{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:20:54Z","timestamp":1773656454252,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,11,28]],"date-time":"2020-11-28T00:00:00Z","timestamp":1606521600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,28]],"date-time":"2020-11-28T00:00:00Z","timestamp":1606521600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["DP190100555"],"award-info":[{"award-number":["DP190100555"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1816386"],"award-info":[{"award-number":["1816386"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1816449"],"award-info":[{"award-number":["1816449"]}],"id":[{"id":"10.13039\/501100008982","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":[[2021,2]]},"DOI":"10.1007\/s10957-020-01787-7","type":"journal-article","created":{"date-parts":[[2020,11,28]],"date-time":"2020-11-28T07:02:31Z","timestamp":1606546951000},"page":"378-401","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["On the Linear Convergence of Forward\u2013Backward Splitting Method: Part I\u2014Convergence Analysis"],"prefix":"10.1007","volume":"188","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7877-5688","authenticated-orcid":false,"given":"Yunier","family":"Bello-Cruz","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2099-7974","authenticated-orcid":false,"given":"Guoyin","family":"Li","sequence":"additional","affiliation":[]},{"given":"Tran T. A.","family":"Nghia","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,11,28]]},"reference":[{"key":"1787_CR1","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, D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413\u20131457 (2004)","journal-title":"Commun. Pure Appl. Math."},{"key":"1787_CR2","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1137\/070698920","volume":"19","author":"ET Hale","year":"2008","unstructured":"Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for $$\\ell _1$$-minimization: methodology and convergence. SIAM J. Optim. 19, 1107\u20131130 (2008)","journal-title":"SIAM J. Optim."},{"key":"1787_CR3","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. R. Stat. Soc. 58, 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc."},{"key":"1787_CR4","doi-asserted-by":"publisher","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, New York (2011)"},{"key":"1787_CR5","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1007\/s00041-008-9041-1","volume":"14","author":"K Bredies","year":"2008","unstructured":"Bredies, K., Lorenz, D.A.: Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14, 813\u2013837 (2008)","journal-title":"J. Fourier Anal. Appl."},{"key":"1787_CR6","doi-asserted-by":"crossref","unstructured":"Beck, A., Teboulle, M.: Gradient-Based Algorithms with Applications to Signal Recovery Problems. In: Palomar, D., Eldar, Y. (Eds.), Convex Optimization in Signal Processing and Communications, pp. 42\u201388. University Press, Cambribge (2010)","DOI":"10.1017\/CBO9780511804458.003"},{"key":"1787_CR7","doi-asserted-by":"crossref","unstructured":"Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems. Science and Engineering. Springer Optimization and Its Applications, vol. 49, pp. 185\u2013212. Springer, New York (2011)","DOI":"10.1007\/978-1-4419-9569-8_10"},{"key":"1787_CR8","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\u2013backward splitting. Multiscale Model. Simul. 4, 1168\u20131200 (2005)","journal-title":"Multiscale Model. Simul."},{"key":"1787_CR9","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1561\/2400000003","volume":"1","author":"P Neal","year":"2014","unstructured":"Neal, P., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1, 127\u2013239 (2014)","journal-title":"Found. Trends Optim."},{"key":"1787_CR10","doi-asserted-by":"crossref","unstructured":"Davis, D., Yin, W.: Convergence rate analysis of several splitting schemes. In: Splitting Methods in Communications, Image Science, and Engineering. Scientific Computation, Springer, Cham (2016)","DOI":"10.1007\/978-3-319-41589-5_4"},{"key":"1787_CR11","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modified forward\u2013backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431\u2013446 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"1787_CR12","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.1080\/10556788.2016.1214959","volume":"31","author":"JY Bello Cruz","year":"2016","unstructured":"Bello Cruz, J.Y., Nghia, T.T.A.: On the convergence of the proximal forward\u2013backward splitting method with linesearches. Optim. Method Softw. 