{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:19:35Z","timestamp":1772515175521,"version":"3.50.1"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"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":["12371299"],"award-info":[{"award-number":["12371299"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s11590-025-02227-z","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T04:56:05Z","timestamp":1754369765000},"page":"307-332","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A superlinear convergence framework for Kurdyka\u2013\u0141ojasiewicz optimization"],"prefix":"10.1007","volume":"20","author":[{"given":"Yitian","family":"Qian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaohua","family":"Pan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,5]]},"reference":[{"key":"2227_CR1","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116, 5\u201316 (2009)","journal-title":"Math. Program."},{"key":"2227_CR2","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka\u2013\u0141ojasiewicz inequality. Math. Oper. Res. 35, 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"2227_CR3","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss\u2013Seidel methods. Math. Program. 137, 91\u2013129 (2013)","journal-title":"Math. Program."},{"key":"2227_CR4","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"2227_CR5","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s10107-009-0286-5","volume":"127","author":"C Cartis","year":"2011","unstructured":"Cartis, C., Gould, N.I.M., Toint, P.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math. Program. 127, 245\u2013295 (2011)","journal-title":"Math. Program."},{"key":"2227_CR6","doi-asserted-by":"crossref","unstructured":"Cartis, C., Gould, N.I.M., Toint, Ph.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part II: Worst-case function- and derivative-evaluation complexity. Math. Program. 130, 295\u2013319 (2011)","DOI":"10.1007\/s10107-009-0337-y"},{"key":"2227_CR7","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1007\/s10107-022-01809-4","volume":"198","author":"G Garrigos","year":"2023","unstructured":"Garrigos, G., Rosasco, L., Villa, S.: Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry. Math. Program. 198, 937\u2013996 (2023)","journal-title":"Math. Program."},{"key":"2227_CR8","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/s10107-020-01606-x","volume":"193","author":"N Doikov","year":"2022","unstructured":"Doikov, N., Nesterov, Y.: Local convergence of tensor methods. Math. Program. 193, 315\u2013336 (2022)","journal-title":"Math. Program."},{"key":"2227_CR9","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1137\/17M1142077","volume":"29","author":"GN Grapiglia","year":"2019","unstructured":"Grapiglia, G.N., Nesterov, Y.: Accelerated regularized Newton methods for minimizing composite convex functions. SIAM J. Optim. 29, 77\u201399 (2019)","journal-title":"SIAM J. Optim."},{"key":"2227_CR10","unstructured":"Griewank, A.: The modification of Newton\u2019s method for unconstrained optimization by bounding cubic terms. Technical report, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK (1981)"},{"key":"2227_CR11","doi-asserted-by":"publisher","first-page":"2897","DOI":"10.1137\/19M1286025","volume":"30","author":"B Jiang","year":"2020","unstructured":"Jiang, B., Lin, T.Y., Zhang, S.Z.: A unified adaptive tensor approximation scheme to accelerate composite convex optimization. SIAM J. Optim. 30, 2897\u20132926 (2020)","journal-title":"SIAM J. Optim."},{"key":"2227_CR12","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/s10589-020-00243-6","volume":"78","author":"C Kanzow","year":"2021","unstructured":"Kanzow, C., Lechner, T.: Globalized inexact proximal Newton-type methods for nonconvex composite functions. Comput. Optim. Appl. 78, 377\u2013410 (2021)","journal-title":"Comput. Optim. Appl."},{"key":"2227_CR13","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s10957-022-02101-3","volume":"195","author":"C Kanzow","year":"2022","unstructured":"Kanzow, C., Mehlitz, P.: Convergence properties of monotone and nonmonotone proximal gradient methods revisited. J. Optim. Theory Appl. 195, 624\u2013646 (2022)","journal-title":"J. Optim. Theory Appl."},{"key":"2227_CR14","doi-asserted-by":"publisher","first-page":"1420","DOI":"10.1137\/130921428","volume":"24","author":"JD Lee","year":"2014","unstructured":"Lee, J.D., Sun, Y.K., Saunders, M.A.: Proximal Newton-type methods for minimizing composite functions. SIAM J. Optim. 24, 1420\u20131443 (2014)","journal-title":"SIAM J. Optim."},{"key":"2227_CR15","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1007\/s10208-017-9366-8","volume":"18","author":"GY Li","year":"2018","unstructured":"Li, G.Y., Pong, T.K.: Calculus of the exponent of Kurdyka\u2013\u0141ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comput. Math. 18, 1199\u20131232 (2018)","journal-title":"Found. Comput. Math."},{"key":"2227_CR16","volume-title":"Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Springer, New York (2006)"},{"key":"2227_CR17","doi-asserted-by":"publisher","unstructured":"Mordukhovich, B.S., Yuan, X.M., Zeng, S.Z., Zhang, J.: A globally convergent proximal Newton-type method in nonsmooth convex optimization. Math. Program. https:\/\/doi.org\/10.1007\/s10107-022-01797-5 (2022)","DOI":"10.1007\/s10107-022-01797-5"},{"key":"2227_CR18","doi-asserted-by":"publisher","unstructured":"Nabou, Y., Necoara, I.: Efficiency of higher-order algorithms for minimizing general composite optimization. Comput. Optim. Appl. https:\/\/doi.org\/10.1007\/s10589-023-00533-9 (2023)","DOI":"10.1007\/s10589-023-00533-9"},{"key":"2227_CR19","unstructured":"Necoara, I., Lupu, D.: General higher-order majorization-minimization algorithms for (non)convex optimization. arXiv:2010.13893v3 (2021)"},{"key":"2227_CR20","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, 177\u2013205 (2006)","journal-title":"Math. Program."