{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T07:16:07Z","timestamp":1758093367752,"version":"3.44.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSF of China","doi-asserted-by":"crossref","award":["12171062"],"award-info":[{"award-number":["12171062"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10957-025-02822-1","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T06:14:41Z","timestamp":1755756881000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Analysis of Modified Bregman Extragradient Method for Variational Inequality Problems"],"prefix":"10.1007","volume":"207","author":[{"given":"Qingqing","family":"Fu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0659-0203","authenticated-orcid":false,"given":"Gang","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Kunrada","family":"Kankam","sequence":"additional","affiliation":[]},{"given":"Prasit","family":"Cholamjiak","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,21]]},"reference":[{"key":"2822_CR1","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s10957-018-1228-2","volume":"178","author":"G Cai","year":"2018","unstructured":"Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces. J. Optim. Theory Appl. 178, 219\u2013239 (2018)","journal-title":"J. Optim. Theory Appl."},{"key":"2822_CR2","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","volume":"148","author":"Y Censor","year":"2011","unstructured":"Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318\u2013335 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"2822_CR3","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1080\/02331934.2010.539689","volume":"61","author":"Y Censor","year":"2012","unstructured":"Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevichs extragradient method for variational inequality problems in Euclidean space. Optimization 61, 119\u20131132 (2012)","journal-title":"Optimization"},{"key":"2822_CR4","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/s11590-011-0407-y","volume":"7","author":"Z Chbani","year":"2013","unstructured":"Chbani, Z., Riahi, H.: Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities. Optim. Lett. 7, 185\u2013206 (2013)","journal-title":"Optim. Lett."},{"key":"2822_CR5","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1007\/s10898-017-0506-0","volume":"70","author":"QL Dong","year":"2018","unstructured":"Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Global Optim. 70, 687\u2013704 (2018)","journal-title":"J. Global Optim."},{"key":"2822_CR6","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s10915-024-02784-y","volume":"102","author":"QQ Fu","year":"2025","unstructured":"Fu, Q.Q., Cai, G., Cholamjiak, P., Inkrong, P.: Modified Extragradient Methods with Inertial Technique for Solving Pseudo-Monotone Variational Inequality Problems. J. Sci. Comput. 102, 59 (2025)","journal-title":"J. Sci. Comput."},{"key":"2822_CR7","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1023\/A:1022679020242","volume":"95","author":"D Goeleven","year":"1997","unstructured":"Goeleven, D., Stavroulakis, G.E., Salmon, G., Panagiotopoulos, P.D.: Solvability Theory and Projection Methods for a Class of Singular Variational Inequalities: Elastostatic Unilateral Contact Applications. J. Optim. Theory Appl. 95, 263\u2013293 (1997)","journal-title":"J. Optim. Theory Appl."},{"key":"2822_CR8","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1007\/s11590-022-01909-2","volume":"17","author":"ST Hu","year":"2023","unstructured":"Hu, S.T., Wang, Y.H., Jing, P., Dong, Q.L.: Strong convergence theorem for a new Bregman extragradient method with a different line-search process for solving variational inequality problems in reflexive Banach spaces. Optim. Lett. 17, 935\u2013954 (2023)","journal-title":"Optim. Lett."},{"key":"2822_CR9","doi-asserted-by":"crossref","unstructured":"Izuchukwu, C., Shehu, Y., Yao, J.C.: New strong convergence analysis for variational inequalities and fixed point problems. Optimization 1-22 (2024)","DOI":"10.1080\/02331934.2024.2424446"},{"issue":"9","key":"2822_CR10","doi-asserted-by":"publisher","first-page":"1732","DOI":"10.1080\/00036811.2024.2432527","volume":"104","author":"C Izuchukwu","year":"2024","unstructured":"Izuchukwu, C., Shehu, Y., Yao, J.C.: Convergence results for proximal point algorithm with inertial and correction terms. Appl. Anal. 104(9), 1732\u20131752 (2024)","journal-title":"Appl. Anal."},{"key":"2822_CR11","doi-asserted-by":"publisher","first-page":"773","DOI":"10.3934\/jimo.2020178","volume":"18","author":"LO Jolaoso","year":"2022","unstructured":"Jolaoso, L.O., Aphane, M.: Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. J. Ind. Manag. Optim. 