31, 1209\u20131238 (2016)","journal-title":"Optim. Method Softw."},{"key":"1787_CR13","doi-asserted-by":"publisher","first-page":"2153","DOI":"10.1137\/16M1073741","volume":"27","author":"S Salzo","year":"2017","unstructured":"Salzo, S.: The variable metric forward\u2013backward splitting algorithm under mild differentiability assumptions. SIAM J. Optim. 27, 2153\u20132181 (2017)","journal-title":"SIAM J. Optim."},{"key":"1787_CR14","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, 330\u2013348 (2017)","journal-title":"Math. Oper. Res."},{"key":"1787_CR15","doi-asserted-by":"publisher","first-page":"2032","DOI":"10.1214\/aos\/1176348385","volume":"19","author":"I Csisz\u00e1r","year":"1991","unstructured":"Csisz\u00e1r, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist. 19, 2032\u20132066 (1991)","journal-title":"Ann. Statist."},{"key":"1787_CR16","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1080\/01621459.1985.10477119","volume":"80","author":"Y Vardi","year":"1985","unstructured":"Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography. J. Amer. Statist. Assoc. 80, 8\u201337 (1985)","journal-title":"J. Amer. Statist. Assoc."},{"key":"1787_CR17","doi-asserted-by":"crossref","unstructured":"Bello-Cruz, J.Y., Li, G., Nghia, T.T.A.: On the linear convergence of forward-backward splitting method. Quadratic growth condition and uniqueness of optimal solution to Lasso, preprint, Part II (2020)","DOI":"10.1007\/s10957-020-01787-7"},{"key":"1787_CR18","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s10107-016-1091-6","volume":"165","author":"J Bolte","year":"2017","unstructured":"Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program. 165, 471\u2013507 (2017)","journal-title":"Math. Program."},{"key":"1787_CR19","unstructured":"Garrigos, G., Rosasco, L., Villa, S.: Convergence of the forward-backward algorithm: beyond the worst case with the help of geometry, arXiv:1703.09477 (2017)"},{"key":"1787_CR20","unstructured":"Garrigos, G., Rosasco, L., Villa, S.: Thresholding gradient methods in Hilbert spaces: support identification and linear convergence, arXiv:1712.00357 (2017)"},{"key":"1787_CR21","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1007\/s10208-017-9366-8","volume":"18","author":"G Li","year":"2018","unstructured":"Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka-\u0141ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comp. Math. 18, 1199\u20131232 (2018)","journal-title":"Found. Comp. Math."},{"key":"1787_CR22","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s10107-016-1014-6","volume":"168","author":"G Li","year":"2018","unstructured":"Li, G., Mordukhovich, B.S., Nghia, T.T.A., Pham, T.S.: Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. Math. Program. 168, 313\u2013346 (2018)","journal-title":"Math. Program."},{"key":"1787_CR23","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s10107-016-1100-9","volume":"165","author":"Z Zhou","year":"2017","unstructured":"Zhou, Z., So, A.M.-C.: A unified approach to error bounds for structured convex optimization. Math. Program. 165, 689\u2013728 (2017)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1787_CR24","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10107-011-0481-z","volume":"137","author":"G Li","year":"2013","unstructured":"Li, G.: Global error bounds for piecewise convex polynomials. Math. Program. 137(1\u20132), 37\u201364 (2013)","journal-title":"Math. Program."},{"key":"1787_CR25","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1287\/moor.2017.0889","volume":"43","author":"D Drusvyatskiy","year":"2018","unstructured":"Drusvyatskiy, D., Lewis, A.: Error bounds, quadratic growth, and linear convergence of proximal methods. Math. Oper. Res. 43, 693\u20131050 (2018)","journal-title":"Math. Oper. Res."},{"key":"1787_CR26","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s10107-018-1232-1","volume":"175","author":"I Necoara","year":"2019","unstructured":"Necoara, I., Nesterov, Yu., Glineur, F.: Linear convergence of first order methods for non-strongly convex optimization. Math. Program. 