{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T07:36:47Z","timestamp":1771486607048,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:00:00Z","timestamp":1649203200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:00:00Z","timestamp":1649203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2022,5]]},"DOI":"10.1007\/s10915-022-01819-6","type":"journal-article","created":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T16:07:37Z","timestamp":1649261257000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Cubic Regularized Newton Method for the Saddle Point Models: A Global and Local Convergence Analysis"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8680-8761","authenticated-orcid":false,"given":"Kevin","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junyu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuzhong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,4,6]]},"reference":[{"key":"1819_CR1","unstructured":"Abadeh, S.S., Mohajerin Esfahani, P., Kuhn, D.: Distributionally robust logistic regression. In: Advances in Neural Information Processing Systems, pp. 1576\u20131584 (2015)"},{"key":"1819_CR2","unstructured":"Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia (2017)"},{"key":"1819_CR3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971132","volume-title":"Dynamic Noncooperative Game Theory","author":"T Ba\u015far","year":"1998","unstructured":"Ba\u015far, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1998)"},{"key":"1819_CR4","volume-title":"Generalized Inverses: Theory and Applications","author":"A Ben-Israel","year":"2003","unstructured":"Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)","edition":"2"},{"key":"1819_CR5","doi-asserted-by":"publisher","DOI":"10.1515\/9781400831050","volume-title":"Robust Optimization","author":"A Ben-Tal","year":"2009","unstructured":"Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization, vol. 28. Princeton University Press, Princeton (2009)"},{"key":"1819_CR6","unstructured":"Gao, R., Kleywegt, A.J.: Distributionally robust stochastic optimization with Wasserstein distance. arXiv preprint arXiv:1604.02199 (2016)"},{"key":"1819_CR7","unstructured":"Gidel, G., Berard, H., Vignoud, G., Vincent, P., Lacoste-Julien, S.: A variational inequality perspective on generative adversarial networks. arXiv preprint arXiv:1802.10551 (2018)"},{"key":"1819_CR8","unstructured":"Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Farley, D.W., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Advances in Neural Information Processing Systems, pp. 2672\u20132680 (2014)"},{"issue":"4","key":"1819_CR9","doi-asserted-by":"publisher","first-page":"2897","DOI":"10.1137\/19M1286025","volume":"30","author":"B Jiang","year":"2020","unstructured":"Jiang, B., Lin, T., Zhang, S.: A unified adaptive tensor approximation scheme to accelerate composite convex optimization. SIAM J. Optim. 30(4), 2897\u20132926 (2020)","journal-title":"SIAM J. Optim."},{"key":"1819_CR10","volume-title":"Numerical Methods in Economics","author":"KL Judd","year":"1998","unstructured":"Judd, K.L.: Numerical Methods in Economics. The MIT Press, Cambridge (1998)"},{"key":"1819_CR11","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":"1819_CR12","unstructured":"Lin, T., Jin, C., Jordan, M.: Near-optimal algorithms for minimax optimization. arXiv preprint arXiv:2002.02417 (2020)"},{"key":"1819_CR13","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/S0898-1221(01)00278-4","volume":"43","author":"TT Lu","year":"2003","unstructured":"Lu, T.T., Shiou, S.H.: Inverse of $$2\\times 2$$ block matrices. Comput. Math. Appl. 43, 119\u2013129 (2003)","journal-title":"Comput. Math. Appl."},{"key":"1819_CR14","unstructured":"Mokhtari, A., Ozdaglar, A., Pattathil, S.: A unified analysis of extra-gradient and optimistic gradient methods for saddle point problems: proximal point approach. arXiv preprint arXiv:1901.08511 (2019)"},{"issue":"1","key":"1819_CR15","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1137\/S1052623403425629","volume":"15","author":"A Nemirovski","year":"2004","unstructured":"Nemirovski, A.: Prox-method with rate of convergence $$o(1\/t)$$ for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15(1), 229\u2013251 (2004)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1819_CR16","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1137\/18M1167498","volume":"29","author":"M-C Yue","year":"2019","unstructured":"Yue, M.-C., Zhou, Z., Man-Cho So, A.: On the quadratic convergence of the cubic regularization method under a local error bound condition. SIAM J. Optim. 29(1), 904\u2013932 (2019)","journal-title":"SIAM J. Optim."},{"key":"1819_CR17","doi-asserted-by":"crossref","unstructured":"Nesterov, Y.: Accelerating the cubic regularization of Newton\u2019s method on convex problems. Math. Program. 112(1), 159\u2013181 (2008)","DOI":"10.1007\/s10107-006-0089-x"},{"issue":"2\u20133","key":"1819_CR18","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10107-006-0034-z","volume":"109","author":"Y Nesterov","year":"2007","unstructured":"Nesterov, Y.: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109(2\u20133), 319\u2013344 (2007)","journal-title":"Math. Program."