{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T18:22:36Z","timestamp":1776363756498,"version":"3.51.2"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T00:00:00Z","timestamp":1726185600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T00:00:00Z","timestamp":1726185600000},"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":["12071324"],"award-info":[{"award-number":["12071324"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s12190-024-02219-9","type":"journal-article","created":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T10:08:11Z","timestamp":1727345291000},"page":"103-131","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extension of the subgradient extragradient algorithm for solving variational inequalities without monotonicity"],"prefix":"10.1007","volume":"71","author":[{"given":"Jiaxin","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zunjie","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Yongle","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,9,13]]},"reference":[{"key":"2219_CR1","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1090\/S0002-9904-1964-11178-2","volume":"70","author":"AA Goldstein","year":"1964","unstructured":"Goldstein, A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70, 709\u2013710 (1964)","journal-title":"Bull. Amer. Math. Soc."},{"key":"2219_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0041-5553(66)90114-5","volume":"6","author":"ES Levitin","year":"1966","unstructured":"Levitin, E.S., Polyak, B.T.: Constrained minimization methods. USSR Comput. Math. Math. Phys. 6, 1\u201350 (1966)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"2219_CR3","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, 877\u2013898 (1976)","journal-title":"SIAM J. Control. Optim."},{"key":"2219_CR4","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1137\/S1052623495286302","volume":"8","author":"RS Burachik","year":"1998","unstructured":"Burachik, R.S., Iusem, A.N.: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. 8, 197\u2013216 (1998)","journal-title":"SIAM J. Optim."},{"key":"2219_CR5","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon. 12, 747\u2013756 (1976)","journal-title":"Matecon."},{"key":"2219_CR6","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1023\/A:1012606212823","volume":"111","author":"YJ Wang","year":"2001","unstructured":"Wang, Y.J., Xiu, N.H., Wang, C.Y.: Unified framework of extragradient-type methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 111, 641\u2013656 (2001)","journal-title":"J. Optim. Theory Appl."},{"key":"2219_CR7","doi-asserted-by":"publisher","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":"2219_CR8","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1137\/S0363012997317475","volume":"37","author":"MV Solodov","year":"1999","unstructured":"Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control. Optim. 37, 765\u2013776 (1999)","journal-title":"SIAM J. Control. Optim."},{"key":"2219_CR9","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.cam.2005.01.031","volume":"185","author":"YR He","year":"2006","unstructured":"He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166\u2013173 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"2219_CR10","doi-asserted-by":"crossref","unstructured":"Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalites in Hilbert space. J. Optim. Theory Appl. 148, 318\u2013335 (2011)","DOI":"10.1007\/s10957-010-9757-3"},{"key":"2219_CR11","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1137\/14097238X","volume":"25","author":"Y Malitsky","year":"2015","unstructured":"Malitsky, Y.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502\u2013520 (2015)","journal-title":"SIAM J. Optim."},{"key":"2219_CR12","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":"2219_CR13","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s10589-014-9659-7","volume":"60","author":"ML Ye","year":"2015","unstructured":"Ye, M.L., He, Y.R.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60, 141\u2013150 (2015)","journal-title":"Comput. Optim. Appl."},{"key":"2219_CR14","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/s10957-012-0002-0","volume":"154","author":"AS Brito","year":"2012","unstructured":"Brito, A.S., da Cruz Neto, J.X., Lopes, J.O., Oliveira, P.R.: Interior proximal algorithm for quasiconvex programming problems and variational inequalities with linear constraints. J. Optim. Theory Appl. 154, 217\u2013234 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"2219_CR15","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1007\/s10957-012-0111-9","volume":"155","author":"N Langenberg","year":"2012","unstructured":"Langenberg, N.: An interior proximal method for a class of quasimonotone variational inequalities. J. Optim. Theory Appl. 155, 902\u2013922 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"2219_CR16","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1007\/s10589-020-00217-8","volume":"77","author":"H Liu","year":"2020","unstructured":"Liu, H., Yang, J.: Weak convergence of iterative methods for solving quasimonotone variational inequalities. Comput. Optim. Appl. 77, 491\u2013508 (2020)","journal-title":"Comput. Optim. Appl."},{"key":"2219_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01670-1","volume":"90","author":"GN Ogwo","year":"2022","unstructured":"Ogwo, G.N., Izuchukwu, C., Shehu, Y., Mewomo, O.T.: Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems. J. Sci. Comput. 