{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T21:17:49Z","timestamp":1782767869007,"version":"3.54.5"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T00:00:00Z","timestamp":1706486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T00:00:00Z","timestamp":1706486400000},"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":["Numer Algor"],"published-print":{"date-parts":[[2024,11]]},"DOI":"10.1007\/s11075-024-01749-4","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T07:02:33Z","timestamp":1706511753000},"page":"1267-1300","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator"],"prefix":"10.1007","volume":"97","author":[{"given":"Duong Viet","family":"Thong","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0780-1559","authenticated-orcid":false,"given":"Simeon","family":"Reich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Prasit","family":"Cholamjiak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Le Dinh","family":"Long","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,1,29]]},"reference":[{"key":"1749_CR1","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1016\/j.jmaa.2016.06.025","volume":"457","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Peypouquet, J., Redont, P.: Backward-forward algorithms for structured monotone inclusions in Hilbert spaces. J. Math. Anal. Appl. 457, 1095\u20131117 (2018)","journal-title":"J. Math. Anal. Appl."},{"key":"1749_CR2","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. Springer (2011). Second Edition (2017).","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"1749_CR3","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.na.2004.07.054","volume":"60","author":"HH Bauschke","year":"2005","unstructured":"Bauschke, H.H., Combettes, P.L., Reich, S.: The asymptotic behavior of the composition of two resolvents. Nonlinear Anal. 60, 283\u2013301 (2005)","journal-title":"Nonlinear Anal."},{"key":"1749_CR4","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/s11075-015-0007-5","volume":"71","author":"RI Bot","year":"2016","unstructured":"Bot, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms. 71, 519\u2013540 (2016)","journal-title":"Numer. Algorithms."},{"key":"1749_CR5","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0304-0208(08)72383-1","volume":"5","author":"H Br\u00e9zis","year":"1973","unstructured":"Br\u00e9zis, H., Chapitre, I.I.: Operateurs maximaux monotones. North-Holland Math Stud. 5, 19\u201351 (1973)","journal-title":"Operateurs maximaux monotones. North-Holland Math Stud."},{"key":"1749_CR6","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","volume":"20","author":"C Byrne","year":"2004","unstructured":"Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103\u2013120 (2004)","journal-title":"Inverse Prob."},{"key":"1749_CR7","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s10589-013-9599-7","volume":"57","author":"XJ Cai","year":"2014","unstructured":"Cai, X.J., Gu, G.Y., He, B.S.: On the $$O(1\/t)$$ convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput Optim Appl. 57, 339\u2013363 (2014)","journal-title":"Comput Optim Appl."},{"key":"1749_CR8","first-page":"117","volume":"6","author":"P Combettes","year":"2005","unstructured":"Combettes, P., Hirstoaga, S.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117\u2013136 (2005)","journal-title":"J. Nonlinear Convex Anal."},{"key":"1749_CR9","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/050626090","volume":"4","author":"PL Combettes","year":"2005","unstructured":"Combettes, P.L., Wajs, V.: Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model. Simul. 4, 1168\u20131200 (2005)","journal-title":"SIAM Multiscale Model. Simul."},{"key":"1749_CR10","doi-asserted-by":"crossref","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, 1168\u20131200 (2005)","journal-title":"Multiscale Model. Simul."},{"key":"1749_CR11","volume-title":"Fixed-Point Algorithms for Inverse Problems in Science and Engineering","author":"PL Combettes","year":"2010","unstructured":"Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York (2010)"},{"key":"1749_CR12","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1007\/s11784-018-0526-5","volume":"20","author":"W Cholamjiak","year":"2018","unstructured":"Cholamjiak, W., Cholamjiak, P., Suantai, S.: An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J. Fixed Point Theory Appl. 20, 42 (2018). https:\/\/doi.org\/10.1007\/s11784-018-0526-5","journal-title":"J. Fixed Point Theory Appl."},{"key":"1749_CR13","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1007\/s11075-015-0030-6","volume":"71","author":"P Cholamjiak","year":"2016","unstructured":"Cholamjiak, P.: A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numer. Algor. 71, 915\u2013932 (2016)","journal-title":"Numer. Algor."},{"key":"1749_CR14","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1080\/10556788.2016.1214959","volume":"31","author":"JYB Cruz","year":"2016","unstructured":"Cruz, J.Y.B., Nghia, T.T.A.: On the convergence of the forward-backward splitting method with linesearches. Optim Method Softw. 31, 1209\u20131238 (2016)","journal-title":"Optim Method Softw."},{"key":"1749_CR15","doi-asserted-by":"crossref","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. Commun. Pure Appl. Math. 57, 1413\u20131457 (2004)","journal-title":"Commun. Pure Appl. Math."