{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:54:24Z","timestamp":1776786864368,"version":"3.51.2"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T00:00:00Z","timestamp":1658102400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T00:00:00Z","timestamp":1658102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2023,3]]},"DOI":"10.1007\/s11590-022-01906-5","type":"journal-article","created":{"date-parts":[[2022,7,18]],"date-time":"2022-07-18T17:02:36Z","timestamp":1658163756000},"page":"399-412","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Resolvent of the parallel composition and the proximity operator of the infimal postcomposition"],"prefix":"10.1007","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9609-2015","authenticated-orcid":false,"given":"Luis M.","family":"Brice\u00f1o-Arias","sequence":"first","affiliation":[]},{"given":"Fernando","family":"Rold\u00e1n","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,18]]},"reference":[{"issue":"2","key":"1906_CR1","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1287\/moor.2016.0817","volume":"42","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Bolte, J., Teboulle, M.: A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications. Math. Oper. Res. 42(2), 330\u2013348 (2017). https:\/\/doi.org\/10.1287\/moor.2016.0817","journal-title":"Math. Oper. Res."},{"key":"1906_CR2","doi-asserted-by":"publisher","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, second edn. CMS Books in Mathematics\/Ouvrages de Math\u00e9matiques de la SMC. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-48311-5","DOI":"10.1007\/978-3-319-48311-5"},{"issue":"1","key":"1906_CR3","first-page":"137","volume":"15","author":"SR Becker","year":"2014","unstructured":"Becker, S.R., Combettes, P.L.: An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery. J. Nonlin. Convex Anal. 15(1), 137\u2013159 (2014)","journal-title":"J. Nonlin. Convex Anal."},{"issue":"4","key":"1906_CR4","doi-asserted-by":"publisher","first-page":"1239","DOI":"10.1137\/050641491","volume":"17","author":"RI Bo\u0163","year":"2006","unstructured":"Bo\u0163, R.I., Grad, S.M., Wanka, G.: Maximal monotonicity for the precomposition with a linear operator. SIAM J. Optim. 17(4), 1239\u20131252 (2006). https:\/\/doi.org\/10.1137\/050641491","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1906_CR5","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1007\/s10957-017-1112-5","volume":"173","author":"K Bredies","year":"2017","unstructured":"Bredies, K., Sun, H.: A proximal point analysis of the preconditioned alternating direction method of multipliers. J. Optim. Theory Appl. 173(3), 878\u2013907 (2017). https:\/\/doi.org\/10.1007\/s10957-017-1112-5","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"1906_CR6","doi-asserted-by":"publisher","first-page":"2987","DOI":"10.1137\/21M1395144","volume":"31","author":"LM Brice\u00f1o-Arias","year":"2021","unstructured":"Brice\u00f1o-Arias, L.M., Rold\u00e1n, F.: Split-Douglas-Rachford algorithm for composite monotone inclusions and split-ADMM. SIAM J. Optim. 31(4), 2987\u20133013 (2021). https:\/\/doi.org\/10.1137\/21M1395144","journal-title":"SIAM J. Optim."},{"key":"1906_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2020.124315","author":"MN B\u00f9i","year":"2020","unstructured":"B\u00f9i, M.N., Combettes, P.L.: Warped proximal iterations for monotone inclusions. J. Math. Anal. Appl. (2020). https:\/\/doi.org\/10.1016\/j.jmaa.2020.124315","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1906_CR8","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","volume":"40","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vision 40(1), 120\u2013145 (2011). https:\/\/doi.org\/10.1007\/s10851-010-0251-1","journal-title":"J. Math. Imag. Vision"},{"issue":"9","key":"1906_CR9","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1080\/02331934.2012.733883","volume":"63","author":"PL Combettes","year":"2014","unstructured":"Combettes, P.L., V\u0169, B.C.: Variable metric forward-backward splitting with applications to monotone inclusions in duality. Optimization 63(9), 1289\u20131318 (2014). https:\/\/doi.org\/10.1080\/02331934.2012.733883","journal-title":"Optimization"},{"issue":"4","key":"1906_CR10","doi-asserted-by":"publisher","first-page":"1168","DOI":"10.1137\/050626090","volume":"4","author":"PL Combettes","year":"2005","unstructured":"Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168\u20131200 (2005). https:\/\/doi.org\/10.1137\/050626090","journal-title":"Multiscale Model. Simul."