{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T16:28:42Z","timestamp":1750177722888,"version":"3.37.3"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,1,30]],"date-time":"2017-01-30T00:00:00Z","timestamp":1485734400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["Sigma-Vision"],"award-info":[{"award-number":["Sigma-Vision"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004795","name":"Institut Universitaire de France","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004795","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2017,3]]},"DOI":"10.1007\/s10957-017-1061-z","type":"journal-article","created":{"date-parts":[[2017,1,30]],"date-time":"2017-01-30T20:28:00Z","timestamp":1485808080000},"page":"874-913","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Local Convergence Properties of Douglas\u2013Rachford and Alternating Direction Method of Multipliers"],"prefix":"10.1007","volume":"172","author":[{"given":"Jingwei","family":"Liang","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8165-7578","authenticated-orcid":false,"given":"Jalal","family":"Fadili","sequence":"additional","affiliation":[]},{"given":"Gabriel","family":"Peyr\u00e9","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,1,30]]},"reference":[{"issue":"2","key":"1061_CR1","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","volume":"82","author":"J Douglas","year":"1956","unstructured":"Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421\u2013439 (1956)","journal-title":"Trans. Am. Math. Soc."},{"issue":"6","key":"1061_CR2","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","volume":"16","author":"PL Lions","year":"1979","unstructured":"Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964\u2013979 (1979)","journal-title":"SIAM J. Numer. Anal."},{"key":"1061_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0964-4","author":"J Liang","year":"2015","unstructured":"Liang, J., Fadili, M.J., Peyr\u00e9, G.: Convergence rates with inexact non-expansive operators. Math Program (2015). doi: 10.1007\/s10107-015-0964-4","journal-title":"Math Program"},{"key":"1061_CR4","unstructured":"Davis, D., Yin, W.: Convergence rate analysis of several splitting schemes. Technical Report arXiv:1406.4834 (2014a)"},{"key":"1061_CR5","unstructured":"Davis, D., Yin, W.: Convergence rates of relaxed Peaceman\u2013Rachford and ADMM under regularity assumptions. Technical Report arXiv:1407.5210 (2014b)"},{"key":"1061_CR6","doi-asserted-by":"crossref","unstructured":"Giselsson, P., Boyd, S.: Metric selection in Douglas\u2013Rachford Splitting and ADMM. arXiv preprint arXiv:1410.8479 (2014)","DOI":"10.1109\/CDC.2014.7040175"},{"key":"1061_CR7","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0168-2024(08)70034-1","volume-title":"Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems","author":"D Gabay","year":"1983","unstructured":"Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, pp. 299\u2013331. Elsevier, North-Holland, Amsterdam (1983)"},{"issue":"1\u20133","key":"1061_CR8","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., Bertsekas, D.P.: On the Douglas\u2013Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1\u20133), 293\u2013318 (1992)","journal-title":"Math. Program."},{"issue":"3","key":"1061_CR9","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1137\/S1052623401387623","volume":"13","author":"AS Lewis","year":"2003","unstructured":"Lewis, A.S.: Active sets, nonsmoothness, and sensitivity. SIAM J. Optim. 13(3), 702\u2013725 (2003)","journal-title":"SIAM J. Optim."},{"issue":"297","key":"1061_CR10","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1090\/mcom\/2965","volume":"85","author":"L Demanet","year":"2016","unstructured":"Demanet, L., Zhang, X.: Eventual linear convergence of the Douglas\u2013Rachford iteration for basis pursuit. Math. Comput. 85(297), 209\u2013238 (2016)","journal-title":"Math. Comput."},{"issue":"4","key":"1061_CR11","doi-asserted-by":"crossref","first-page":"2183","DOI":"10.1137\/120878951","volume":"23","author":"D Boley","year":"2013","unstructured":"Boley, D.: Local linear convergence of the alternating direction method of multipliers on quadratic or linear programs. SIAM J. Optim. 23(4), 2183\u20132207 (2013)","journal-title":"SIAM J. Optim."},{"key":"1061_CR12","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.jat.2014.06.002","volume":"185","author":"H Bauschke","year":"2014","unstructured":"Bauschke, H., Cruz, J., Nghia, T., Phan, H., Wang, X.: The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the Friedrichs angle. J. Approx. Theory 185, 63\u201379 (2014)","journal-title":"J. Approx. Theory"},{"key":"1061_CR13","doi-asserted-by":"crossref","unstructured":"Liang, J., Fadili, M.J., Peyr\u00e9, G., Luke, R.: Activity identification and local linear convergence of Douglas\u2013Rachford\/ADMM under partial smoothness. In: Scale Space and Variational Methods in Computer Vision, pp. 642\u2013653. Springer (2015)","DOI":"10.1007\/978-3-319-18461-6_51"},{"key":"1061_CR14","first-page":"93","volume-title":"Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications","author":"JM Borwein","year":"2011","unstructured":"Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence of convexity. