{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T08:08:57Z","timestamp":1758874137290,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T00:00:00Z","timestamp":1655510400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T00:00:00Z","timestamp":1655510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"The Ministry of Education and Training of Viet Nam","award":["B2022-TCT-02"],"award-info":[{"award-number":["B2022-TCT-02"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2023,4]]},"DOI":"10.1007\/s11590-022-01899-1","type":"journal-article","created":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T12:04:16Z","timestamp":1655553856000},"page":"699-720","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Regularization of vector equilibrium problems"],"prefix":"10.1007","volume":"17","author":[{"given":"Lam Quoc","family":"Anh","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9524-4179","authenticated-orcid":false,"given":"Tran Quoc","family":"Duy","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,18]]},"reference":[{"issue":"9\u201310","key":"1899_CR1","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1080\/01630568908816342","volume":"10","author":"LD Muu","year":"1989","unstructured":"Muu, L.D., Oettli, W.: A lagrangian penalty function method for monotone variational inequalities. Numer. Func. Anal. Optim. 10(9\u201310), 1003\u20131017 (1989)","journal-title":"Numer. Func. Anal. Optim."},{"issue":"12","key":"1899_CR2","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(12), 1159\u20131166 (1992)","journal-title":"Nonlinear Anal."},{"issue":"1\u20132","key":"1899_CR3","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s10107-007-0125-5","volume":"116","author":"AN Iusem","year":"2009","unstructured":"Iusem, A.N., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116(1\u20132), 259\u2013273 (2009)","journal-title":"Math. Program."},{"issue":"2","key":"1899_CR4","first-page":"489","volume":"18","author":"AN Iusem","year":"2011","unstructured":"Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18(2), 489\u2013503 (2011)","journal-title":"J. Convex Anal."},{"issue":"1","key":"1899_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ejor.2012.11.037","volume":"227","author":"G Bigi","year":"2013","unstructured":"Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1\u201311 (2013)","journal-title":"Eur. J. Oper. Res."},{"key":"1899_CR6","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/978-981-13-3059-9_6","volume-title":"Mathematical Programming and Game Theory","author":"D Aussel","year":"2018","unstructured":"Aussel, D., Dutta, J., Pandit, T.: About the links between equilibrium problems and variational inequalities. In: Neogy, S., Bapat, R., Dubey, D. (eds.) Mathematical Programming and Game Theory, pp. 115\u2013130. Springer, Berlin (2018)"},{"key":"1899_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-00205-3","volume-title":"Nonlinear Programming Techniques for Equilibria","author":"G Bigi","year":"2019","unstructured":"Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer, Switzerland (2019)"},{"key":"1899_CR8","volume-title":"Equilibrium Problems and Applications","author":"G Kassay","year":"2019","unstructured":"Kassay, G., R\u0103dulescu, V.D.: Equilibrium Problems and Applications. Academic Press, London (2019)"},{"key":"1899_CR9","first-page":"151","volume-title":"Variational Inequalities and Complementarity Problems","author":"F Giannessi","year":"1980","unstructured":"Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Gianneesi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151\u2013186. John Wiley and Sons, New York (1980)"},{"issue":"1","key":"1899_CR10","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10898-014-0264-1","volume":"63","author":"G Kassay","year":"2015","unstructured":"Kassay, G., Miholca, M.: Existence results for vector equilibrium problems given by a sum of two functions. J. Glob. Optim. 63(1), 195\u2013211 (2015)","journal-title":"J. Glob. Optim."},{"issue":"11","key":"1899_CR11","doi-asserted-by":"publisher","first-page":"1929","DOI":"10.1080\/02331934.2016.1209673","volume":"65","author":"LQ Anh","year":"2016","unstructured":"Anh, L.Q., Duy, T.Q.: Tykhonov well-posedness for lexicographic equilibrium problems. Optimization 65(11), 1929\u20131948 (2016)","journal-title":"Optimization"},{"issue":"4","key":"1899_CR12","doi-asserted-by":"publisher","first-page":"2405","DOI":"10.1137\/17M111883X","volume":"27","author":"C Guti\u00e9rrez","year":"2017","unstructured":"Guti\u00e9rrez, C., Kassay, G., Novo, V., R\u00f3denas-Pedregosa, J.: Ekeland variational principles in vector equilibrium problems. SIAM J. Optim. 