},{"key":"1906_CR11","unstructured":"Condat, L., Kitahara, D., Contreras, A., Hirabayashi, A.: Proximal splitting algorithms: A tour of recent advances, with new twists (2020). arXiv:1912.00137"},{"key":"1906_CR12","doi-asserted-by":"publisher","unstructured":"C\u00f4t\u00e9, F.D., Psaromiligkos, I.N., Gross, W.J.: A theory of generalized proximity for ADMM. In: 2017 IEEE Global conference on signal and information processing (GlobalSIP), pp. 578\u2013582 (2017). https:\/\/doi.org\/10.1109\/GlobalSIP.2017.8309025","DOI":"10.1109\/GlobalSIP.2017.8309025"},{"issue":"11","key":"1906_CR13","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1002\/cpa.20042","volume":"57","author":"I Daubechies","year":"2004","unstructured":"Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57(11), 1413\u20131457 (2004). https:\/\/doi.org\/10.1002\/cpa.20042","journal-title":"Comm. Pure Appl. Math."},{"key":"1906_CR14","doi-asserted-by":"publisher","first-page":"421","DOI":"10.2307\/1993056","volume":"82","author":"J Douglas Jr","year":"1956","unstructured":"Douglas, J., Jr., Rachford, H.H., Jr.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc. 82, 421\u2013439 (1956). https:\/\/doi.org\/10.2307\/1993056","journal-title":"Trans. Amer. Math. Soc."},{"key":"1906_CR15","doi-asserted-by":"publisher","unstructured":"Fadili, M.J., Starck, J.L.: Monotone operator splitting for optimization problems in sparse recovery. In: 2009 16th IEEE International conference on image processing (ICIP), pp. 1461\u20131464 (2009). https:\/\/doi.org\/10.1109\/ICIP.2009.5414555","DOI":"10.1109\/ICIP.2009.5414555"},{"key":"1906_CR16","doi-asserted-by":"publisher","DOI":"10.1109\/jstsp.2007.910281","author":"MAT Figueiredo","year":"2007","unstructured":"Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. Select. Topics Sig. Process. IEEE J. (2007). https:\/\/doi.org\/10.1109\/jstsp.2007.910281","journal-title":"Select. Topics Sig. Process. IEEE J."},{"key":"1906_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5610(95)00012-7","author":"M Fukushima","year":"1996","unstructured":"Fukushima, M.: The primal Douglas\u2013Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem. Math. Program. (1996). https:\/\/doi.org\/10.1016\/0025-5610(95)00012-7","journal-title":"Math. Program."},{"key":"1906_CR18","doi-asserted-by":"publisher","unstructured":"Gabay, D.: Chapter IX applications of the method of multipliers to variational inequalities. In: M.\u00a0Fortin, R.\u00a0Glowinski (eds.) Augmented Lagrangian methods: applications to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and Its Applications, vol.\u00a015, pp. 299 \u2013 331. Elsevier, New York (1983). https:\/\/doi.org\/10.1016\/S0168-2024(08)70034-1","DOI":"10.1016\/S0168-2024(08)70034-1"},{"issue":"3","key":"1906_CR19","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1137\/20M1345062","volume":"31","author":"P Giselsson","year":"2021","unstructured":"Giselsson, P.: Nonlinear forward-backward splitting with projection correction. SIAM J. Optim. 31(3), 2199\u20132226 (2021). https:\/\/doi.org\/10.1137\/20M1345062","journal-title":"SIAM J. Optim."},{"key":"1906_CR20","doi-asserted-by":"crossref","unstructured":"Jiang, X., Vandenberghe, L.: Bregman primal\u2013dual first-order method and application to sparse semidefinite programming (2021). http:\/\/www.seas.ucla.edu\/~vandenbe\/publications\/sdp-bregman.pdf","DOI":"10.1007\/s10589-021-00339-7"},{"key":"1906_CR21","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/cpa.3160200302","volume":"20","author":"JL Lions","year":"1967","unstructured":"Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493\u2013519 (1967). https:\/\/doi.org\/10.1002\/cpa.3160200302","journal-title":"Comm. Pure Appl. Math."},{"key":"1906_CR22","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/27\/4\/045009","author":"CA Micchelli","year":"2011","unstructured":"Micchelli, C.A., Shen, L., Xu, Y.: Proximity algorithms for image models: denoising. Inverse Probl. (2011). https:\/\/doi.org\/10.1088\/0266-5611\/27\/4\/045009","journal-title":"Inverse Probl."