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, vol. 49, pp. 93\u2013109. Springer, New York (2011)"},{"issue":"4","key":"1061_CR15","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s10208-008-9036-y","volume":"9","author":"AS Lewis","year":"2009","unstructured":"Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485\u2013513 (2009)","journal-title":"Found. Comput. Math."},{"key":"1061_CR16","unstructured":"Hesse, R., Luke, D.R., Neumann, P.: Projection methods for sparse affine feasibility: results and counterexamples. Technical Report (2013)"},{"issue":"4","key":"1061_CR17","doi-asserted-by":"crossref","first-page":"2397","DOI":"10.1137\/120902653","volume":"23","author":"R Hesse","year":"2013","unstructured":"Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4), 2397\u20132419 (2013)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1061_CR18","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1080\/02331934.2015.1051532","volume":"65","author":"HM Phan","year":"2016","unstructured":"Phan, H.M.: Linear convergence of the Douglas\u2013Rachford method for two closed sets. Optimization 65(2), 369\u2013385 (2016)","journal-title":"Optimization"},{"key":"1061_CR19","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Dao, M.N., Noll, D., Phan, H.M.: On Slater\u2019s condition and finite convergence of the Douglas\u2013Rachford algorithm for solving convex feasibility problems in Euclidean spaces. J. Global Optim. pp. 1\u201321 (2015) (In press)","DOI":"10.1007\/s10898-015-0373-5"},{"key":"1061_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)"},{"issue":"1","key":"1061_CR21","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.jmaa.2014.11.044","volume":"425","author":"PL Combettes","year":"2015","unstructured":"Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425(1), 55\u201370 (2015)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1061_CR22","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s11075-015-0085-4","volume":"73","author":"HH Bauschke","year":"2016","unstructured":"Bauschke, H.H., Bello Cruz, J.Y., Nghia, T.T.A., Pha, H.M., Wang, X.: Optimal rates of linear convergence of relaxed alternating projections and generalized Douglas-Rachford methods for two subspaces. Numer. Algorithms 73(1), 33\u201376 (2016). doi: 10.1007\/s11075-015-0085-4","journal-title":"Numer. Algorithms"},{"key":"1061_CR23","doi-asserted-by":"publisher","first-page":"1016","DOI":"10.1007\/978-0-387-74759-0_179","volume-title":"Encyclopedia of Optimization","author":"PL Combettes","year":"2001","unstructured":"Combettes, P.L.: Fej\u00e9r monotonicity in convex optimization. In: Floudas, A.C., Pardalos, M.P. (eds.) Encyclopedia of Optimization, pp. 1016\u20131024. Springer, Boston (2001). doi: 10.1007\/978-0-387-74759-0_179"},{"issue":"5\u20136","key":"1061_CR24","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1080\/02331930412331327157","volume":"53","author":"PL Combettes","year":"2004","unstructured":"Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5\u20136), 475\u2013504 (2004)","journal-title":"Optimization"},{"key":"1061_CR25","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Moursi, W.: On the order of the operators in the Douglas\u2013Rachford algorithm. Optim. Lett. (2016). In press ( arXiv:1505.02796v1 )","DOI":"10.1007\/s10107-016-1086-3"},{"key":"1061_CR26","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S1570-579X(01)80010-0","volume":"8","author":"PL Combettes","year":"2001","unstructured":"Combettes, P.L.: Quasi\u2013Fej\u00e9rian analysis of some optimization algorithms. Stud. Comput Math 8, 115\u2013152 (2001)","journal-title":"Stud. Comput Math"},{"issue":"4","key":"1061_CR27","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1137\/0331048","volume":"31","author":"SJ Wright","year":"1993","unstructured":"Wright, S.J.: Identifiable surfaces in constrained optimization. SIAM J. Control Optim. 31(4), 1063\u20131079 (1993)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1061_CR28","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1090\/S0002-9947-99-02243-6","volume":"352","author":"C Lemar\u00e9chal","year":"2000","unstructured":"Lemar\u00e9chal, C., Oustry, F., Sagastiz\u00e1bal, C.: The U-lagrangian of a convex function. Trans. Am. Math. Soc. 352(2), 711\u2013729 (2000)","journal-title":"Trans. Am. Math. Soc."