{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T14:39:03Z","timestamp":1768487943482,"version":"3.49.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T00:00:00Z","timestamp":1751760000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T00:00:00Z","timestamp":1751760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201517"],"award-info":[{"award-number":["12201517"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012669","name":"Natural Science Foundation Project of Chongqing, Chongqing Science and Technology Commission","doi-asserted-by":"publisher","award":["CSTB2022NSCQ-MSX0580"],"award-info":[{"award-number":["CSTB2022NSCQ-MSX0580"]}],"id":[{"id":"10.13039\/501100012669","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University","doi-asserted-by":"publisher","award":["Y304024909"],"award-info":[{"award-number":["Y304024909"]}],"id":[{"id":"10.13039\/501100004706","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University","doi-asserted-by":"publisher","award":["12471299"],"award-info":[{"award-number":["12471299"]}],"id":[{"id":"10.13039\/501100004706","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":[[2025,10]]},"DOI":"10.1007\/s10957-025-02764-8","type":"journal-article","created":{"date-parts":[[2025,7,6]],"date-time":"2025-07-06T02:18:46Z","timestamp":1751768326000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Proximal-Type Algorithm with Bregman Distance for Solving Equilibrium Problems"],"prefix":"10.1007","volume":"207","author":[{"given":"Grace Nnennaya","family":"Ogwo","sequence":"first","affiliation":[]},{"given":"Olawale Kazeem","family":"Oyewole","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5603-3640","authenticated-orcid":false,"given":"Liya","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,6]]},"reference":[{"key":"2764_CR1","first-page":"14987","volume":"1344","author":"E Amid","year":"2019","unstructured":"Amid, E., Warmuth, M.K., Anil, R., Koren, T.: Robust bi-tempered logistic loss based on Bregman divergences. Conference on neural information processing systems 1344, 14987\u201314996 (2019)","journal-title":"Conference on neural information processing systems"},{"issue":"1","key":"2764_CR2","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s11075-017-0431-9","volume":"79","author":"PN Anh","year":"2017","unstructured":"Anh, P.N., Anh, T.T.H., Hien, N.D.: Modified basic projection methods for a class of equilibrium problems. Numer. Algorithms 79(1), 139\u2013152 (2017)","journal-title":"Numer. Algorithms"},{"issue":"5","key":"2764_CR3","doi-asserted-by":"publisher","first-page":"2835","DOI":"10.1007\/s40840-018-0632-y","volume":"42","author":"PN Anh","year":"2018","unstructured":"Anh, P.N., Hien, N.D., Tuan, P.M.: Computational errors of the extragradient method for equilibrium problems. Bull. Malays. Math. Sci. Soc. 42(5), 2835\u20132858 (2018)","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"2764_CR4","first-page":"1705","volume":"6","author":"A Banerjee","year":"2005","unstructured":"Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J. Mach. Learn. Res. 6, 1705\u20131749 (2005)","journal-title":"J. Mach. Learn. Res."},{"issue":"4","key":"2764_CR5","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1142\/S0219199701000524","volume":"3","author":"HH Bauschke","year":"2001","unstructured":"Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Comm. Contemp. Math. 3(4), 615\u2013647 (2001)","journal-title":"Comm. Contemp. Math."},{"key":"2764_CR6","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2010","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer Nature, Switzerland (2010)"},{"issue":"3","key":"2764_CR7","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","volume":"7","author":"LM Bregman","year":"1967","unstructured":"Bregman, L.M.: The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200\u2013217 (1967)","journal-title":"USSR Comput. Math. Math. Phys."},{"issue":"1","key":"2764_CR8","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.ejor.2020.04.035","volume":"287","author":"RI Bot","year":"2020","unstructured":"Bot, R.I., Csetnek, E.R., Vuong, P.T.: The forward-backward-forward method from continuous and discrete perspective for pseudomonotone variational inequalities in Hilbert spaces. Eur. J. Oper. Res. 287(1), 49\u201360 (2020)","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"2764_CR9","first-page":"117","volume":"6","author":"PL Combettes","year":"2005","unstructured":"Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117\u2013136 (2005)","journal-title":"J. Nonlinear Convex Anal."