{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T18:51:16Z","timestamp":1784659876041,"version":"3.55.0"},"reference-count":68,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T00:00:00Z","timestamp":1649289600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T00:00:00Z","timestamp":1649289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100014718","name":"Innovative Research Group Project of the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61977038"],"award-info":[{"award-number":["61977038"]}],"id":[{"id":"10.13039\/100014718","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/research","name":"Thousand Talents Plan of Jiangxi Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/research","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Grants Council of Hong Kong","award":["11307319"],"award-info":[{"award-number":["11307319"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2022,5]]},"DOI":"10.1007\/s10915-022-01815-w","type":"journal-article","created":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T19:29:20Z","timestamp":1649359760000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints"],"prefix":"10.1007","volume":"91","author":[{"given":"Jinshan","family":"Zeng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6697-9731","authenticated-orcid":false,"given":"Wotao","family":"Yin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ding-Xuan","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,4,7]]},"reference":[{"issue":"4","key":"1815_CR1","doi-asserted-by":"crossref","first-page":"1286","DOI":"10.1137\/060654797","volume":"18","author":"R Andreani","year":"2007","unstructured":"Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286\u20131309 (2007)","journal-title":"SIAM J. Optim."},{"key":"1815_CR2","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10107-006-0077-1","volume":"111","author":"R Andreani","year":"2008","unstructured":"Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5\u201332 (2008)","journal-title":"Math. Program."},{"issue":"2","key":"1815_CR3","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10589-009-9240-y","volume":"45","author":"R Andreani","year":"2010","unstructured":"Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: Second-order negative-curvature methods for box-constrained and general constrained optimization. Comput. Optim. Appl. 45(2), 209\u2013236 (2010)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"1815_CR4","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1137\/17M1147330","volume":"29","author":"R Andreani","year":"2019","unstructured":"Andreani, R., Fazzio, N., Schuverdt, M.L., Secchin, L.: A sequential optimality condition related to the quasi-normality constraint qualification and its algorithmic consequences. SIAM J. Optim. 29(1), 743\u2013766 (2019)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1815_CR5","doi-asserted-by":"crossref","first-page":"2574","DOI":"10.1137\/17M1125698","volume":"28","author":"R Andreani","year":"2018","unstructured":"Andreani, R., Secchin, L., Silva, P.: Convergence properties of a second order augmented Lagrangian method for mathematical programs with complementarity constraints. SIAM J. Optim. 28(3), 2574\u20132600 (2018)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1815_CR6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10556788.2015.1025401","volume":"32","author":"P Armand","year":"2017","unstructured":"Armand, P., Omheni, R.: A globally and quadratically convergent primal-dual augmented Lagrangian algorithm for equality constrained optimization. Optim. Methods Softw. 32(1), 1\u201321 (2017)","journal-title":"Optim. Methods Softw."},{"key":"1815_CR7","doi-asserted-by":"crossref","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":"1815_CR8","doi-asserted-by":"crossref","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\u2013219 (2013)","journal-title":"Math. Program."},{"key":"1815_CR9","doi-asserted-by":"crossref","unstructured":"Bertsekas, D.P.: In: Convergence rate of penalty and multiplier methods, pp. 260\u2013264. SanDiego, California (1973)","DOI":"10.1109\/CDC.1973.269172"},{"issue":"2","key":"1815_CR10","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1137\/0314017","volume":"14","author":"DP Bertsekas","year":"1976","unstructured":"Bertsekas, D.P.: On penalty and multiplier methods for constrained minimization. SIAM J. Control. Optim. 14(2), 216\u2013235 (1976)","journal-title":"SIAM J. Control. Optim."},{"key":"1815_CR11","volume-title":"Constrained Optimization and Lagrange Multiplier Methods","author":"DP Bertsekas","year":"1982","unstructured":"Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, London (1982)"},{"issue":"1","key":"1815_CR12","first-page":"301","volume":"149","author":"W Bian","year":"2005","unstructured":"Bian, W., Chen, X., Ye, Y.: Complexity analysis of interior point algorithms for non-Lipschitz and nonconvex minimization. Math. Program. 