{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T20:41:16Z","timestamp":1771620076779,"version":"3.50.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,7]],"date-time":"2025-01-07T00:00:00Z","timestamp":1736208000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,7]],"date-time":"2025-01-07T00:00:00Z","timestamp":1736208000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"crossref","award":["DMS-1808978"],"award-info":[{"award-number":["DMS-1808978"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"crossref","award":["DMS-2204519"],"award-info":[{"award-number":["DMS-2204519"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s10898-024-01460-7","type":"journal-article","created":{"date-parts":[[2025,1,7]],"date-time":"2025-01-07T02:05:21Z","timestamp":1736215521000},"page":"611-646","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Inexact proximal methods for weakly convex functions"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7156-1380","authenticated-orcid":false,"given":"Pham Duy","family":"Khanh","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6071-6049","authenticated-orcid":false,"given":"Boris S.","family":"Mordukhovich","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1211-8025","authenticated-orcid":false,"given":"Vo Thanh","family":"Phat","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6711-7471","authenticated-orcid":false,"given":"Dat Ba","family":"Tran","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,7]]},"reference":[{"key":"1460_CR1","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1137\/040605266","volume":"16","author":"P-A Absil","year":"2005","unstructured":"Absil, P.-A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16, 531\u2013547 (2005)","journal-title":"SIAM J. Optim."},{"key":"1460_CR2","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolt\u00e9, 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":"1460_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., Bolt\u00e9, 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":"1460_CR4","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 property. Math. Oper. Res. 35, 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"1460_CR5","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1364\/OL.13.000547","volume":"13","author":"GR Ayers","year":"1988","unstructured":"Ayers, G.R., Dainty, J.C.: Iterative blind deconvolution method and its applications. Optim. Lett. 13, 547\u2013549 (1988)","journal-title":"Optim. Lett."},{"key":"1460_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume-title":"First-Order Methods in Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-Order Methods in Optimization. SIAM, Philadelphia (2017)"},{"key":"1460_CR7","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. Imaging Sci. 2, 183\u2013202 (2009)","journal-title":"SIAM J. Imaging Sci."},{"key":"1460_CR8","unstructured":"Bednarchuk, E., Bruccola, G., Serivanti, G., Tran, T.H.: Convergence analysis of an inexact forward-backward algorithm for problems involving weakly convex functions. arXiv:2303.14021 (2023)"},{"key":"1460_CR9","unstructured":"Benedetti, R., Risler, J.-J.: Real algebraic and semialgebraic sets. In: Hermann, \u00c9diteur des Sciences et des Arts, Paris (1990)"},{"key":"1460_CR10","volume-title":"Nonlinear Programming","author":"DP Bertsekas","year":"2016","unstructured":"Bertsekas, D.P.: Nonlinear Programming, 3rd edn. Athena Scientific, Belmont (2016)","edition":"3"},{"key":"1460_CR11","doi-asserted-by":"publisher","first-page":"628","DOI":"10.1007\/s10957-020-01800-z","volume":"188","author":"A Bohm","year":"2021","unstructured":"Bohm, A., Wright, S.J.: Variable smoothing for weakly convex composite functions. J. Optim. Theory Appl. 188, 628\u2013649 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"1460_CR12","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, 1205\u20131223 (2007)","journal-title":"SIAM J. Optim."},{"key":"1460_CR13","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1137\/060670080","volume":"18","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18, 556\u2013572 (2007)","journal-title":"SIAM J. Optim."},{"key":"1460_CR14","doi-asserted-by":"publisher","first-page":"3069","DOI":"10.1137\/19M1254155","volume":"30","author":"S Bonettini","year":"2020","unstructured":"Bonettini, S., Prato, M., Rebegoldi, S.: Convergence of inexact forward\u2013backward algorithms using the forward\u2013backward envelope. SIAM J. Optim. 30, 3069\u20133097 (2020)","journal-title":"SIAM J. Optim."},{"key":"1460_CR15","doi-asserted-by":"publisher","first-page":"1956","DOI":"10.1137\/080738970","volume":"20","author":"J-F Cai","year":"2010","unstructured":"Cai, J.-F., Cand\u00e9s, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956\u20131982 (2010)","journal-title":"SIAM J. Optim."},{"key":"1460_CR16","unstructured":"Chierchia, G., Chouzenoux, E., Combettes, P.L., Pesquet, J.