},{"key":"2227_CR21","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s10107-006-0089-x","volume":"112","author":"Y Nesterov","year":"2008","unstructured":"Nesterov, Y.: Accelerating the cubic regularization of Newton\u2019s method on convex problems. Math. Program. 112, 159\u2013181 (2008)","journal-title":"Math. Program."},{"key":"2227_CR22","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1080\/10556788.2020.1854252","volume":"37","author":"Y Nesterov","year":"2022","unstructured":"Nesterov, Y.: Inexact basic tensor methods for some classes of convex optimization problems. Optim. Methods Softw. 37, 878\u2013906 (2022)","journal-title":"Optim. Methods Softw."},{"key":"2227_CR23","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s10957-018-1272-y","volume":"177","author":"P Ochs","year":"2018","unstructured":"Ochs, P.: Local convergence of the heavy-ball method and iPiano for non-convex optimization. J. Optim. Theory Methods 177, 153\u2013180 (2018)","journal-title":"J. Optim. Theory Methods"},{"key":"2227_CR24","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s10589-010-9383-x","volume":"50","author":"T Pham Dinh","year":"2011","unstructured":"Pham Dinh, T., Niu, Y.S.: An efficient DC programming approach for portfolio decision with higher moments. Comput. Optim. Appl. 50, 525\u2013554 (2011)","journal-title":"Comput. Optim. Appl."},{"key":"2227_CR25","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T., Wets, R.J-B.: Variational Analysis. Springer (1998)","DOI":"10.1007\/978-3-642-02431-3"},{"key":"2227_CR26","doi-asserted-by":"publisher","first-page":"505","DOI":"10.2307\/2330125","volume":"6","author":"JH William","year":"1971","unstructured":"William, J.H.: The extension of portfolio analysis to three or more parameters. J. Financ. Quantitative Anal. 6, 505\u2013515 (1971)","journal-title":"J. Financ. Quantitative Anal."},{"key":"2227_CR27","doi-asserted-by":"publisher","first-page":"2479","DOI":"10.1109\/TSP.2009.2016892","volume":"57","author":"SJ Wright","year":"2009","unstructured":"Wright, S.J., Nowak, R., Figueiredo, M.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57, 2479\u20132493 (2009)","journal-title":"IEEE Trans. Signal Process."},{"key":"2227_CR28","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1007\/s10957-020-01779-7","volume":"188","author":"YQ Wu","year":"2021","unstructured":"Wu, Y.Q., Pan, S.H., Bi, S.J.: Kurdyka\u2013\u0141ojasiewicz property of zero-norm composite functions. J. Optim. Theory Appl. 188, 94\u2013112 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"2227_CR29","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.1007\/s10208-021-09528-6","volume":"22","author":"PR Yu","year":"2022","unstructured":"Yu, P.R., Li, G.Y., Pong, T.K.: Kurdyka\u2013\u0141ojasiewicz exponent via inf-projection. Found. Comput. Math. 22, 1171\u20131217 (2022)","journal-title":"Found. Comput. Math."},{"key":"2227_CR30","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s10107-018-1280-6","volume":"174","author":"MC Yue","year":"2019","unstructured":"Yue, M.C., Zhou, Z.R., So, A.M.-C.: A family of inexact SQA methods for non-smooth convex minimization with provable convergence guarantees based on the Luo-Tseng error bound property. Math. Program. 174, 327\u2013358 (2019)","journal-title":"Math. Program."},{"key":"2227_CR31","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1137\/18M1167498","volume":"29","author":"MC Yue","year":"2019","unstructured":"Yue, M.C., Zhou, Z.R., So, A.M.-C.: On the quadratic convergence of the cubic regularization method under a local error bound condition. SIAM J. Optim. 29, 904\u2013932 (2019)","journal-title":"SIAM J. Optim."},{"key":"2227_CR32","unstructured":"Zhou, Y., Wang, Z., Liang, Y.B.: Convergence of cubic regularization for nonconvex optimization under KL property. Neural Information Processing Systems Conference (2018)"},{"key":"2227_CR33","doi-asserted-by":"publisher","first-page":"892","DOI":"10.1109\/TSP.2021.3051369","volume":"69","author":"R Zhou","year":"2021","unstructured":"Zhou, R., Palomar, D.P.: Solving high-order portfolios via successive convex approximation algorithms. IEEE Trans. Signal Process. 69, 892\u2013904 (2021)","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-025-02227-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-025-02227-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-025-02227-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T04:00:48Z","timestamp":1772510448000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-025-02227-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,5]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2227"],"URL":"https:\/\/doi.org\/10.1007\/s11590-025-02227-z","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,5]]},"assertion":[{"value":"13 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 August 2025","order":3,"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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}