149(1), 301\u2013327 (2005)","journal-title":"Math. Program."},{"key":"1815_CR13","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10589-005-1066-7","volume":"31","author":"EG Birgin","year":"2005","unstructured":"Birgin, E.G., Castillo, R., Martinez, J.M.: Numerical comparison of augmented Lagrangian algorithms for nonconvex problems. Comput. Optim. Appl. 31, 31\u201356 (2005)","journal-title":"Comput. Optim. Appl."},{"key":"1815_CR14","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s10107-009-0264-y","volume":"125","author":"EG Birgin","year":"2010","unstructured":"Birgin, E.G., Floudas, C.A., Martinez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Program. 125, 139\u2013162 (2010)","journal-title":"Math. Program."},{"issue":"6","key":"1815_CR15","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1080\/10556788.2011.556634","volume":"27","author":"EG Birgin","year":"2012","unstructured":"Birgin, E.G., Floudas, C.A., Martinez, J.M.: The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems. Optim. Methods Softw. 27(6), 1001\u20131024 (2012)","journal-title":"Optim. Methods Softw."},{"issue":"1","key":"1815_CR16","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10589-017-9937-2","volume":"69","author":"EG Birgin","year":"2018","unstructured":"Birgin, E.G., Haeser, G., Ramos, A.: Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points. Comput. Optim. Appl. 69(1), 51\u201375 (2018)","journal-title":"Comput. Optim. Appl."},{"key":"1815_CR17","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611973365","volume-title":"Practical Augmented Lagrangian Methods for Constrained Optimization","author":"EG Birgin","year":"2014","unstructured":"Birgin, E.G., Martinez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization, vol. 10. SIAM, Philadelphia (2014)"},{"key":"1815_CR18","volume-title":"Complexity and performance of an augmented Lagrangian algorithm","author":"EG Birgin","year":"2020","unstructured":"Birgin, E.G., Martinez, J.M.: Complexity and performance of an augmented Lagrangian algorithm. Optim, Methods Softw (2020)"},{"key":"1815_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03718-8","volume-title":"Real Algebraic Geometry","author":"J Bochnak","year":"1998","unstructured":"Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry, vol. 36. Springer, Berlin (1998)"},{"issue":"4","key":"1815_CR20","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, A., Lewis, A.: The \u0142ojasiewicz 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":"2","key":"1815_CR21","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1137\/060670080","volume":"18","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clark subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556\u2013572 (2007)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1815_CR22","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1), 459\u2013494 (2014)","journal-title":"Math. Program."},{"key":"1815_CR23","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex Optimization","author":"S Boyd","year":"2004","unstructured":"Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)"},{"key":"1815_CR24","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1137\/S1052623493251463","volume":"6","author":"AR Conn","year":"1996","unstructured":"Conn, A.R., Gould, N.I.M., Startenaer, A., Toint, P.L.: Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints. SIAM J. Optim. 6, 674\u2013703 (1996)","journal-title":"SIAM J. Optim."},{"key":"1815_CR25","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1137\/0728030","volume":"28","author":"AR Conn","year":"1991","unstructured":"Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545\u2013572 (1991)","journal-title":"SIAM J. Numer. Anal."},{"key":"1815_CR26","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719857","volume-title":"Trust-Region Methods","author":"AR Conn","year":"2000","unstructured":"Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)"},{"issue":"1","key":"1815_CR27","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s10107-014-0784-y","volume":"152","author":"FE Curtis","year":"2015","unstructured":"Curtis, F.E., Jiang, H., Robinson, D.P.: An adaptive augmented Lagrangian method for large-scale constrained optimization. Math. Program. 152(1), 201\u2013245 (2015)","journal-title":"Math. Program."},{"issue":"1","key":"1815_CR28","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1137\/18M1178244","volume":"29","author":"D Davis","year":"2019","unstructured":"Davis, D., Drusvyatskiy, D.: Stochastic model-based minimization of weakly convex functions. SIAM J. Optim. 29(1), 207\u2013239 (2019)","journal-title":"SIAM J. Optim."