18, 773\u2013794 (2022)","journal-title":"J. Ind. Manag. Optim."},{"key":"2822_CR12","doi-asserted-by":"publisher","first-page":"4807","DOI":"10.1080\/00036811.2020.1869947","volume":"101","author":"LO Jolaoso","year":"2022","unstructured":"Jolaoso, L.O., Shehu, Y.: Single Bregman projection method for solving variational inequalities in reflexive Banach spaces. Appl. Anal. 101, 4807\u20134828 (2022)","journal-title":"Appl. Anal."},{"key":"2822_CR13","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1007\/s40314-022-02006-x","volume":"41","author":"LO Jolaoso","year":"2022","unstructured":"Jolaoso, L.O., Sunthrayuth, P., Cholamjiak, P., Cho, J.Y.: Analysis of two versions of relaxed inertial algorithms with Bregman divergences for solving variational inequalities. Comput. Appl. Math. 41, 300 (2022)","journal-title":"Comput. Appl. Math."},{"key":"2822_CR14","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1007\/s11228-004-4379-2","volume":"12","author":"A Kaplan","year":"2004","unstructured":"Kaplan, A., Tichatschke, R.: Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators. Set-Valued Var. Anal. 12, 357\u2013382 (2004)","journal-title":"Set-Valued Var. Anal."},{"key":"2822_CR15","volume-title":"An Introduction to Variational Inequalities and Their Applications","author":"D Kinderlehrer","year":"1980","unstructured":"Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)"},{"key":"2822_CR16","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Mat. Metody. 12, 747\u2013756 (1976)","journal-title":"Ekonomikai Mat. Metody."},{"key":"2822_CR17","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1007\/s11228-008-0102-z","volume":"16","author":"PE Maing\u00e9","year":"2008","unstructured":"Maing\u00e9, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Var. Anal. 16, 899\u2013912 (2008)","journal-title":"Set-Valued Var. Anal."},{"key":"2822_CR18","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/s10559-014-9614-8","volume":"50","author":"YV Malitsky","year":"2014","unstructured":"Malitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybernet. Systems Anal. 50, 271\u2013277 (2014)","journal-title":"Cybernet. Systems Anal."},{"key":"2822_CR19","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/BF01584381","volume":"33","author":"P Marcotte","year":"1985","unstructured":"Marcotte, P.: A new algorithm for solving variational inequalities with application to the traffic assignment problem. Math. Program. 33, 339\u2013351 (1985)","journal-title":"Math. Program."},{"key":"2822_CR20","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.3934\/dcdss.2013.6.1043","volume":"6","author":"V Mart\u00edn-M\u00e1rquez","year":"2013","unstructured":"Mart\u00edn-M\u00e1rquez, V., Reich, S., Sabach, S.: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst. Ser. S 6, 1043 (2013)","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"2822_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1687-1812-2013-141","volume":"2013","author":"E Naraghirad","year":"2013","unstructured":"Naraghirad, E., Yao, J.C.: Bregman weak relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2013, 1\u201343 (2013)","journal-title":"Fixed Point Theory Appl."},{"key":"2822_CR22","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":"2822_CR23","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1007\/s11075-021-01133-6","volume":"89","author":"OK Oyewole","year":"2022","unstructured":"Oyewole, O.K., Abass, H.A., Mebawondu, A.A., Aremu, K.O.: A Tseng extragradient method for solving variational inequality problems in Banach spaces. Numer. Algorithms 89, 769\u2013789 (2022)","journal-title":"Numer. Algorithms"},{"key":"2822_CR24","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1080\/01630560903499852","volume":"31","author":"S Reich","year":"2010","unstructured":"Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22\u201344 (2010)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"2822_CR25","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1007\/s10473-020-0412-2","volume":"40","author":"Y Shehu","year":"2020","unstructured":"Shehu, Y.: Single projection algorithm for variational inequalities in Banach spaces with application to contact problem. Acta Math. Sci. Ser. B Engl. Ed. 40, 1045\u20131063 (2020)","journal-title":"Acta Math. Sci. Ser. B Engl. Ed."