},{"issue":"1","key":"1819_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10107-021-01727-x","volume":"186","author":"Y Nesterov","year":"2018","unstructured":"Nesterov, Y.: Implementable tensor methods in unconstrained convex optimization. Math. Program. 186(1), 1\u201327 (2018)","journal-title":"Math. Program."},{"key":"1819_CR20","first-page":"2019","volume":"23","author":"Y Nesterov","year":"2019","unstructured":"Nesterov, Y.: Inexact basic tensor methods. CORE DP 23, 2019 (2019)","journal-title":"CORE DP"},{"issue":"1","key":"1819_CR21","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10107-006-0706-8","volume":"108","author":"Y Nesterov","year":"2006","unstructured":"Nesterov, Y., Polyak, B.T.: Cubic regularization of newton method and its global performance. Math. Program. 108(1), 177\u2013205 (2006)","journal-title":"Math. Program."},{"key":"1819_CR22","doi-asserted-by":"crossref","unstructured":"Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Available at SSRN 970903 (2006)","DOI":"10.2139\/ssrn.970903"},{"key":"1819_CR23","volume-title":"Numerical Optimization","author":"J Nocedal","year":"2006","unstructured":"Nocedal, J., Wright, S.: Numerical Optimization. Springer Science & Business Media, New York (2006)"},{"key":"1819_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-019-01420-0","volume":"185","author":"Y Ouyang","year":"2019","unstructured":"Ouyang, Y., Xu, Y.: Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems. Math. Program. 185, 1\u201335 (2019)","journal-title":"Math. Program."},{"key":"1819_CR25","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718812","volume-title":"A Mathematical View of Interior-Point Methods in Convex Optimization","author":"J Renegar","year":"2001","unstructured":"Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization, vol. 3. SIAM, Philadelphia (2001)"},{"issue":"5","key":"1819_CR26","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."},{"key":"1819_CR27","volume-title":"Algorithmic Game Theory","year":"2007","unstructured":"Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.): Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)"},{"issue":"1\u20133","key":"1819_CR28","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/BF01581276","volume":"58","author":"K Taji","year":"1993","unstructured":"Taji, K., Fukushima, M., Ibaraki, T.: A globally convergent Newton method for solving strongly monotone variational inequalities. Math. Program. 58(1\u20133), 369\u2013383 (1993)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1819_CR29","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/0377-0427(94)00094-H","volume":"60","author":"P Tseng","year":"1995","unstructured":"Tseng, P.: On linear convergence of iterative methods for the variational inequality problem. J. Comput. Appl. Math. 60(1\u20132), 237\u2013252 (1995)","journal-title":"J. Comput. Appl. Math."},{"key":"1819_CR30","unstructured":"Tseng, P.: On accelerated proximal gradient methods for convex-concave optimization. Unpublished Manuscript (2008)"},{"key":"1819_CR31","volume-title":"Theory of Games and Economic Behavior (Commemorative Edition)","author":"J von Neumann","year":"2007","unstructured":"von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior (Commemorative Edition). Princeton University Press, Princeton (2007)"},{"key":"1819_CR32","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971453","volume-title":"Primal-Dual Interior-Point Methods","author":"SJ Wright","year":"1997","unstructured":"Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)"},{"key":"1819_CR33","unstructured":"Zhang, G., Wu, K., Poupart, P., Yu, Y.: Newton-type methods for minimax optimization. arXiv preprint arXiv:2006.14592 (2020)"},{"key":"1819_CR34","unstructured":"Zhang, J., Hong, M., Zhang, S.: On lower iteration complexity bounds for the saddle point problems. arXiv preprint arXiv:1912.07481 (2018)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01819-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-022-01819-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01819-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T21:14:23Z","timestamp":1675199663000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-022-01819-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,6]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,5]]}},"alternative-id":["1819"],"URL":"https:\/\/doi.org\/10.1007\/s10915-022-01819-6","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,6]]},"assertion":[{"value":"14 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 February 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2022","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 declares that they have no conflicts of interest\/competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The codes analyzed during the current study are available from the corresponding author on reasonable request.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code Availability"}}],"article-number":"60"}}