},{"key":"1815_CR29","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1007\/s10915-016-0318-2","volume":"71","author":"W Deng","year":"2017","unstructured":"Deng, W., Lai, M.J., Peng, Z., Yin, W.: Parallel multi-block admm with $$o(1\/k)$$ convergence. J. Sci. Comput. 71, 712\u2013736 (2017)","journal-title":"J. Sci. Comput."},{"key":"1815_CR30","first-page":"1","volume":"26","author":"D Drusvyatskiy","year":"2018","unstructured":"Drusvyatskiy, D.: The proximal point method revisited. SIAG\/OPT Views and News 26, 1\u20138 (2018)","journal-title":"SIAG\/OPT Views and News"},{"key":"1815_CR31","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s10107-018-1311-3","volume":"178","author":"D Drusvyatskiy","year":"2019","unstructured":"Drusvyatskiy, D., Paquette, C.: Efficiency of minimizing compositions of convex functions and smooth maps. Math. Program. 178, 503\u2013558 (2019)","journal-title":"Math. Program."},{"key":"1815_CR32","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348\u20131360 (2001)","journal-title":"J. Am. Stat. Assoc."},{"issue":"2","key":"1815_CR33","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1137\/10081085X","volume":"22","author":"D Fernadez","year":"2012","unstructured":"Fernadez, D., Solodov, M.V.: Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition. SIAM J. Optim. 22(2), 384\u2013407 (2012)","journal-title":"SIAM J. Optim."},{"key":"1815_CR34","unstructured":"Grapiglia, G.N., Yuan, Y.X.: On the complexity of an augmented Lagrangian method for nonconvex optimization. ArXiv e-prints (2019)"},{"key":"1815_CR35","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s10107-018-1290-4","volume":"178","author":"G Haeser","year":"2019","unstructured":"Haeser, G., Liu, H., Ye, Y.: Optimality condition and complexity analysis for linearly-constrained optimization without differentiability on the boundary. Math. Program. 178, 263\u2013299 (2019)","journal-title":"Math. Program."},{"key":"1815_CR36","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s10107-019-01365-4","volume":"176","author":"D Hajinezhad","year":"2019","unstructured":"Hajinezhad, D., Hong, M.: Perturbed proximal primal-dual algorithm for nonconvex nonsmooth optimization. Math. Program. 176, 207\u2013245 (2019)","journal-title":"Math. Program."},{"key":"1815_CR37","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF00927673","volume":"4","author":"MR Hestenes","year":"1969","unstructured":"Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303\u2013320 (1969)","journal-title":"J. Optim. Theory Appl."},{"key":"1815_CR38","unstructured":"Hong, M., Hajinezhad, D., Zhao, M.M.: Prox-pda,: In: The proximal primal-dual algorithm for fast dostributed nonconvex optimization and learning over networks, pp. 1529\u20131538. , Sydney, Australia (2017)"},{"issue":"1","key":"1815_CR39","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10589-018-0034-y","volume":"72","author":"B Jiang","year":"2019","unstructured":"Jiang, B., Lin, T., Ma, S., Zhang, S.: Structured nonconvex and nonsmooth otpmization: algorithms and iteration complexity analysis. Comput. Optim. Appl. 72(1), 115\u2013157 (2019)","journal-title":"Comput. Optim. Appl."},{"key":"1815_CR40","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-8134-0","volume-title":"A Primer of Real Analytic Functions","author":"S Krantz","year":"2002","unstructured":"Krantz, S., Parks, H.R.: A Primer of Real Analytic Functions, 2nd edn. Birkhauser, Basel (2002)","edition":"2"},{"key":"1815_CR41","doi-asserted-by":"crossref","unstructured":"Kurdyka, K.: On gradients of functions definable in o-minimal structures. Annales de l\u2019institut Fourier 48(3), 769\u2013783 (1998)","DOI":"10.5802\/aif.1638"},{"key":"1815_CR42","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1007\/s10208-017-9366-8","volume":"18","author":"G Li","year":"2018","unstructured":"Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka\u2013\u0142ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comput. Math. 18, 1199\u20131232 (2018)","journal-title":"Found. Comput. Math."},{"key":"1815_CR43","unstructured":"\u0141ojasiewicz, S.: Une propri\u00e9t\u00e9 topologique des sous-ensembles analytiques r\u00e9els. In: Les \u00c9quations aux d\u00e9riv\u00e9es partielles. \u00c9ditions du centre National de la Recherche Scientifique, Paris pp. 87\u201389 (1963)"},{"key":"1815_CR44","doi-asserted-by":"crossref","unstructured":"\u0141ojasiewicz, S.: Sur la geometrie semi-et sous-analytique. Annales de l\u2019institut Fourier 43(5), 1575\u20131595 (1993)","DOI":"10.5802\/aif.1384"},{"key":"1815_CR45","volume-title":"Variational Analysis and Generalized Differentiation I: Basic Theory","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer-Verlag, New York (2006)"},{"key":"1815_CR46","doi-asserted-by":"crossref","first-page":"273","DOI":"10.24033\/bsmf.1625","volume":"93","author":"J