},{"key":"1906_CR23","unstructured":"Moreau, J.J.: D\u00e9composition orthogonale d\u2019un espace hilbertien selon deux c\u00f4nes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238\u2013240 (1962). https:\/\/hal.archives-ouvertes.fr\/hal-01867187\/document"},{"key":"1906_CR24","doi-asserted-by":"crossref","unstructured":"Moreau, J.J.: Proximit\u00e9 et dualit\u00e9 dans un espace hilbertien. Bull. Soc. Math. France 93, 273\u2013299 (1965). http:\/\/www.numdam.org\/item\/10.24033\/bsmf.1625.pdf","DOI":"10.24033\/bsmf.1625"},{"issue":"3","key":"1906_CR25","doi-asserted-by":"publisher","first-page":"87","DOI":"10.4067\/s0719-06462014000300007","volume":"16","author":"A Moudafi","year":"2014","unstructured":"Moudafi, A.: Computing the resolvent of composite operators. Cubo 16(3), 87\u201396 (2014). https:\/\/doi.org\/10.4067\/s0719-06462014000300007","journal-title":"Cubo"},{"issue":"3","key":"1906_CR26","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1007\/s10013-016-0238-3","volume":"45","author":"QV Nguyen","year":"2017","unstructured":"Nguyen, Q.V.: Forward-backward splitting with Bregman distances. Vietnam J. Math. 45(3), 519\u2013539 (2017). https:\/\/doi.org\/10.1007\/s10013-016-0238-3","journal-title":"Vietnam J. Math."},{"key":"1906_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-018-1321-1","author":"D O\u2019Connor","year":"2020","unstructured":"O\u2019Connor, D., Vandenberghe, L.: On the equivalence of the primal-dual hybrid gradient method and Douglas\u2013Rachford splitting. Math. Program. (2020). https:\/\/doi.org\/10.1007\/s10107-018-1321-1","journal-title":"Math. Program."},{"issue":"1","key":"1906_CR28","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1137\/18M1163993","volume":"30","author":"A Themelis","year":"2020","unstructured":"Themelis, A., Patrinos, P.: Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results. SIAM J. Optim. 30(1), 149\u2013181 (2020). https:\/\/doi.org\/10.1137\/18M1163993","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1906_CR29","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58(1), 267\u2013288 (1996)","journal-title":"J. Roy. Statist. Soc. Ser. B"},{"key":"1906_CR30","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2005.00490.x","author":"R Tibshirani","year":"2005","unstructured":"Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., Knight, K.: Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B Stat. Methodol. (2005). https:\/\/doi.org\/10.1111\/j.1467-9868.2005.00490.x","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"issue":"3","key":"1906_CR31","doi-asserted-by":"publisher","first-page":"1335","DOI":"10.1214\/11-AOS878","volume":"39","author":"RJ Tibshirani","year":"2011","unstructured":"Tibshirani, R.J., Taylor, J.: The solution path of the generalized lasso. Ann. Statist. 39(3), 1335\u20131371 (2011). https:\/\/doi.org\/10.1214\/11-AOS878","journal-title":"Ann. Statist."},{"issue":"3","key":"1906_CR32","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1007\/s10444-011-9254-8","volume":"38","author":"BC V\u0169","year":"2013","unstructured":"V\u0169, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667\u2013681 (2013). https:\/\/doi.org\/10.1007\/s10444-011-9254-8","journal-title":"Adv. Comput. Math."},{"key":"1906_CR33","doi-asserted-by":"publisher","DOI":"10.3390\/math7020131","author":"Y Yang","year":"2019","unstructured":"Yang, Y., Tang, Y., Zhu, C.: Iterative methods for computing the resolvent of composed operators in Hilbert spaces. Mathematics (2019). https:\/\/doi.org\/10.3390\/math7020131","journal-title":"Mathematics"}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01906-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-022-01906-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01906-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,29]],"date-time":"2024-09-29T10:03:27Z","timestamp":1727604207000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-022-01906-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,18]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["1906"],"URL":"https:\/\/doi.org\/10.1007\/s11590-022-01906-5","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,18]]},"assertion":[{"value":"5 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 July 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}