-C.: The proximity operator repository. http:\/\/proximity-operator.net\/index.html, (2016)"},{"key":"1460_CR17","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1007\/s10208-021-09516-w","volume":"22","author":"D Davis","year":"2022","unstructured":"Davis, D., Drusviatskiy, D.: Proximal methods avoid active strict saddles of weakly convex functions. Found. Comput. Math. 22, 561\u2013606 (2022)","journal-title":"Found. Comput. Math."},{"key":"1460_CR18","doi-asserted-by":"crossref","unstructured":"Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. vol. I. Springer, New York, II (2003)","DOI":"10.1007\/b97544"},{"key":"1460_CR19","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1109\/TIP.2010.2072512","volume":"20","author":"J Fadili","year":"2011","unstructured":"Fadili, J., Peyr\u00e9, G.: Total variation projection with first order schemes. IEEE Trans. Image Process. 20, 657\u2013669 (2011)","journal-title":"IEEE Trans. Image Process."},{"key":"1460_CR20","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.-B., Lemar\u00e9chal, C.: Convex Analysis and Minimization Algorithms. vols. I and II, Springer, Berlin (1993)","DOI":"10.1007\/978-3-662-02796-7"},{"key":"1460_CR21","doi-asserted-by":"publisher","first-page":"1894","DOI":"10.1137\/080722059","volume":"19","author":"AD Ioffe","year":"2009","unstructured":"Ioffe, A.D.: An invitation to tame optimization. SIAM J. Optim. 19, 1894\u20131917 (2009)","journal-title":"SIAM J. Optim."},{"key":"1460_CR22","doi-asserted-by":"publisher","first-page":"1811","DOI":"10.1287\/moor.2022.1320","volume":"48","author":"PD Khanh","year":"2023","unstructured":"Khanh, P.D., Mordukhovich, B.S., Phat, V.T.: A generalized Newton method for subgradient systems. Math. Oper. Res. 48, 1811\u20131845 (2023)","journal-title":"Math. Oper. Res."},{"key":"1460_CR23","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10898-022-01248-7","volume":"86","author":"PD Khanh","year":"2023","unstructured":"Khanh, P.D., Mordukhovich, B.S., Phat, V.T., Tran, D.B.: Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials. J. Global Optim. 86, 93\u2013122 (2023)","journal-title":"J. Global Optim."},{"key":"1460_CR24","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s10107-023-01980-2","volume":"205","author":"PD Khanh","year":"2024","unstructured":"Khanh, P.D., Mordukhovich, B.S., Phat, V.T., Tran, D.B.: Globally convergent coderivative-based generalized Newton methods in nonsmooth optimization. Math. Program. 205, 373\u2013429 (2024)","journal-title":"Math. Program."},{"key":"1460_CR25","doi-asserted-by":"publisher","first-page":"2138","DOI":"10.1007\/s10957-023-02319-9","volume":"203","author":"PD Khanh","year":"2024","unstructured":"Khanh, P.D., Mordukhovich, B.S., Tran, D.B.: Inexact reduced gradient methods in smooth nonconvex optimization. J. Optim. Theory Appl. 203, 2138\u20132178 (2024)","journal-title":"J. Optim. Theory Appl."},{"key":"1460_CR26","doi-asserted-by":"publisher","DOI":"10.1080\/10556788.2024.2322700","author":"PD Khanh","year":"2024","unstructured":"Khanh, P.D., Mordukhovich, B.S., Tran, D.B.: A new inexact gradient descent method with applications to nonsmooth convex optimization. Optim. Methods Softw. (2024). https:\/\/doi.org\/10.1080\/10556788.2024.2322700","journal-title":"Optim. Methods Softw."},{"key":"1460_CR27","doi-asserted-by":"crossref","unstructured":"Khanh, P.D., Luong, H.C., Mordukhovich, B.S., Tran, D.B.: Fundamental convergence analysis of sharpness-aware minimization. In: Advances in Neural Information Processing Systems 2024. arXiv:2401.08060 (2024)","DOI":"10.1007\/978-3-031-73420-5_11"},{"key":"1460_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8134-0","volume-title":"A Primer of Real Analytic Functions","author":"SG Krantz","year":"2002","unstructured":"Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, 2nd edn. Birkh\u00e4user, Basel (2002)","edition":"2"},{"key":"1460_CR29","unstructured":"\u0141ojasiewicz, S.: Ensembles semi-analytiques. Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette (Seine-et-Oise), France (1965)"},{"key":"1460_CR30","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet, B.: Regularisation d\u2019indquations variationelles par approximations successives. Rev. Francaise Inf. Rech. Oper. 4, 154\u2013159 (1970)","journal-title":"Rev. Francaise Inf. Rech. Oper."},{"key":"1460_CR31","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, Berlin (2006)"},{"key":"1460_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-92775-6","volume-title":"Variational Analysis and Applications","author":"BS Mordukhovich","year":"2018","unstructured":"Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)"},{"key":"1460_CR33","doi-asserted-by":"crossref","unstructured":"Mordukhovich, B.S.: Second-order variational analysis in optimization, variational stability, and control: theory. In: Algorithms, Applications, Springer, Cham (2024)","DOI":"10.1007\/978-3-031-53476-8"},{"key":"1460_CR34","unstructured":"Mordukhovich, B.S., Nam, N.M.: Convex analysis and beyond. In: Basic Theory, vol.