# 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 & 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. doi: 10.1137/090754297. Google Scholar [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces,, CMS Books in Mathematics, (2011). doi: 10.1007/978-1-4419-9467-7. Google Scholar [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. Google Scholar [4] R. I. Boţ, Conjugate Duality in Convex Optimization,, Lecture Notes in Economics and Mathematical Systems, (2010). doi: 10.1007/978-3-642-04900-2. Google Scholar [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. doi: 10.1137/12088255X. Google Scholar [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. doi: 10.1007/s10107-014-0766-0. Google Scholar [7] R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization,, Springer, (2009). doi: 10.1007/978-3-642-02886-1. Google Scholar [8] R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems,, TOP, 23 (2015), 124. doi: 10.1007/s11750-014-0326-z. Google Scholar [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. doi: 10.1007/s10851-013-0486-8. Google Scholar [10] R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems,, Optimization, 64 (2015), 265. doi: 10.1080/02331934.2012.745530. Google Scholar [11] R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems,, Comput. Optim. Appl., 54 (2013), 239. doi: 10.1007/s10589-012-9523-6. Google Scholar [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. doi: 10.1137/120901106. Google Scholar [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. doi: 10.1137/10081602X. Google Scholar [14] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems,, Numer. Math., 76 (1997), 167. doi: 10.1007/s002110050258. Google Scholar [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. doi: 10.1007/s10851-010-0251-1. Google Scholar [16] P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms,, In: D. Butnariu, 8 (2001), 115. doi: 10.1016/S1570-579X(01)80010-0. Google Scholar [17] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators,, Optimization, 53 (2004), 475. doi: 10.1080/02331930412331327157. Google Scholar [18] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, J. Convex Anal., 16 (2009), 727. Google Scholar [19] P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications,, SIAM J. Optim., 23 (2013), 2420. doi: 10.1137/130904160. Google Scholar [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. doi: 10.1007/s11228-011-0191-y. Google Scholar [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. doi: 10.1007/s10957-012-0245-9. Google Scholar [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. doi: 10.1090/S0002-9947-1956-0084194-4. Google Scholar [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. doi: 10.1007/BF01581204. Google Scholar [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, 7893 (2013), 125. doi: 10.1007/978-3-642-38267-3_11. Google Scholar [25] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings,, Pacific J. Math., 33 (1970), 209. doi: 10.2140/pjm.1970.33.209. Google Scholar [26] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [27] S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals,, Commun. Math. Sci., 9 (2011), 797. doi: 10.4310/CMS.2011.v9.n3.a7. Google Scholar [28] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings,, SIAM J. Control Optim., 38 (2000), 431. doi: 10.1137/S0363012998338806. Google Scholar [29] B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators,, Adv. Comp. Math., 38 (2013), 667. doi: 10.1007/s10444-011-9254-8. Google Scholar [30] C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar

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. doi: 10.1137/090754297. Google Scholar [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces,, CMS Books in Mathematics, (2011). doi: 10.1007/978-1-4419-9467-7. Google Scholar [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. Google Scholar [4] R. I. Boţ, Conjugate Duality in Convex Optimization,, Lecture Notes in Economics and Mathematical Systems, (2010). doi: 10.1007/978-3-642-04900-2. Google Scholar [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. doi: 10.1137/12088255X. Google Scholar [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. doi: 10.1007/s10107-014-0766-0. Google Scholar [7] R. I. Boţ, S. M. Grad and G. Wanka, Duality in Vector Optimization,, Springer, (2009). doi: 10.1007/978-3-642-02886-1. Google Scholar [8] R. I. Boţ and C. Hendrich, A variable smoothing algorithm for solving convex optimization problems,, TOP, 23 (2015), 124. doi: 10.1007/s11750-014-0326-z. Google Scholar [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. doi: 10.1007/s10851-013-0486-8. Google Scholar [10] R. I. Boţ and C. Hendrich, On the acceleration of the double smoothing technique for unconstrained convex optimization problems,, Optimization, 64 (2015), 265. doi: 10.1080/02331934.2012.745530. Google Scholar [11] R. I. Boţ and C. Hendrich, A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems,, Comput. Optim. Appl., 54 (2013), 239. doi: 10.1007/s10589-012-9523-6. Google Scholar [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. doi: 10.1137/120901106. Google Scholar [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. doi: 10.1137/10081602X. Google Scholar [14] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems,, Numer. Math., 76 (1997), 167. doi: 10.1007/s002110050258. Google Scholar [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. doi: 10.1007/s10851-010-0251-1. Google Scholar [16] P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms,, In: D. Butnariu, 8 (2001), 115. doi: 10.1016/S1570-579X(01)80010-0. Google Scholar [17] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators,, Optimization, 53 (2004), 475. doi: 10.1080/02331930412331327157. Google Scholar [18] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, J. Convex Anal., 16 (2009), 727. Google Scholar [19] P. L. Combettes, Systems of structured monotone inclusions: Duality, algorithms, and applications,, SIAM J. Optim., 23 (2013), 2420. doi: 10.1137/130904160. Google Scholar [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. doi: 10.1007/s11228-011-0191-y. Google Scholar [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. doi: 10.1007/s10957-012-0245-9. Google Scholar [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. doi: 10.1090/S0002-9947-1956-0084194-4. Google Scholar [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. doi: 10.1007/BF01581204. Google Scholar [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, 7893 (2013), 125. doi: 10.1007/978-3-642-38267-3_11. Google Scholar [25] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings,, Pacific J. Math., 33 (1970), 209. doi: 10.2140/pjm.1970.33.209. Google Scholar [26] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [27] S. Setzer, G. Steidl and T. Teuber, Infimal convolution regularizations with discrete $l_1$-type functionals,, Commun. Math. Sci., 9 (2011), 797. doi: 10.4310/CMS.2011.v9.n3.a7. Google Scholar [28] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings,, SIAM J. Control Optim., 38 (2000), 431. doi: 10.1137/S0363012998338806. Google Scholar [29] B. C. Vũ, A splitting algorithm for dual monotone inclusions involving cocoercive operators,, Adv. Comp. Math., 38 (2013), 667. doi: 10.1007/s10444-011-9254-8. Google Scholar [30] C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar
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