27(4), 2405\u20132425 (2017)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1899_CR13","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/s11117-017-0496-7","volume":"22","author":"LQ Anh","year":"2018","unstructured":"Anh, L.Q., Duy, T.Q.: On penalty method for equilibrium problems in lexicographic order. Positivity 22(1), 39\u201357 (2018)","journal-title":"Positivity"},{"key":"1899_CR14","first-page":"231","volume":"1","author":"N Hebestreit","year":"2019","unstructured":"Hebestreit, N.: Vector variational inequalities and related topics: a survey of theory and applications. Appl. Set-Valued Anal. Optim. 1, 231\u2013305 (2019)","journal-title":"Appl. Set-Valued Anal. Optim."},{"key":"1899_CR15","first-page":"45","volume":"2019","author":"T Chen","year":"2019","unstructured":"Chen, T., Zou, S., Zhang, Y.: New existence theorems for vector equilibrium problems with set-valued mappings. J. Nonlinear Funct. Anal. 2019, 45 (2019)","journal-title":"J. Nonlinear Funct. Anal."},{"issue":"3","key":"1899_CR16","first-page":"427","volume":"4","author":"NV Hung","year":"2020","unstructured":"Hung, N.V., Dai, L.X., Kobis, E., Yao, J.C.: The generic stability of solutions for vector quasi-equilibrium problems on Hadamard manifolds. J. Nonlinear Var. Anal. 4(3), 427\u2013438 (2020)","journal-title":"J. Nonlinear Var. Anal."},{"issue":"4","key":"1899_CR17","doi-asserted-by":"publisher","first-page":"1323","DOI":"10.1007\/s11117-021-00818-5","volume":"25","author":"LQ Anh","year":"2021","unstructured":"Anh, L.Q., Duy, T.Q., Khanh, P.Q.: Levitin-Polyak well-posedness for equilibrium problems with the lexicographic order. Positivity 25(4), 1323\u20131349 (2021)","journal-title":"Positivity"},{"issue":"1","key":"1899_CR18","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s11117-015-0341-9","volume":"20","author":"LQ Anh","year":"2016","unstructured":"Anh, L.Q., Duy, T.Q., Khanh, P.Q.: Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity 20(1), 61\u201380 (2016)","journal-title":"Positivity"},{"key":"1899_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-63049-6","volume-title":"Vector Variational Inequalities and Vector Optimization","author":"QH Ansari","year":"2018","unstructured":"Ansari, Q.H., K\u00f6bis, E., Yao, J.C.: Vector Variational Inequalities and Vector Optimization. Springer, Berlin (2018)"},{"issue":"4","key":"1899_CR20","doi-asserted-by":"publisher","first-page":"1326","DOI":"10.1137\/060675320","volume":"18","author":"MP Friedlander","year":"2007","unstructured":"Friedlander, M.P., Tseng, P.: Exact regularization of convex programs. SIAM J. Optim. 18(4), 1326\u20131350 (2007)","journal-title":"SIAM J. Optim."},{"key":"1899_CR21","first-page":"226","volume-title":"Complementarity and Variational Problems: State of the Art","author":"OL Mangasarian","year":"1997","unstructured":"Mangasarian, O.L.: The ill-posed linear complementarity problem. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 226\u2013233. SIAM Publications, Philadelphia (1997)"},{"key":"1899_CR22","first-page":"501","volume":"152","author":"AN Tikhonov","year":"1963","unstructured":"Tikhonov, A.N.: On the solution of ill-posed problems and regularization method. Dokl. Akad. Nauk SSSR 152, 501\u2013504 (1963)","journal-title":"Dokl. Akad. Nauk SSSR"},{"issue":"8","key":"1899_CR23","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1080\/02331934.2015.1028935","volume":"64","author":"AA Khan","year":"2015","unstructured":"Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64(8), 1703\u20131724 (2015)","journal-title":"Optimization"},{"issue":"2","key":"1899_CR24","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1080\/02331934.2019.1607339","volume":"69","author":"J Chen","year":"2020","unstructured":"Chen, J., Ju, X., K\u00f6bis, E., Liou, Y.C.: Tikhonov type regularization methods for inverse mixed variational inequalities. Optimization 69(2), 401\u2013413 (2020)","journal-title":"Optimization"},{"key":"1899_CR25","first-page":"374","volume":"28","author":"J Gwinner","year":"1978","unstructured":"Gwinner, J.: On the regularization of monotone variational inequalities. Oper. Res. Verfahren. 28, 374\u2013386 (1978)","journal-title":"Oper. Res. Verfahren."},{"key":"1899_CR26","first-page":"93","volume":"10","author":"IV Konnov","year":"2009","unstructured":"Konnov, I.V.: Regularization method for nonmonotone equilibrium problems. J. Nonlinear Convex Anal. 10, 93\u2013101 (2009)","journal-title":"J. Nonlinear Convex Anal."