},{"key":"2822_CR26","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/s11075-017-0449-z","volume":"79","author":"Y Shehu","year":"2018","unstructured":"Shehu, Y., Iyiola, O.S.: Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer. Algorithms 79, 529\u2013553 (2018)","journal-title":"Numer. Algorithms"},{"key":"2822_CR27","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s11075-019-00758-y","volume":"84","author":"Y Shehu","year":"2020","unstructured":"Shehu, Y., Li, X.H., Dong, Q.L.: An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numer. Algorithms 84, 365\u2013388 (2020)","journal-title":"Numer. Algorithms"},{"key":"2822_CR28","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10898-021-01095-y","volume":"82","author":"B Tan","year":"2022","unstructured":"Tan, B., Qin, X., Yao, J.C.: Strong convergence of inertial projection and contraction methods for pseudomonotone variational inequalities with applications to optimal control problems. J. Global Optim. 82, 523\u2013557 (2022)","journal-title":"J. Global Optim."},{"key":"2822_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114260","volume":"410","author":"DV Thong","year":"2022","unstructured":"Thong, D.V., Dong, Q.L., Liu, L.L., Triet, N.A., Lan, N.P.: Two fast converging inertial subgradient extragradient algorithms with variable stepsizes for solving pseudo-monotone VIPs in Hilbert spaces. J. Comput. Appl. Math. 410, 114260 (2022)","journal-title":"J. Comput. Appl. Math."},{"key":"2822_CR30","doi-asserted-by":"crossref","unstructured":"Thong, D.V., Gibali, A.: Extragradient methods for solving non-Lipschitzian pseudo-monotone variational inequalities. J. Fixed Point Theory Appl. 21, 20 (2019)","DOI":"10.1007\/s11784-018-0656-9"},{"key":"2822_CR31","doi-asserted-by":"crossref","unstructured":"Thong, D.V., Hieu, D.V.: New extragradient methods for solving variational inequality problems and fixed point problems. J. Fixed Point Theory Appl. 20, 129 (2018)","DOI":"10.1007\/s11784-018-0610-x"},{"key":"2822_CR32","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s11590-019-01511-z","volume":"14","author":"DV Thong","year":"2020","unstructured":"Thong, D.V., Hieu, D.V., Rassias, T.M.: Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim. Lett. 14, 115\u2013144 (2020)","journal-title":"Optim. Lett."},{"key":"2822_CR33","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.1007\/s11075-022-01457-x","volume":"93","author":"DV Thong","year":"2023","unstructured":"Thong, D.V., Reich, S., Shehu, Y., Iyiola, O.S.: Novel projection methods for solving variational inequality problems and applications. Numer. Algorithms 93, 1105\u20131135 (2023)","journal-title":"Numer. Algorithms"},{"key":"2822_CR34","doi-asserted-by":"publisher","first-page":"2207","DOI":"10.1080\/02331934.2019.1616191","volume":"68","author":"DV Thong","year":"2019","unstructured":"Thong, D.V., Vuong, P.T.: Modified Tseng\u2019s extragradient methods for solving pseudo-monotone variational inequalities. Optimization 68, 2207\u20132226 (2019)","journal-title":"Optimization"},{"key":"2822_CR35","doi-asserted-by":"crossref","unstructured":"Thung, K.H., Raveendran, P.: A survey of image quality measures. In: 2009 International Conference for Technical Postgraduates (TECHPOS), pp. 1-4. IEEE (2009)","DOI":"10.1109\/TECHPOS.2009.5412098"},{"key":"2822_CR36","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modifed forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38, 431\u2013446 (2000)","journal-title":"SIAM J. Control. Optim."},{"issue":"4","key":"2822_CR37","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1109\/TIP.2003.819861","volume":"13","author":"Z Wang","year":"2004","unstructured":"Wang, Z., Bovik, A.C., Sheikh, H.R.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600\u2013612 (2004)","journal-title":"IEEE Trans. Image Process."},{"key":"2822_CR38","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/s11075-022-01414-8","volume":"93","author":"ZB Xie","year":"2023","unstructured":"Xie, Z.B., Cai, G., Dong, Q.L.: Strong convergence of Bregman projection method for solving variational inequality problems in reflexive Banach spaces. Numer. Algorithms 93, 269\u2013294 (2023)","journal-title":"Numer. Algorithms"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02822-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02822-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02822-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T07:44:24Z","timestamp":1758008664000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02822-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,21]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["2822"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02822-1","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2025,8,21]]},"assertion":[{"value":"23 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"60"}}