},{"key":"1749_CR16","doi-asserted-by":"crossref","first-page":"3097","DOI":"10.1007\/s11784-017-0472-7","volume":"19","author":"Q Dong","year":"2017","unstructured":"Dong, Q., Jiang, D., Cholamjiak, P., Shehu, Y.: A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions. J. Fixed Point Theory Appl. 19, 3097\u20133118 (2017)","journal-title":"J. Fixed Point Theory Appl."},{"key":"1749_CR17","first-page":"2899","volume":"10","author":"J Duchi","year":"2009","unstructured":"Duchi, J., Singer, Y.: Efficient online and batch learning using forward-backward splitting. J. Mach. Learn. Res. 10, 2899\u20132934 (2009)","journal-title":"J. Mach. Learn. Res."},{"key":"1749_CR18","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10107-006-0070-8","volume":"111","author":"J Eckstein","year":"2008","unstructured":"Eckstein, J., Svaiter, B.F.: A family of projective splitting methods for the sum of two maximal monotone operators. Math. Program. Ser. B 111, 173\u2013199 (2008)","journal-title":"Math. Program. Ser. B"},{"key":"1749_CR19","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1137\/070698816","volume":"48","author":"J Eckstein","year":"2009","unstructured":"Eckstein, J., Svaiter, B.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Optim. 48, 787\u2013811 (2009)","journal-title":"SIAM J. Control Optim."},{"key":"1749_CR20","volume-title":"Uniform convexity, hyperbolic geometry, and nonexpansive mappings","author":"K Goebel","year":"1984","unstructured":"Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)"},{"key":"1749_CR21","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10092-018-0292-1","volume":"55","author":"A Gibali","year":"2018","unstructured":"Gibali, A., Thong, D.V.: Tseng type methods for solving inclusion problems and its applications. Calcolo. 55, 49 (2018). https:\/\/doi.org\/10.1007\/s10092-018-0292-1","journal-title":"Calcolo."},{"key":"1749_CR22","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s002459900037","volume":"35","author":"BS He","year":"1997","unstructured":"He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69\u201376 (1997)","journal-title":"Appl. Math. Optim."},{"key":"1749_CR23","doi-asserted-by":"crossref","first-page":"1020","DOI":"10.1016\/j.jmaa.2005.12.064","volume":"324","author":"S Hirstoaga","year":"2006","unstructured":"Hirstoaga, S.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020\u20131035 (2006)","journal-title":"J. Math. Anal. Appl."},{"key":"1749_CR24","first-page":"60","volume":"237","author":"YY Huang","year":"2014","unstructured":"Huang, Y.Y., Dong, Y.D.: New properties of forward-backward splitting and a practical proximal-descent algorithm. Appl Math Comput. 237, 60\u201368 (2014)","journal-title":"Appl Math Comput."},{"key":"1749_CR25","doi-asserted-by":"crossref","unstructured":"Khan, S.A., Suantai, S., Cholamjiak., W.: Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems. RACSAM 113, 645-656 (2019)","DOI":"10.1007\/s13398-018-0504-1"},{"key":"1749_CR26","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1002\/mma.5420","volume":"42","author":"K Kankam","year":"2019","unstructured":"Kankam, K., Pholasa, N., Cholamjiak, P.: On the convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization. Math. Meth. Appl. Sci. 42, 1352\u20131362 (2019)","journal-title":"Math. Meth. Appl. Sci."},{"key":"1749_CR27","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/j.cam.2018.04.057","volume":"354","author":"D Kitkuan","year":"2019","unstructured":"Kitkuan, D., Kumam, P., Padcharoen, A., Kumam, W., Thounthong, P.: Algorithms for zeros of two accretive operators for solving convex minimization problems and its application to image restoration problems. J. Comput. Appl. Math. 354, 471\u2013495 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"1749_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-020-00217-8","author":"H Liu","year":"2020","unstructured":"Liu, H., Yang, J.: Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput. Optim. Appl. (2020). https:\/\/doi.org\/10.1007\/s10589-020-00217-8","journal-title":"Comput. Optim. Appl."},{"key":"1749_CR29","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, 964\u2013979 (1979)","journal-title":"SIAM J. Numer. Anal."},{"key":"1749_CR30","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10851-014-0523-2","volume":"51","author":"DA Lorenz","year":"2015","unstructured":"Lorenz, D.A., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311\u2013325 (2015)","journal-title":"J. Math. Imaging Vis."},{"key":"1749_CR31","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","volume":"4","author":"WR Mann","year":"1953","unstructured":"Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506\u2013510 (1953)","journal-title":"Proc. Am. Math. Soc."},{"issue":"527","key":"1749_CR32","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1006\/jmaa.1999.6615","volume":"241","author":"A Moudafi","year":"2000","unstructured":"Moudafi, A.: Viscosity approximating methods for fixed point problems. J. Math. Anal. Appl. 241(527), 46\u201355 (2000)","journal-title":"J. Math. Anal. Appl."