},{"issue":"4","key":"1899_CR27","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s10898-010-9551-7","volume":"49","author":"IV Konnov","year":"2011","unstructured":"Konnov, I.V., Dyabilkin, D.A.: Nonmonotone equilibrium problems: coercivity conditions and weak regularization. J. Glob. Optim. 49(4), 575\u2013587 (2011)","journal-title":"J. Glob. Optim."},{"issue":"17","key":"1899_CR28","doi-asserted-by":"publisher","first-page":"6121","DOI":"10.1016\/j.na.2011.05.091","volume":"74","author":"PG Hung","year":"2011","unstructured":"Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74(17), 6121\u20136129 (2011)","journal-title":"Nonlinear Anal."},{"issue":"1","key":"1899_CR29","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.jmaa.2012.12.034","volume":"401","author":"PR Oliveira","year":"2013","unstructured":"Oliveira, P.R., Santos, P.S.M., Silva, A.N.: A Tikhonov-type regularization for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 401(1), 336\u2013342 (2013)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"1899_CR30","doi-asserted-by":"publisher","first-page":"726","DOI":"10.1007\/s10957-017-1062-y","volume":"172","author":"O Chadli","year":"2017","unstructured":"Chadli, O., Ansari, Q.H., Al-Homidan, S.: Existence of solutions and algorithms for bilevel vector equilibrium problems: an auxiliary principle technique. J. Optim. Theory Appl. 172(3), 726\u2013758 (2017)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1899_CR31","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1007\/s10957-014-0588-5","volume":"164","author":"IV Konnov","year":"2015","unstructured":"Konnov, I.V.: Regularized penalty method for general equilibrium problems in banach spaces. J. Optim. Theory Appl. 164(2), 500\u2013513 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1899_CR32","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1007\/s10589-020-00258-z","volume":"78","author":"LQ Anh","year":"2021","unstructured":"Anh, L.Q., Duy, T.Q., Van Tri, T., Muu, L.D.: The Tikhonov regularization for vector equilibrium problems. Comput. Optim. Appl. 78(3), 769\u2013792 (2021)","journal-title":"Comput. Optim. Appl."},{"key":"1899_CR33","volume-title":"Functional Analysis","author":"W Rudin","year":"1991","unstructured":"Rudin, W.: Functional Analysis. McGraw-Hill, New York (1991)"},{"key":"1899_CR34","doi-asserted-by":"publisher","first-page":"69","DOI":"10.7146\/math.scand.a-11507","volume":"34","author":"J Zowe","year":"1974","unstructured":"Zowe, J.: Subdifferentiability of convex functions with values in an ordered vector space. Math. Scand. 34, 69\u201383 (1974)","journal-title":"Math. Scand."},{"issue":"1","key":"1899_CR35","first-page":"123","volume":"63","author":"E Blum","year":"1994","unstructured":"Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. India 63(1), 123\u2013145 (1994)","journal-title":"Math. Stud. India"},{"issue":"1","key":"1899_CR36","first-page":"213","volume":"22","author":"W Oettli","year":"1997","unstructured":"Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22(1), 213\u2013221 (1997)","journal-title":"Acta Math. Vietnam."},{"issue":"1","key":"1899_CR37","doi-asserted-by":"publisher","first-page":"165","DOI":"10.11650\/twjm\/1500406534","volume":"16","author":"Q Ansari","year":"2012","unstructured":"Ansari, Q., Farajzadeh, A., Schaible, S.: Existence of solutions of strong vector equilibrium problems. Taiwan. J. Math. 16(1), 165\u2013178 (2012)","journal-title":"Taiwan. J. Math."},{"key":"1899_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18351-5","volume-title":"Vector Optimization with Infimum and Supremum","author":"A L\u00f6hne","year":"2011","unstructured":"L\u00f6hne, A.: Vector Optimization with Infimum and Supremum. Springer, Berlin (2011)"}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01899-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-022-01899-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01899-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,8]],"date-time":"2023-03-08T12:16:35Z","timestamp":1678277795000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-022-01899-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,18]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["1899"],"URL":"https:\/\/doi.org\/10.1007\/s11590-022-01899-1","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"type":"print","value":"1862-4472"},{"type":"electronic","value":"1862-4480"}],"subject":[],"published":{"date-parts":[[2022,6,18]]},"assertion":[{"value":"20 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}