\u00a0I, Springer, Cham"},{"key":"1460_CR35","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s10107-012-0629-5","volume":"140","author":"Yu Nesterov","year":"2013","unstructured":"Nesterov, Yu.: Gradient methods for minimizing composite functions. Math. Program. 140, 125\u2013161 (2013)","journal-title":"Math. Program."},{"key":"1460_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-91578-4","volume-title":"Lectures on Convex Optimization","author":"Yu Nesterov","year":"2018","unstructured":"Nesterov, Yu.: Lectures on Convex Optimization, 2nd edn. Springer, Cham (2018)","edition":"2"},{"key":"1460_CR37","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF01068677","volume":"9","author":"EA Nurminskii","year":"1973","unstructured":"Nurminskii, E.A.: The quasigradient method for the solving of the nonlinear programming problems. Cybernetics 9, 145\u2013150 (1973)","journal-title":"Cybernetics"},{"key":"1460_CR38","doi-asserted-by":"publisher","first-page":"1388","DOI":"10.1137\/130942954","volume":"7","author":"P Ochs","year":"2014","unstructured":"Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7, 1388\u20131419 (2014)","journal-title":"SIAM J. Imaging Sci."},{"key":"1460_CR39","volume-title":"Solution of Equations and Systems of Equations","author":"A Ostrowski","year":"1966","unstructured":"Ostrowski, A.: Solution of Equations and Systems of Equations, 2nd edn. Academic Press, New York (1966)","edition":"2"},{"key":"1460_CR40","doi-asserted-by":"crossref","unstructured":"Patrinos, P., Bemporad, A.: Proximal Newton methods for convex composite optimization. In: IEEE Conference on Decision and Control, pp. 2358\u20132363 (2013)","DOI":"10.1109\/CDC.2013.6760233"},{"key":"1460_CR41","unstructured":"Pelckmans, K., De Brabanter, J., De Moor, B., Suykens, J.A.K.: Convex clustering shrinkage, In: PASCAL Workshop on Statistics and Optimization of Clustering, pp. 1\u20136. London, UK, (2005)"},{"key":"1460_CR42","first-page":"72","volume":"7","author":"BT Polyak","year":"1966","unstructured":"Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. Dokl. 7, 72\u201375 (1966)","journal-title":"Soviet Math. Dokl."},{"key":"1460_CR43","doi-asserted-by":"publisher","first-page":"3990","DOI":"10.1109\/TSP.2010.2049997","volume":"58","author":"M Raginsky","year":"2010","unstructured":"Raginsky, M., Willett, R., Harmany, Z., Marcia, R.: Compressed sensing performance bounds under Poisson noise. IEEE Trans. Signal Process. 58, 3990\u20134002 (2010)","journal-title":"IEEE Trans. Signal Process."},{"key":"1460_CR44","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":"1460_CR45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"key":"1460_CR46","first-page":"1458","volume":"24","author":"M Schmidt","year":"2011","unstructured":"Schmidt, M., Roux, N.L., Bach, F.R.: Convergence rates of inexact proximal-gradient methods for convex optimization. Adv. Neural. Inf. Process. Syst. 24, 1458\u20131466 (2011)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"1460_CR47","unstructured":"Sidford, A.: Introduction to optimization theory, lecture notes. https:\/\/web.stanford.edu\/~sidford\/courses\/19fa_opt_theory\/fa19_opt_theory.html (2019)"},{"key":"1460_CR48","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s10589-017-9912-y","volume":"67","author":"A Themelis","year":"2017","unstructured":"Themelis, A., Stella, L., Patrinos, P.: Forward-backward quasi-Newton methods for nonsmooth optimization problems. Comput. Optim. Appl. 67, 443\u2013487 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"1460_CR49","doi-asserted-by":"publisher","first-page":"2274","DOI":"10.1137\/16M1080240","volume":"28","author":"A Themelis","year":"2018","unstructured":"Themelis, A., Stella, L., Patrinos, P.: Forward-backward envelope for the sum of two nonconvex functions, further properties and nonmonotone linesearch algorithms. SIAM J. Optim. 28, 2274\u20132303 (2018)","journal-title":"SIAM J. Optim."},{"key":"1460_CR50","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. 58, 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc."},{"key":"1460_CR51","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/j.jmaa.2010.03.041","volume":"368","author":"X Wang","year":"2010","unstructured":"Wang, X.: On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368, 293\u2013310 (2010)","journal-title":"J. Math. Anal. Appl."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-024-01460-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-024-01460-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-024-01460-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T06:12:08Z","timestamp":1740550328000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-024-01460-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,7]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["1460"],"URL":"https:\/\/doi.org\/10.1007\/s10898-024-01460-7","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,7]]},"assertion":[{"value":"31 July 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 December 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}