## Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators

 1 University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria 2 Chemnitz University of Technology, Department of Mathematics, D-09107 Chemnitz, Germany

Received  June 2014 Revised  April 2016 Published  August 2016

The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some elaborated splitting techniques, all of the operators occurring in the problem formulation are processed individually via forward or backward steps. The treatment of parallel sums of linearly composed maximally monotone operators is motivated by applications in imaging which involve first- and second-order total variation functionals, to which a special attention is given.
Citation: Radu Ioan Boţ, Christopher Hendrich. Solving monotone inclusions involving parallel sums of linearly composed maximally monotone operators. Inverse Problems and Imaging, 2016, 10 (3) : 617-640. doi: 10.3934/ipi.2016014
##### References:
 [1] H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297. [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [3] S. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery, J. Nonlinear Convex A., 15 (2014), 137-159. [4] R. I. Boţ, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 637, Springer, Berlin, 2010. doi: 10.1007/978-3-642-04900-2. [5] R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X. [6] R. I. Boţ, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279. doi: 10.1007/s10107-014-0766-0. [7] R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer, Berlin, 2009. doi: 10.1007/978-3-642-02886-1. [8] R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems, TOP, 23 (2015), 124-150. doi: 10.1007/s11750-014-0326-z. [9] R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568. doi: 10.1007/s10851-013-0486-8. [10] R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems, Optimization, 64 (2015), 265-288. doi: 10.1080/02331934.2012.745530. [11] R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems, Comput. Optim. Appl., 54 (2013), 239-262. doi: 10.1007/s10589-012-9523-6. [12] R. I. Boţ and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106. [13] L. M. Briceño-Arias and P. L. Combettes, A monotone + skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250. doi: 10.1137/10081602X. [14] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [15] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [16] P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, In: D. Butnariu, Y. Censor and S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Elsevier, New York, 8 (2001), 115-152. doi: 10.1016/S1570-579X(01)80010-0. [17] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504. doi: 10.1080/02331930412331327157. [18] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. [19] P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447. doi: 10.1137/130904160. [20] P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330. doi: 10.1007/s11228-011-0191-y. [21] L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479. doi: 10.1007/s10957-012-0245-9. [22] J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Trans. of the Amer. Math. Soc., 82 (1956), 421-439. doi: 10.1090/S0002-9947-1956-0084194-4. [23] J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), 293-318. doi: 10.1007/BF01581204. [24] S. Harizanov, J.-C. Pesquet and G. Steidl, Epigraphical projection for solving least squares anscombe transformed constrained optimization problems, In: Scale Space and Variational Methods in Computer Vision, Springer, Berlin Heidelberg, 7893 (2013), 125-136. doi: 10.1007/978-3-642-38267-3_11. [25] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216. doi: 10.2140/pjm.1970.33.209. [26] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898. doi: 10.1137/0314056. [27] S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals, Commun. Math. Sci., 9 (2011), 797-827. doi: 10.4310/CMS.2011.v9.n3.a7. [28] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806. [29] B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comp. Math., 38 (2013), 667-681. doi: 10.1007/s10444-011-9254-8. [30] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.

show all references

##### References:
 [1] H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297. [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7. [3] S. Becker and P. L. Combettes, An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery, J. Nonlinear Convex A., 15 (2014), 137-159. [4] R. I. Boţ, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 637, Springer, Berlin, 2010. doi: 10.1007/978-3-642-04900-2. [5] R. I. Boţ, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X. [6] R. I. Boţ, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279. doi: 10.1007/s10107-014-0766-0. [7] R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer, Berlin, 2009. doi: 10.1007/978-3-642-02886-1. [8] R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems, TOP, 23 (2015), 124-150. doi: 10.1007/s11750-014-0326-z. [9] R. I. Boţ and C. Hendrich, Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vis., 49 (2014), 551-568. doi: 10.1007/s10851-013-0486-8. [10] R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems, Optimization, 64 (2015), 265-288. doi: 10.1080/02331934.2012.745530. [11] R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems, Comput. Optim. Appl., 54 (2013), 239-262. doi: 10.1007/s10589-012-9523-6. [12] R. I. Boţ and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106. [13] L. M. Briceño-Arias and P. L. Combettes, A monotone + skew splitting model for composite monotone inclusions in duality, SIAM J. Optim., 21 (2011), 1230-1250. doi: 10.1137/10081602X. [14] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [15] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [16] P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms, In: D. Butnariu, Y. Censor and S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Elsevier, New York, 8 (2001), 115-152. doi: 10.1016/S1570-579X(01)80010-0. [17] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504. doi: 10.1080/02331930412331327157. [18] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727-748. [19] P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim., 23 (2013), 2420-2447. doi: 10.1137/130904160. [20] P. L. Combettes and J.-C. Pesquet, Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), 307-330. doi: 10.1007/s11228-011-0191-y. [21] L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479. doi: 10.1007/s10957-012-0245-9. [22] J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Trans. of the Amer. Math. Soc., 82 (1956), 421-439. doi: 10.1090/S0002-9947-1956-0084194-4. [23] J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), 293-318. doi: 10.1007/BF01581204. [24] S. Harizanov, J.-C. Pesquet and G. Steidl, Epigraphical projection for solving least squares anscombe transformed constrained optimization problems, In: Scale Space and Variational Methods in Computer Vision, Springer, Berlin Heidelberg, 7893 (2013), 125-136. doi: 10.1007/978-3-642-38267-3_11. [25] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216. doi: 10.2140/pjm.1970.33.209. [26] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898. doi: 10.1137/0314056. [27] S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals, Commun. Math. Sci., 9 (2011), 797-827. doi: 10.4310/CMS.2011.v9.n3.a7. [28] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806. [29] B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comp. Math., 38 (2013), 667-681. doi: 10.1007/s10444-011-9254-8. [30] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.
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