175, 69\u2013107 (2019)","journal-title":"Math. Program."},{"key":"1787_CR27","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1137\/151004549","volume":"26","author":"S Tao","year":"2016","unstructured":"Tao, S., Boley, D., Zhang, S.: Local linear convergence of ISTA and FISTA on the Lasso problem. SIAM J. Optim. 26, 313\u2013336 (2016)","journal-title":"SIAM J. Optim."},{"key":"1787_CR28","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/BF02096261","volume":"46","author":"Z-Q Luo","year":"1993","unstructured":"Luo, Z.-Q., Tseng, P.: Error bounds and convergence analysis of feasible descent methods: a general approach. Ann. Oper. Res. 46, 157\u2013178 (1993)","journal-title":"Ann. Oper. Res."},{"key":"1787_CR29","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s11784-015-0220-9","volume":"16","author":"D Az\u00e9","year":"2014","unstructured":"Az\u00e9, D., Corvellec, J.-N.: Nonlinear local error bounds via a change of metric. J. Fixed Point Theory Appl. 16, 251\u2013372 (2014)","journal-title":"J. Fixed Point Theory Appl."},{"key":"1787_CR30","first-page":"365","volume":"15","author":"FJ Arag\u00f3n Artacho","year":"2008","unstructured":"Arag\u00f3n Artacho, F.J., Geoffroy, M.H.: Characterizations of metric regularity of subdifferentials. J. Convex Anal. 15, 365\u2013380 (2008)","journal-title":"J. Convex Anal."},{"key":"1787_CR31","first-page":"1165","volume":"21","author":"D Drusvyatskiy","year":"2014","unstructured":"Drusvyatskiy, D., Mordukhovich, B.S., Nghia, T.T.A.: Second-order growth, tilt stability, and metric regularity of the subdifferential. J. Convex Anal. 21, 1165\u20131192 (2014)","journal-title":"J. Convex Anal."},{"key":"1787_CR32","unstructured":"Liang, J., Fadili, J., Peyr\u00e9, G.: Local linear convergence of forward-backward under partial smoothness. Adv. Neural Inf. Process Syst. (2014)"},{"key":"1787_CR33","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1137\/16M106340X","volume":"27","author":"J Liang","year":"2017","unstructured":"Liang, J., Fadili, J., Peyr\u00e9, G.: Activity identification and local linear convergence of forward\u2013backward type methods. SIAM J. Optim. 27, 408\u2013437 (2017)","journal-title":"SIAM J. Optim."},{"key":"1787_CR34","first-page":"35","volume":"15","author":"FJ Arag\u00f3n Artacho","year":"2014","unstructured":"Arag\u00f3n Artacho, F.J., Geoffroy, M.H.: Metric subregularity of the convex subdifferential in Banach spaces. J. Nonlinear Convex Anal. 15, 35\u201347 (2014)","journal-title":"J. Nonlinear Convex Anal."},{"key":"1787_CR35","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/s10107-007-0170-0","volume":"117","author":"P Tseng","year":"2000","unstructured":"Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117, 387\u2013423 (2000)","journal-title":"Math. Program."},{"key":"1787_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"key":"1787_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-87821-8","volume-title":"Implicit Functions and Solution Mappings. A View from Variational Analysis","author":"AL Dontchev","year":"2009","unstructured":"Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Dordrecht (2009)"},{"key":"1787_CR38","volume-title":"Set-Valued Mappings and Enlargements of Monotone Operators","author":"RS Burachik","year":"2008","unstructured":"Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008)"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-020-01787-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-020-01787-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-020-01787-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,18]],"date-time":"2024-08-18T05:53:32Z","timestamp":1723960412000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-020-01787-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,28]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["1787"],"URL":"https:\/\/doi.org\/10.1007\/s10957-020-01787-7","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,28]]},"assertion":[{"value":"19 March 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 November 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}