},{"key":"2764_CR10","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s10915-021-01452-9","volume":"87","author":"XK Chang","year":"2021","unstructured":"Chang, X.K., Yang, J.F.: A golden ratio primal-dual algorithm for structured convex optimization. J. Sci. Comput. 87, 47 (2021)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"2764_CR11","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1109\/TPWRS.2003.820692","volume":"19","author":"J Contreras","year":"2004","unstructured":"Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets. EEE Trans. Power Syst. 19(1), 195\u2013206 (2004)","journal-title":"EEE Trans. Power Syst."},{"issue":"4","key":"2764_CR12","doi-asserted-by":"publisher","first-page":"749","DOI":"10.4134\/BKMS.2012.49.4.749","volume":"49","author":"JK Kim","year":"2012","unstructured":"Kim, J.K., Anh, P.N., Hyun, H.G.: A proximal point-type algorithm for pseudomonotone equilibrium problems. Bull. Korean Math. Soc. 49(4), 749\u2013759 (2012)","journal-title":"Bull. Korean Math. Soc."},{"key":"2764_CR13","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1007\/s10589-017-9899-4","volume":"67","author":"DV Hieu","year":"2017","unstructured":"Hieu, D.V.: New subgradient methods for common solutions to equilibrium problems. Comput. Optim. Appl. 67, 571\u2013594 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"2764_CR14","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1007\/s11075-018-0532-0","volume":"80","author":"DV Hieu","year":"2019","unstructured":"Hieu, D.V.: New inertial algorithm for a class of equilibrium problems. Numer. Algorithms 80, 1413\u20131436 (2019)","journal-title":"Numer. Algorithms"},{"key":"2764_CR15","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s00186-018-0640-6","volume":"88","author":"DV Hieu","year":"2018","unstructured":"Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper. Res. 88, 399\u2013415 (2018)","journal-title":"Math. Methods Oper. Res."},{"key":"2764_CR16","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s10107-007-0125-5","volume":"116","author":"A Iusem","year":"2009","unstructured":"Iusem, A., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116, 259\u2013273 (2009)","journal-title":"Math. Program."},{"issue":"3","key":"2764_CR17","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1080\/0233193031000120039","volume":"52","author":"A Iusem","year":"2003","unstructured":"Iusem, A., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301\u2013316 (2003)","journal-title":"Optimization"},{"key":"2764_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2024.108217","volume":"138","author":"C Izuchukwu","year":"2024","unstructured":"Izuchukwu, C., Shehu, Y.: A Golden Ratio Algorithm with Backward Inertial Step for Variational Inequalities. Commun. Nonlinear Sci. and Numer. Simul. 138, 108217 (2024)","journal-title":"Commun. Nonlinear Sci. and Numer. Simul."},{"issue":"14","key":"2764_CR19","doi-asserted-by":"publisher","first-page":"4807","DOI":"10.1080\/00036811.2020.1869947","volume":"101","author":"LO Jolaoso","year":"2021","unstructured":"Jolaoso, L.O., Shehu, Y.: Single Bregman projection method for solving variational inequalities in reflexive Banach spaces. Appl. Anal. 101(14), 4807\u20134828 (2021)","journal-title":"Appl. Anal."},{"key":"2764_CR20","doi-asserted-by":"crossref","unstructured":"Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer Nature, Berlin, 495, (2001)","DOI":"10.1007\/978-3-642-56886-2"},{"issue":"4","key":"2764_CR21","first-page":"747","volume":"12","author":"G Korpelevich","year":"1976","unstructured":"Korpelevich, G.: The extragradient method for finding saddle points and other problems. Ekon. Math. Meto. 12(4), 747\u2013756 (1976)","journal-title":"Ekon. Math. Meto."},{"key":"2764_CR22","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1007\/s10559-011-9343-1","volume":"47","author":"SI Lyashko","year":"2011","unstructured":"Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich\u2019s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631\u201364 (2011)","journal-title":"Cybern. Syst. Anal."