Primal-dual splittings as fixed point iterations in the range of linear operators

Aubin, J.P., Frankowska, H.: Set-valued Analysis. Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA (2009) https://doi.org/10.1007/978-0-8176-4848-0 Bauschke, H.H.: New demiclosedness principles for (firmly) nonexpansive operators. In: Bailey, D.H., Bauschke, H.H., Borwein, P., Garvan, F., Théra, M., Vanderwerff, J.D., Wolkowicz, H. (eds.) Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics 50, pp. 19–28. Springer, New York (2013) https://doi.org/10.1007/978-1-4614-7621-4_2 Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. Springer, New York (2017) Bauschke, H.H., Moursi, W.M.: On the Douglas-Rachford algorithm. Math. Program. 164, 263–284 (2017). https://doi.org/10.1007/s10107-016-1086-3 Boţ, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators. SIAM J. Optim. 23, 2011–2036 (2013). https://doi.org/10.1137/12088255X 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). https://doi.org/10.1137/120901106 Briceño, L., Cominetti, R., Cortés, C.E., Martínez, F.: An integrated behavioral model of land use and transport system: a hyper-network equilibrium approach. Netw. Spat. Econ. 8, 201–224 (2008). https://doi.org/10.1007/s11067-007-9052-5 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). https://doi.org/10.1137/10081602X 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., Théra, M., Vanderwerff, J., and Wolkowicz, H. (eds.) Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics 50, pp. 143–159. Springer, New York (2013) https://doi.org/10.1007/978-1-4614-7621-4_9 Briceño-Arias, L.M., Davis, D.: Forward-backward-half forward algorithm for solving monotone inclusions. SIAM J. Optim. 28, 2839–2871 (2018). https://doi.org/10.1137/17M1120099 Briceño-Arias, L.M., Deride, J., Vega, C.: Random activations in primal-dual splittings for monotone inclusions with a priori information. J. Optim. Theory Appl. 192, 56–81 (2022). https://doi.org/10.1007/s10957-021-01944-6 Briceño-Arias, L.M., López Rivera, S.: A projected primal-dual method for solving constrained monotone inclusions. J. Optim. Theory Appl 180, 907–924 (2019). https://doi.org/10.1007/s10957-018-1430-2 Briceño-Arias, L.M., Roldán, F.: Split-Douglas-Rachford algorithm for composite monotone inclusions and split-ADMM. SIAM J. Optim. 31, 2987–3013 (2021). https://doi.org/10.1137/21M1395144 Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20, 89–97 (2004). https://doi.org/10.1023/B:JMIV.0000011320.81911.38 Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery. Radon Ser. Comput. Appl. Math. 9., pp. 263–340. Walter de Gruyter, Berlin (2010) https://doi.org/10.1515/9783110226157.263 Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997). https://doi.org/10.1007/s002110050258 Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40, 120–145 (2011). https://doi.org/10.1007/s10851-010-0251-1 Colas, J., Pustelnik, N., Oliver, C., Abry, P., Géminard, J.-C., Vidal, V.: Nonlinear denoising for characterization of solid friction under low confinement pressure. Phys. Rev. E 100, 032803 (2019). https://doi.org/10.1103/PhysRevE.100.032803 Combettes, P.L.: Quasi-Fejérian analysis of some optimization algorithms. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently parallel algorithms in feasibility and optimization and their applications. Stud. Comput. Math. 8, pp. 115–152. North-Holland, Amsterdam (2001) https://doi.org/10.1016/S1570-579X(01)80010-0 Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004). https://doi.org/10.1080/02331930412331327157 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). https://doi.org/10.1007/s11228-011-0191-y Combettes, P.L., Vũ, B.C.: Variable metric quasi-Fejér monotonicity. Nonlinear Anal. 78, 17–31 (2013). https://doi.org/10.1016/j.na.2012.09.008 Combettes, P.L., Vũ, B.C.: Variable metric forward-backward splitting with applications to monotone inclusions in duality. Optimization 63, 1289–1318 (2014). https://doi.org/10.1080/02331934.2012.733883 Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158, 460–479 (2013). https://doi.org/10.1007/s10957-012-0245-9 Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413–1457 (2004). https://doi.org/10.1002/cpa.20042 Davis, D.: Convergence rate analysis of primal-dual splitting schemes. SIAM J. Optim. 25, 1912–1943 (2015). https://doi.org/10.1137/151003076 Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992). https://doi.org/10.1007/BF01581204 Fukushima, M.: The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem. Math. Program. 72, 1–15 (1996). https://doi.org/10.1016/0025-5610(95)00012-7 Gabay, D.: Chapter IX applications of the method of multipliers to variational inequalities. In: Fortin, M. and Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and Its Applications 15, pp. 299–331. Elsevier (1983) https://doi.org/10.1016/S0168-2024(08)70034-1 Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984). https://doi.org/10.1137/0322061 Glowinski, R., Marrocco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9, 41–76 (1975) Goldstein, A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70, 709–710 (1964). https://doi.org/10.1090/S0002-9904-1964-11178-2 Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2006) https://doi.org/10.1137/1.9780898718874 He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012). https://doi.org/10.1137/100814494 Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979). https://doi.org/10.1137/0716071 Mallat, S.: A wavelet tour of signal processing: the sparse way. Elsevier/Academic Press, Amsterdam (2009) Martinet, B.: Brève communication. régularisation d’inéquations variationnelles par approximations successives. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 4, 154–158 (1970) http://www.numdam.org/item/M2AN_1970__4_3_154_0 Mises, R.V., Pollaczek-Geiringer, H.: Praktische verfahren der gleichungsauflösung. Z. Angew. Math. Mech. (1929). https://doi.org/10.1002/zamm.19290090206 Molinari, C., Peypouquet, J., Roldan, F.: Alternating forward-backward splitting for linearly constrained optimization problems. Optim. Lett. 14, 1071–1088 (2020). https://doi.org/10.1007/s11590-019-01388-y Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: 2011 international conference on computer vision, pp. 1762–1769 (2011) https://doi.org/10.1109/ICCV.2011.6126441 Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976). https://doi.org/10.1137/0314056 Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D. 60, 259–268 (1992). https://doi.org/10.1016/0167-2789(92)90242-F Showalter, R.E.: Monotone operators in banach space and nonlinear partial differential equations. Mathematical surveys and monographs 49, American mathematical society, Providence, RI (1997) https://doi.org/10.1090/surv/049 Svaiter, B.F.: On weak convergence of the Douglas-Rachford method. SIAM J. Control Optim. 49, 280–287 (2011). https://doi.org/10.1137/100788100 Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667–681 (2013). https://doi.org/10.1007/s10444-011-9254-8 Yang, Y., Tang, Y., Wen, M., Zeng, T.: Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Probl. Imaging 15, 787–825 (2021). https://doi.org/10.3934/ipi.2021014