{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:56:27Z","timestamp":1757620587688,"version":"3.44.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T00:00:00Z","timestamp":1753920000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T00:00:00Z","timestamp":1753920000000},"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":["12371303","12301406"],"award-info":[{"award-number":["12371303","12301406"]}],"id":[{"id":"10.13039\/501100001809","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":[[2025,11]]},"DOI":"10.1007\/s10957-025-02799-x","type":"journal-article","created":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T08:38:10Z","timestamp":1753951090000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Implementable Proximal Extragradient Method for Structured Fractional Programming"],"prefix":"10.1007","volume":"207","author":[{"given":"Jiajun","family":"Hao","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7672-4943","authenticated-orcid":false,"given":"Hongjin","family":"He","sequence":"additional","affiliation":[]},{"given":"Liangshao","family":"Hou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,31]]},"reference":[{"key":"2799_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."},{"issue":"2","key":"2799_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-\u0141ojasiewicz inequality. Math. Oper. Res. 35(2), 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"2799_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-Seidel methods. Math. Program. 137, 91\u2013129 (2013)","journal-title":"Math. Program."},{"key":"2799_CR4","unstructured":"Azizian, W., Mitliagkas, I., Lacoste-Julien, S., Gidel, G.: A tight and unified analysis of gradient-based methods for a whole spectrum of differentiable games. In: International Conference on Artificial Intelligence and Statistics, pp. 2863\u20132873. PMLR (2020)"},{"issue":"2","key":"2799_CR5","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1287\/trsc.2019.0929","volume":"54","author":"R Baldacci","year":"2020","unstructured":"Baldacci, R., Lim, A., Traversi, E., Wolfler Calvo, R.: Optimal solution of vehicle routing problems with fractional objective function. Transp. Sci. 54(2), 434\u2013452 (2020)","journal-title":"Transp. Sci."},{"issue":"1","key":"2799_CR6","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(1), 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"issue":"3","key":"2799_CR7","doi-asserted-by":"publisher","first-page":"1480","DOI":"10.1137\/120869778","volume":"23","author":"A Beck","year":"2013","unstructured":"Beck, A., Eldar, Y.C.: Sparsity constrained nonlinear optimization: optimality conditions and algorithms. SIAM J. Optim. 23(3), 1480\u20131509 (2013)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"2799_CR8","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2(1), 183\u2013202 (2009)","journal-title":"SIAM J. Imag. Sci."},{"issue":"2","key":"2799_CR9","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.ejor.2005.02.069","volume":"173","author":"HP Benson","year":"2006","unstructured":"Benson, H.P.: Fractional programming with convex quadratic forms and functions. Eur. J. Oper. Res. 173(2), 351\u2013369 (2006)","journal-title":"Eur. J. Oper. Res."},{"issue":"4","key":"2799_CR10","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, A., Lewis, A.: The \u0141ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205\u20131223 (2007)","journal-title":"SIAM J. Optim."},{"issue":"8","key":"2799_CR11","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1080\/02331934.2017.1294592","volume":"66","author":"RI Bo\u0163","year":"2017","unstructured":"Bo\u0163, R.I., Csetnek, E.R.: Proximal-gradient algorithms for fractional programming. Optimization 66(8), 1383\u20131396 (2017)","journal-title":"Optimization"},{"issue":"3","key":"2799_CR12","doi-asserted-by":"publisher","first-page":"2415","DOI":"10.1287\/moor.2021.1214","volume":"47","author":"RI Bo\u0163","year":"2022","unstructured":"Bo\u0163, R.I., Dao, M.N., Li, G.: Extrapolated proximal subgradient algorithms for nonconvex and nonsmooth fractional programs. Math. Oper. Res. 47(3), 2415\u20132443 (2022)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"2799_CR13","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1137\/S1052623495290179","volume":"7","author":"GH Chen","year":"1997","unstructured":"Chen, G.H., Rockafellar, R.T.: Convergence rates in forward-backward splitting. SIAM J. Optim. 7(2), 421\u2013444 (1997)","journal-title":"SIAM J. Optim."},{"issue":"7","key":"2799_CR14","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1287\/mnsc.13.7.492","volume":"13","author":"W Dinkelbach","year":"1967","unstructured":"Dinkelbach, W.: On nonlinear fractional programming. Manage. Sci. 13(7), 492\u2013498 (1967)","journal-title":"Manage. Sci."},{"key":"2799_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115564","volume":"438","author":"K Gao","year":"2024","unstructured":"Gao, K., Huang, Z.H., Guo, L.: Low-rank matrix recovery problem minimizing a new ratio of two norms approximating the rank function then using an ADMM-type solver with applications. J. Comput. Appl. Math. 438, 115564 (2024)","journal-title":"J. Comput. Appl. Math."