},{"issue":"3","key":"2764_CR23","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1016\/j.ejor.2010.01.042","volume":"205","author":"PE Maing\u00e9","year":"2010","unstructured":"Maing\u00e9, P.E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205(3), 501\u2013506 (2010)","journal-title":"Eur. J. Oper. Res."},{"key":"2764_CR24","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s10107-019-01416-w","volume":"184","author":"Y Malitsky","year":"2020","unstructured":"Malitsky, Y.: Golden ratio algorithms for variational inequalities. Math. Program. 184, 383\u2013410 (2020)","journal-title":"Math. Program."},{"key":"2764_CR25","doi-asserted-by":"crossref","unstructured":"Martinet, B.: Br\u00e9ve communication. R\u00e9gularisation d\u2019in\u00e9quations variationnelles par approximations successives. Revue francaise d\u2019informatique et de recherche op\u00e9rationnelle. S\u00e9rie rouge 4(R3), 154-158 (1970)","DOI":"10.1051\/m2an\/197004R301541"},{"key":"2764_CR26","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1023\/A:1026050425030","volume":"27","author":"G Mastroeni","year":"2003","unstructured":"Mastroeni, G.: Gap function for equilibrium problems. J. Glob. Optim. 27, 411\u2013426 (2003)","journal-title":"J. Glob. Optim."},{"key":"2764_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0239-1_15","volume-title":"On Auxiliary Principle for Equilibrium Problems","author":"G Mastroeni","year":"2003","unstructured":"Mastroeni, G.: On Auxiliary Principle for Equilibrium Problems. Springer Nature, Boston (2003)"},{"key":"2764_CR28","doi-asserted-by":"publisher","first-page":"993","DOI":"10.1007\/s10957-023-02320-2","volume":"199","author":"MK Tam","year":"2023","unstructured":"Tam, M.K., Uteda, D.J.: Bregman golden ratio algorithms for variational inequalities. J. Optim. Theory Appl. 199, 993\u20131021 (2023)","journal-title":"J. Optim. Theory Appl."},{"issue":"1\u20132","key":"2764_CR29","first-page":"91","volume":"15","author":"A Moudafi","year":"1999","unstructured":"Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geom. 15(1\u20132), 91\u2013100 (1999)","journal-title":"J. Nat. Geom."},{"issue":"1","key":"2764_CR30","first-page":"18","volume":"4","author":"A Moudafi","year":"2003","unstructured":"Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4(1), 18 (2003)","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"2764_CR31","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1016\/0362-546X(92)90159-C","volume":"18","author":"LD Muu","year":"1992","unstructured":"Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159\u20131166 (1992)","journal-title":"Nonlinear Anal."},{"key":"2764_CR32","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s10013-014-0115-x","volume":"43","author":"LD Mu","year":"2015","unstructured":"Mu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229\u2013238 (2015)","journal-title":"Vietnam J. Math."},{"key":"2764_CR33","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1007\/s10957-013-0400-y","volume":"160","author":"TTV Nguyen","year":"2014","unstructured":"Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space. J. Optim. Theory Appl. 160, 809\u2013831 (2014)","journal-title":"J. Optim. Theory Appl."},{"key":"2764_CR34","first-page":"543","volume":"269","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method for solving the convex programming problem with convergence rate $$\\cal{O} (1\/k^2).$$ Dokl. Akad. Nauk SSSR 269, 543\u2013547 (1983)","journal-title":"Akad. Nauk SSSR"},{"key":"2764_CR35","doi-asserted-by":"publisher","unstructured":"Ogwo, G.N., Izuchukwu, C., Shehu, Y., Zhao, X.: Inertial Bregman Golden Ratio Algorithm for Solving Variational Inequalities. Authorea, 22, (2024). https:\/\/doi.org\/10.22541\/au.171906736.63358991\/v1","DOI":"10.22541\/au.171906736.63358991\/v1"},{"key":"2764_CR36","doi-asserted-by":"crossref","unstructured":"Oyewole, O.K., Abass, H.A., Moshokoa, S.P.: A simple proximal algorithm based on the golden ratio for equilibrium problem on Hadamard manifolds. Rend.Circ. Mat. Palermo, II. Ser. 74(1), 66 (2025)","DOI":"10.1007\/s12215-024-01183-4"},{"key":"2764_CR37","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","volume":"73","author":"Z Opial","year":"1967","unstructured":"Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591\u2013597 (1967)","journal-title":"Bull. Am. Math. Soc."