90, 1\u201335 (2022)","journal-title":"J. Sci. Comput."},{"key":"2219_CR18","doi-asserted-by":"publisher","first-page":"1723","DOI":"10.1007\/s11075-021-01170-1","volume":"89","author":"ML Ye","year":"2022","unstructured":"Ye, M.L.: An infeasible projection type algorithm for nonmonotone variational inequalities. Numer. Algorithms. 89, 1723\u20131742 (2022)","journal-title":"Numer. Algorithms."},{"key":"2219_CR19","first-page":"44","volume":"37","author":"IV Konnov","year":"1993","unstructured":"Konnov, I.V.: Combined relaxation methods for finding equilibrium points and solving related problems. Russ. Math. (Iz VUZ) 37, 44\u201351 (1993)","journal-title":"Russ. Math. (Iz VUZ)"},{"key":"2219_CR20","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1007\/s10957-020-01791-x","volume":"188","author":"M Lei","year":"2021","unstructured":"Lei, M., He, Y.R.: An extragradient method for solving variational inequalities without monotonicity. J. Optim. Theory Appl. 188, 432\u2013446 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"2219_CR21","doi-asserted-by":"publisher","first-page":"1715","DOI":"10.1007\/s11075-021-01248-w","volume":"90","author":"B Van Dinh","year":"2022","unstructured":"Van Dinh, B., Manh, H.D., Thanh, T.T.H.: A modified Solodov-Svaiter method for solving nonmonotone variational inequality problems. Numer. Algorithms. 90, 1715\u20131734 (2022)","journal-title":"Numer. Algorithms."},{"key":"2219_CR22","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s11075-019-00780-0","volume":"84","author":"DV Thong","year":"2020","unstructured":"Thong, D.V., Shehu, Y., Iyiola, O.S.: Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings. Numer. Algorithms. 84, 795\u2013823 (2020)","journal-title":"Numer. Algorithms."},{"key":"2219_CR23","doi-asserted-by":"publisher","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":"2219_CR24","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2024.2371041","author":"ZJ Huang","year":"2024","unstructured":"Huang, Z.J., Zhang, Y.L., He, Y.R.: A class modified projection algorithms for nonmonotone variational inequalities with continuity. Optimization (2024). https:\/\/doi.org\/10.1080\/02331934.2024.2371041","journal-title":"Optimization"},{"key":"2219_CR25","unstructured":"Huang, K., Zhang, S.: Beyond monotone variational inequalities: Solution methods and iteration complexities. arXiv:2304.04153 (2023)"},{"key":"2219_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-024-02075-2","author":"T Lin","year":"2024","unstructured":"Lin, T., Jordan, M.I.: Perseus: A simple and optimal high-order method for variational inequalities. Math. Program. (2024). https:\/\/doi.org\/10.1007\/s10107-024-02075-2","journal-title":"Math. Program."},{"key":"2219_CR27","doi-asserted-by":"publisher","first-page":"397","DOI":"10.11650\/tjm\/231202","volume":"28","author":"BT Tuyen","year":"2024","unstructured":"Tuyen, B.T., Manh, H.D., Van Dinh, B.: Inertial algorithms for solving nonmonotone variational inequality problems. Taiwan. J. Math. 28, 397\u2013421 (2024)","journal-title":"Taiwan. J. Math."},{"key":"2219_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-48311-5","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)","edition":"2"},{"key":"2219_CR29","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1007\/s10559-015-9768-z","volume":"51","author":"SV Denisov","year":"2015","unstructured":"Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybernet. Syst. Anal. 51, 757\u2013765 (2015)","journal-title":"Cybernet. Syst. Anal."},{"key":"2219_CR30","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1023\/A:1011219303531","volume":"20","author":"P Marcotte","year":"2001","unstructured":"Marcotte, P., Zhu, D.L.: A cutting plane method for solving quasimonotone variational inequalities. Comput. Optim. Appl. 20, 317\u2013324 (2001)","journal-title":"Comput. Optim. Appl."},{"key":"2219_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-2301-7","volume-title":"Projected Dynamical Systems and Variational Inequalities with Applications","author":"A Nagurney","year":"1996","unstructured":"Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Springer, New York (1996)"},{"key":"2219_CR32","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1080\/10556788.2017.1414814","volume":"34","author":"K Tu","year":"2019","unstructured":"Tu, K., Zhang, H.B., Xia, F.Q.: A new alternating projection-based prediction-correction method for structured variational inequalities. Optim. Method. Softw. 34, 707\u2013730 (2019)","journal-title":"Optim. Method. Softw."},{"key":"2219_CR33","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex Optimization","author":"S Boyd","year":"2004","unstructured":"Boyd, S., Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02219-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-024-02219-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02219-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T19:59:50Z","timestamp":1737662390000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-024-02219-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,13]]},"references-count":33,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["2219"],"URL":"https:\/\/doi.org\/10.1007\/s12190-024-02219-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,13]]},"assertion":[{"value":"31 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 September 2024","order":4,"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 Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}}]}}