},{"key":"2799_CR16","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1023\/A:1013096613105","volume":"112","author":"BS He","year":"2002","unstructured":"He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. Journal of Optimization Theory And Applications 112, 111\u2013128 (2002)","journal-title":"Journal of Optimization Theory And Applications"},{"issue":"12","key":"2799_CR17","first-page":"1457","volume":"5","author":"PO Hoyer","year":"2004","unstructured":"Hoyer, P.O.: Non-negative matrix factorization with sparseness constraints. J. Mach. Learn. Res. 5(12), 1457\u20131469 (2004)","journal-title":"J. Mach. Learn. Res."},{"issue":"3","key":"2799_CR18","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF02591871","volume":"26","author":"T Ibaraki","year":"1983","unstructured":"Ibaraki, T.: Parametric approaches to fractional programs. Math. Program. 26(3), 345\u2013362 (1983)","journal-title":"Math. Program."},{"issue":"7","key":"2799_CR19","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1287\/mnsc.12.7.609","volume":"12","author":"R Jagannathan","year":"1966","unstructured":"Jagannathan, R.: On some properties of programming problems in parametric form pertaining to fractional programming. Manage. Sci. 12(7), 609\u2013615 (1966)","journal-title":"Manage. Sci."},{"issue":"2","key":"2799_CR20","doi-asserted-by":"publisher","first-page":"143","DOI":"10.15807\/jorsj.32.143","volume":"32","author":"H Konno","year":"1989","unstructured":"Konno, H., Inori, M.: Bond portfolio optimization by bilinear fractional programming. Journal of the Operations Research Society of Japan 32(2), 143\u2013158 (1989)","journal-title":"Journal of the Operations Research Society of Japan"},{"issue":"3","key":"2799_CR21","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s10957-007-9284-z","volume":"135","author":"H Konno","year":"2007","unstructured":"Konno, H., Tsuchiya, K., Yamamoto, R.: Minimization of the ratio of functions defined as sums of the absolute values. J. Optim. Theory Appl. 135(3), 399\u2013410 (2007)","journal-title":"J. Optim. Theory Appl."},{"key":"2799_CR22","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":"2799_CR23","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.acha.2021.08.004","volume":"56","author":"Q Li","year":"2022","unstructured":"Li, Q., Shen, L., Zhang, N., Zhou, J.: A proximal algorithm with backtracked extrapolation for a class of structured fractional programming. Appl. Comput. Harmon. Anal. 56, 98\u2013122 (2022)","journal-title":"Appl. Comput. Harmon. Anal."},{"issue":"1","key":"2799_CR24","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s10208-015-9282-8","volume":"17","author":"T Lin","year":"2017","unstructured":"Lin, T., Ma, S., Zhang, S.: An extragradient-based alternating direction method for convex minimization. Found. Comput. Math. 17(1), 35\u201359 (2017)","journal-title":"Found. Comput. Math."},{"key":"2799_CR25","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1016\/j.neucom.2017.07.066","volume":"273","author":"T Lin","year":"2018","unstructured":"Lin, T., Qiao, L., Zhang, T., Feng, J., Zhang, B.: Stochastic primal-dual proximal extra gradient descent for compositely regularized optimization. Neurocomputing 273, 516\u2013525 (2018)","journal-title":"Neurocomputing"},{"issue":"2","key":"2799_CR26","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1109\/TIP.2015.2511584","volume":"25","author":"C Lu","year":"2016","unstructured":"Lu, C., Tang, J., Yan, S., Lin, Z.: Nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm. IEEE Trans. Image Process. 25(2), 829\u2013839 (2016)","journal-title":"IEEE Trans. Image Process."},{"issue":"4","key":"2799_CR27","doi-asserted-by":"publisher","first-page":"2725","DOI":"10.1137\/18M1214342","volume":"29","author":"Z Lu","year":"2019","unstructured":"Lu, Z., Zhou, Z.: Nonmonotone enhanced proximal DC algorithms for a class of structured nonsmooth DC programming. SIAM J. Optim. 29(4), 2725\u20132752 (2019)","journal-title":"SIAM J. Optim."},{"key":"2799_CR28","unstructured":"Mokhtari, A., Ozdaglar, A., Pattathil, S.: A unified analysis of extra-gradient and optimistic gradient methods for saddle point problems: Proximal point approach. In: International Conference on Artificial Intelligence and Statistics, pp. 1497\u20131507. PMLR (2020)"},{"key":"2799_CR29","doi-asserted-by":"crossref","unstructured":"Mordukhovich, B.S.: Generalized Differentiation in Banach Spaces, pp. 3\u2013170. Springer Berlin Heidelberg, Berlin, Heidelberg (2006)","DOI":"10.1007\/3-540-31247-1_1"},{"key":"2799_CR30","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s10957-017-1200-6","volume":"176","author":"TP Nguyen","year":"2018","unstructured":"Nguyen, T.P., Pauwels, E., Richard, E., Suter, B.W.: Extragradient method in optimization: convergence and complexity. J. Optim. Theory Appl. 176, 137\u2013162 (2018)","journal-title":"J. Optim. Theory Appl."