},{"issue":"4","key":"2764_CR38","first-page":"493","volume":"5","author":"HA Pham Thi","year":"2021","unstructured":"Pham Thi, H.A., Ngo Thi, T., Nguyen, V.: The Golden ratio algorithms for solving equilibrium problems in Hilbert spaces. J. Nonlinear Var. Anal. 5(4), 493\u2013518 (2021)","journal-title":"J. Nonlinear Var. Anal."},{"issue":"5","key":"2764_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","volume":"4","author":"BT Polyak","year":"1964","unstructured":"Polyak, B.T.: Some methods of speeding up the convergence of iterates methods. USSR Comput. Math. Phys. 4(5), 1\u201317 (1964)","journal-title":"USSR Comput. Math. Phys."},{"issue":"6","key":"2764_CR40","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1080\/02331930601122876","volume":"57","author":"TD Quoc","year":"2008","unstructured":"Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749\u2013766 (2008)","journal-title":"Optimization"},{"key":"2764_CR41","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s10898-011-9693-2","volume":"52","author":"TD Quoc","year":"2011","unstructured":"Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52, 139\u2013159 (2011)","journal-title":"J. Glob. Optim."},{"issue":"5","key":"2764_CR42","doi-asserted-by":"publisher","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."},{"issue":"6","key":"2764_CR43","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1080\/02331930601122876","volume":"57","author":"DQ Tran","year":"2008","unstructured":"Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749\u2013776 (2008)","journal-title":"Optimization"},{"key":"2764_CR44","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1016\/j.apnum.2021.09.013","volume":"171","author":"DV Thong","year":"2022","unstructured":"Thong, D.V., Gibali, A., Vuong, P.T.: An explicit algorithm for solving monotone variational inequalities. Appl. Numer. Math. 171, 408\u2013425 (2022)","journal-title":"Appl. Numer. Math."},{"key":"2764_CR45","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1007\/s40306-019-00338-1","volume":"44","author":"NT Vinh","year":"2019","unstructured":"Vinh, N.T., Muu, L.D.: Inertial extragradient algorithms for solving equilibrium problems. Acta Math. Vietnam 44, 639\u2013663 (2019)","journal-title":"Acta Math. Vietnam"},{"key":"2764_CR46","doi-asserted-by":"publisher","first-page":"1803","DOI":"10.1007\/s11590-019-01474-1","volume":"14","author":"J Yang","year":"2020","unstructured":"Yang, J., Liu, H.: The subgradient extragradient method extended to pseudomonotone equilibrium problems and fixed point problems in Hilbert space. Optim. Lett. 14, 1803\u20131816 (2020)","journal-title":"Optim. Lett."},{"key":"2764_CR47","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/s10589-019-00156-z","volume":"75","author":"J Yang","year":"2020","unstructured":"Yang, J., Liu, H.: A self adaptive method for pseudomonotone equilibrium problems and variational inequalities. Comput. Optim. Appl. 75, 423\u2013440 (2020)","journal-title":"Comput. Optim. Appl."},{"issue":"4","key":"2764_CR48","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1007\/s10898-024-01377-1","volume":"89","author":"Y Yao","year":"2024","unstructured":"Yao, Y., Adamu, A., Shehu, Y., et al.: Simple proximal-type algorithms for equilibrium problems. J. Glob. Optim. 89(4), 1069\u20131098 (2024)","journal-title":"J. Glob. Optim."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02764-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02764-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02764-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,7]],"date-time":"2025-09-07T01:40:30Z","timestamp":1757209230000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02764-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,6]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["2764"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02764-8","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,6]]},"assertion":[{"value":"13 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 June 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"7"}}