},{"key":"1749_CR33","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/S0377-0427(02)00906-8","volume":"155","author":"A Moudafi","year":"2003","unstructured":"Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447\u2013454 (2003)","journal-title":"J. Comput. Appl. Math."},{"key":"1749_CR34","volume-title":"Iterative solution of nonlinear equations in several variables","author":"JM Ortega","year":"1970","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)"},{"key":"1749_CR35","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.cam.2018.09.053","volume":"354","author":"A Padcharoen","year":"2018","unstructured":"Padcharoen, A., Kumam, P., Mart\u00ednez-Moren, J.: Augmented Lagrangian method for TV $$-l_1-l_2$$ based colour image restoration. J. Comput. Appl. Math. 354, 507\u2013519 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"1749_CR36","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0022-247X(79)90234-8","volume":"72","author":"GB Passty","year":"1979","unstructured":"Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383\u2013390 (1979)","journal-title":"J. Math. Anal. Appl."},{"key":"1749_CR37","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1137\/120872802","volume":"6","author":"H Raguet","year":"2013","unstructured":"Raguet, H., Fadili, J., Peyr\u00e9, G.: A generalized forward-backward splitting. SIAM J. Imaging Sci. 6, 1199\u20131226 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"1749_CR38","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/0022-1236(77)90022-2","volume":"26","author":"S Reich","year":"1977","unstructured":"Reich, S.: Extension problems for accretive sets in Banach spaces. J. Functional Analysis 26, 378\u2013395 (1977)","journal-title":"J. Functional Analysis"},{"key":"1749_CR39","doi-asserted-by":"crossref","first-page":"209","DOI":"10.2140\/pjm.1970.33.209","volume":"33","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209\u2013216 (1970)","journal-title":"Pac. J. Math."},{"key":"1749_CR40","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1016\/j.na.2011.09.005","volume":"75","author":"S Saejung","year":"2012","unstructured":"Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742\u2013750 (2012)","journal-title":"Nonlinear Anal."},{"key":"1749_CR41","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF02192286","volume":"91","author":"DF Sun","year":"1996","unstructured":"Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123\u2013140 (1996)","journal-title":"J. Optim. Theory Appl."},{"key":"1749_CR42","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.11650\/twjm\/1500406684","volume":"16","author":"W Takahashi","year":"2012","unstructured":"Takahashi, W., Wong, N.C., Yao, J.C.: Two generalized strong convergence theorems of Halpern-type in Hilbert spaces and applications. Taiwanese J. Math. 16, 1151\u20131172 (2012)","journal-title":"Taiwanese J. Math."},{"key":"1749_CR43","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1007\/s40314-019-0855-z","volume":"38","author":"DV Thong","year":"2019","unstructured":"Thong, D.V., Cholamjiak, P.: Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions. Comp. Appl. Math. 38, 94 (2019). https:\/\/doi.org\/10.1007\/s40314-019-0855-z","journal-title":"Comp. Appl. Math."},{"key":"1749_CR44","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim. 38, 431\u2013446 (2000)","journal-title":"SIAM J Control Optim."},{"key":"1749_CR45","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1080\/02331934.2017.1411485","volume":"67","author":"Y Wang","year":"2018","unstructured":"Wang, Y., Wang, F.: Strong convergence of the forward-backward splitting method with multiple parameters in Hilbert spaces. Optimization 67, 493\u2013505 (2018)","journal-title":"Optimization"},{"key":"1749_CR46","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1080\/02331934.2018.1455832","volume":"67","author":"C Zhang","year":"2018","unstructured":"Zhang, C., Wang, Y.: Proximal algorithm for solving monotone variational inclusion. Optimization 67, 1197\u20131209 (2018)","journal-title":"Optimization"},{"issue":"4","key":"1749_CR47","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1137\/18M1224763","volume":"12","author":"PL Combettes","year":"2019","unstructured":"Combettes, P.L., Glaudin, L.E.: Proximal activation of smooth functions in splitting algorithms for convex image recovery. SIAM J. Imaging Sci. 12(4), 1905\u20131935 (2019)","journal-title":"SIAM J. Imaging Sci."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01749-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01749-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01749-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,9]],"date-time":"2024-11-09T10:57:33Z","timestamp":1731149853000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01749-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,29]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["1749"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01749-4","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,29]]},"assertion":[{"value":"10 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 January 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 January 2024","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 declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}