},{"key":"2799_CR31","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/BF01299454","volume":"7","author":"PM Pardalos","year":"1994","unstructured":"Pardalos, P.M., Sandstr\u00f6m, M., Zopounidis, C.: On the use of optimization models for portfolio selection: a review and some computational results. Comput. Econ. 7, 227\u2013244 (1994)","journal-title":"Comput. Econ."},{"issue":"6","key":"2799_CR32","doi-asserted-by":"publisher","first-page":"A3649","DOI":"10.1137\/18M123147X","volume":"41","author":"Y Rahimi","year":"2019","unstructured":"Rahimi, Y., Wang, C., Dong, H., Lou, Y.: A scale-invariant approach for sparse signal recovery. SIAM J. Sci. Comput. 41(6), A3649\u2013A3672 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"2799_CR33","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational analysis, vol. 317. Springer Science & Business Media (2009)"},{"issue":"4","key":"2799_CR34","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0377-2217(83)90153-4","volume":"12","author":"S Schaible","year":"1983","unstructured":"Schaible, S., Ibaraki, T.: Fractional programming. Eur. J. Oper. Res. 12(4), 325\u2013338 (1983)","journal-title":"Eur. J. Oper. Res."},{"issue":"10","key":"2799_CR35","doi-asserted-by":"publisher","first-page":"2616","DOI":"10.1109\/TSP.2018.2812733","volume":"66","author":"K Shen","year":"2018","unstructured":"Shen, K., Yu, W.: Fractional programming for communication systems-part i: power control and beamforming. IEEE Trans. Signal Process. 66(10), 2616\u20132630 (2018)","journal-title":"IEEE Trans. Signal Process."},{"issue":"5","key":"2799_CR36","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1111\/rssb.12291","volume":"80","author":"KM Tan","year":"2018","unstructured":"Tan, K.M., Wang, Z., Liu, H., Zhang, T.: Sparse generalized eigenvalue problem: optimal statistical rates via truncated rayleigh flow. J. R. Stat. Soc. Ser. B Stat Methodol. 80(5), 1057\u20131086 (2018)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"key":"2799_CR37","unstructured":"Tran-Dinh, Q.: Sublinear convergence rates of extragradient-type methods: A survey on classical and recent developments. arXiv:2303.17192 (2023)"},{"issue":"1","key":"2799_CR38","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1137\/0329006","volume":"29","author":"P Tseng","year":"1991","unstructured":"Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control. Optim. 29(1), 119\u2013138 (1991)","journal-title":"SIAM J. Control. Optim."},{"issue":"2","key":"2799_CR39","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(2), 431\u2013446 (2000)","journal-title":"SIAM J. Control. Optim."},{"issue":"7","key":"2799_CR40","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.D., Figueiredo, M.A.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57(7), 2479\u20132493 (2009)","journal-title":"IEEE Trans. Signal Process."},{"key":"2799_CR41","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1016\/j.acha.2021.06.006","volume":"55","author":"Y Xu","year":"2021","unstructured":"Xu, Y., Narayan, A., Tran, H., Webster, C.G.: Analysis of the ratio of $$\\ell _1$$ and $$\\ell _2$$ norms in compressed sensing. Appl. Comput. Harmon. Anal. 55, 486\u2013511 (2021)","journal-title":"Appl. Comput. Harmon. Anal."},{"issue":"2","key":"2799_CR42","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1007\/s10915-017-0376-0","volume":"72","author":"Y Xu","year":"2017","unstructured":"Xu, Y., Yin, W.: A globally convergent algorithm for nonconvex optimization based on block coordinate update. J. Sci. Comput. 72(2), 700\u2013734 (2017)","journal-title":"J. Sci. Comput."},{"key":"2799_CR43","unstructured":"Yu, P., Zeng, L., Pong, T.K.: Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-\u0141ojasiewicz property. arXiv:2502.06183 (2025)"},{"issue":"11","key":"2799_CR44","doi-asserted-by":"publisher","first-page":"2844","DOI":"10.1109\/TSP.2017.2673813","volume":"65","author":"A Zappone","year":"2017","unstructured":"Zappone, A., Bj\u00f6rnson, E., Sanguinetti, L., Jorswieck, E.: Globally optimal energy-efficient power control and receiver design in wireless networks. IEEE Trans. Signal Process. 65(11), 2844\u20132859 (2017)","journal-title":"IEEE Trans. Signal Process."},{"key":"2799_CR45","doi-asserted-by":"crossref","unstructured":"Zhang, N., Li, Q.: First-order algorithms for a class of fractional optimization problems. SIAM J. Optim. 32(1), 100\u2013129 (2022)","DOI":"10.1137\/20M1325381"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02799-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02799-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02799-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T09:01:05Z","timestamp":1757322065000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02799-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,31]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["2799"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02799-x","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2025,7,31]]},"assertion":[{"value":"15 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"35"}}