},{"key":"1061_CR29","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1137\/130916710","volume":"35","author":"A Daniilidis","year":"2014","unstructured":"Daniilidis, A., Drusvyatskiy, D., Lewis, A.S.: Orthogonal invariance and identifiability. SIAM J. Matrix Anal. Appl. 35, 580\u2013598 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1061_CR30","unstructured":"Liang, J., Fadili, M.J., Peyr\u00e9, G.: Activity identification and local linear convergence of Forward\u2013Backward-type methods (2015). Submitted ( arXiv:1503.03703 )"},{"issue":"2","key":"1061_CR31","first-page":"75","volume":"2","author":"W Hare","year":"2007","unstructured":"Hare, W., Lewis, A.S.: Identifying active manifolds. Algorithm. Oper. Res. 2(2), 75\u201382 (2007)","journal-title":"Algorithm. Oper. Res."},{"key":"1061_CR32","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.: Variational Analysis, vol. 317. Springer, Berlin (1998)"},{"key":"1061_CR33","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1997","unstructured":"Rockafellar, R.T.: Convex Analysis, vol. 28. Princeton University Press, Princeton (1997)"},{"key":"1061_CR34","first-page":"101","volume":"25","author":"N Kim","year":"2000","unstructured":"Kim, N., Luc, D.: Normal cones to a polyhedral convex set and generating efficient faces in multiobjective linear programming. Acta Math. Vietnam. 25, 101\u2013124 (2000)","journal-title":"Acta Math. Vietnam."},{"issue":"5","key":"1061_CR35","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877\u2013898 (1976)","journal-title":"SIAM J. Control Optim."},{"key":"1061_CR36","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1137\/0322019","volume":"22","author":"F Luque","year":"1984","unstructured":"Luque, F.: Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control Optim. 22, 277\u2013293 (1984)","journal-title":"SIAM J. Control Optim."},{"key":"1061_CR37","unstructured":"Combettes, P.L., Pesquet, J.C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24(6), 065,014 (2008). http:\/\/stacks.iop.org\/0266-5611\/24\/i=6\/a=065014"},{"issue":"3","key":"1061_CR38","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1137\/120872802","volume":"6","author":"H Raguet","year":"2013","unstructured":"Raguet, H., Fadili, M.J., Peyr\u00e9, G.: A generalized forward\u2013backward splitting. SIAM J. Imaging Sci. 6(3), 1199\u20131226 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"1061_CR39","doi-asserted-by":"crossref","unstructured":"Vaiter, S., Deledalle, C., Fadili, J.M., Peyr\u00e9, G., Dossal, C.: The degrees of freedom of partly smooth regularizers. Ann. Inst. Stat. Math. (2015) arXiv:1404.5557 . To appear","DOI":"10.1007\/s10463-016-0563-z"},{"key":"1061_CR40","unstructured":"Vaiter, S., Peyr\u00e9, G., Fadili, M.J.: Model consistency of partly smooth regularizers. Technical Report arXiv:1307.2342 , submitted (2015)"},{"key":"1061_CR41","unstructured":"Br\u00e9zis, H.: Op\u00e9rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. In: North-Holland Mathematics Studies. Elsevier, New York (1973)"},{"issue":"2","key":"1061_CR42","first-page":"251","volume":"11","author":"WL Hare","year":"2004","unstructured":"Hare, W.L., Lewis, A.S.: Identifying active constraints via partial smoothness and prox-regularity. J. Convex Anal. 11(2), 251\u2013266 (2004)","journal-title":"J. Convex Anal."},{"key":"1061_CR43","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511616822","volume-title":"Riemannian Geometry: A Modern Introduction","author":"I Chavel","year":"2006","unstructured":"Chavel, I.: Riemannian Geometry: A Modern Introduction, vol. 98. Cambridge University Press, Cambridge (2006)"},{"issue":"2\u20133","key":"1061_CR44","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s10107-005-0631-2","volume":"104","author":"SA Miller","year":"2005","unstructured":"Miller, S.A., Malick, J.: Newton methods for nonsmooth convex minimization: connections among-Lagrangian, Riemannian Newton and SQP methods. Math. Program. 104(2\u20133), 609\u2013633 (2005)","journal-title":"Math. Program."},{"key":"1061_CR45","doi-asserted-by":"crossref","unstructured":"Absil, P.A., Mahony, R., Trumpf, J.: An extrinsic look at the Riemannian Hessian. In: Geometric Science of Information, pp. 361\u2013368. Springer (2013)","DOI":"10.1007\/978-3-642-40020-9_39"},{"key":"1061_CR46","volume-title":"Smooth Manifolds","author":"JM Lee","year":"2003","unstructured":"Lee, J.M.: Smooth Manifolds. Springer, Berlin (2003)"},{"key":"1061_CR47","unstructured":"Liang, J., Fadili, M.J., Peyr\u00e9, G.: Local linear convergence of forward\u2013backward under partial smoothness. In: Advances in Neural Information Processing Systems, pp. 1970\u20131978 (2014)"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-017-1061-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-017-1061-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-017-1061-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,23]],"date-time":"2022-07-23T01:43:34Z","timestamp":1658540614000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-017-1061-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,30]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,3]]}},"alternative-id":["1061"],"URL":"https:\/\/doi.org\/10.1007\/s10957-017-1061-z","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2017,1,30]]}}}