Moreau","year":"1965","unstructured":"Moreau, J.: Proximit\u00e9 et dualit\u00e9 dans un espace hilbertien. Bull. Soc. Math. France 93, 273\u2013299 (1965)","journal-title":"Bull. Soc. Math. France"},{"key":"1815_CR47","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization. Springer-Verlag, New York (1999)"},{"key":"1815_CR48","unstructured":"Nouiehed, M., Lee, J.D., Razaviyayn, M.: Convergence to second-order stationary for constrained non-convex optimization. ArXiv e-prints (2018)"},{"key":"1815_CR49","doi-asserted-by":"crossref","unstructured":"Nurminskii, E.A.: The quasigradient method for the solving of the nonlinear programming problems. Cybernetics 9, 145\u2013150 (1973)","DOI":"10.1007\/BF01068677"},{"key":"1815_CR50","doi-asserted-by":"crossref","unstructured":"O\u2019Neill, M., Wright, S.J.: A log-barrier newton-cg method for bound constrained optimization with complexity guarantees. IMA J. Numer. Anal. 1\u201338 (2020)","DOI":"10.1093\/imanum\/drz074"},{"key":"1815_CR51","first-page":"42","volume":"13","author":"BT Polyak","year":"1973","unstructured":"Polyak, B.T., Tretyakov, N.V.: The method of penalty bounds for constrained extremum problems. Zh. Vych Mat i Mat. Fiz, 13:34\u201346 = U.S.S.R. Computational Mathematics and Mathmatical. Physics 13, 42\u201358 (1973)","journal-title":"Physics"},{"key":"1815_CR52","unstructured":"Powell, M.J.D.: In: Optimization, R. Fletcher. (ed.) A method for nonlinear constraints in minimization problems, pp. 283\u2013298. Academic Press, London (1969)"},{"key":"1815_CR53","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T.: The multiplier method of hestenes and powell applied to convex programming. J. Optim. Theory Appl. 12, 555\u2013562 (1973)","DOI":"10.1007\/BF00934777"},{"issue":"2","key":"1815_CR54","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1287\/moor.1.2.97","volume":"1","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1(2), 97\u2013116 (1976)","journal-title":"Math. Oper. Res."},{"key":"1815_CR55","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1997","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, New York (1997)"},{"key":"1815_CR56","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-2008-4","volume-title":"Geometry of Subanalytic and Semialgebraic Sets (Progress in Mathematics)","author":"M Shiota","year":"1997","unstructured":"Shiota, M.: Geometry of Subanalytic and Semialgebraic Sets (Progress in Mathematics). Birkhauser, Basel (1997)"},{"key":"1815_CR57","first-page":"525","volume":"9","author":"NY Tretykov","year":"1973","unstructured":"Tretykov, N.Y.: The method of penalty estimates of convex programming. Econ. Math. Methods (Russian) 9, 525\u2013540 (1973)","journal-title":"Econ. Math. Methods (Russian)"},{"key":"1815_CR58","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s10915-018-0757-z","volume":"78","author":"Y Wang","year":"2019","unstructured":"Wang, Y., Yin, W., Zeng, J.: Global convergence of admm in nonconvex nonsmooth optimization. J. Sci. Comput. 78, 29\u201363 (2019)","journal-title":"J. Sci. Comput."},{"key":"1815_CR59","unstructured":"Xie, Y., Wright, S.J.: Complexity of proximal augmented Lagrangian for nonconvex optimalization with nonlinear equality constraints. ArXiv e-prints (2019)"},{"issue":"3","key":"1815_CR60","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1137\/120887795","volume":"6","author":"Y Xu","year":"2013","unstructured":"Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimizaton with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758\u20131789 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"1815_CR61","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-021-09528-6","author":"P Yu","year":"2021","unstructured":"Yu, P., Li, G., Pong, T.: Kurdyka\u2013\u0142ojasiewicz exponent via inf-projection. Found. Comput. Math. (2021). https:\/\/doi.org\/10.1007\/s10208-021-09528-6","journal-title":"Found. Comput. Math."},{"key":"1815_CR62","unstructured":"Zeng, J., Lau, T.T.K., Lin, S.B., Yao, Y.: In: Global convergence of block coordinate descent in deep learning. , Long Beach, California (2019) . (PMLR 97)"},{"issue":"199","key":"1815_CR63","first-page":"1","volume":"22","author":"J Zeng","year":"2021","unstructured":"Zeng, J., Lin, S.B., Yao, Y., Zhou, D.X.: On admm in deep learning: convergence and saturation-avoidance. J. Mach. Learn. Res. 22(199), 1\u201367 (2021)","journal-title":"J. Mach. Learn. Res."},{"issue":"11","key":"1815_CR64","doi-asserted-by":"crossref","first-page":"2834","DOI":"10.1109\/TSP.2018.2818081","volume":"66","author":"J Zeng","year":"2018","unstructured":"Zeng, J., Yin, W.: On nonconvex descentralized gradient descent. IEEE Trans. Signal Process. 66(11), 2834\u20132848 (2018)","journal-title":"IEEE Trans. Signal Process."},{"issue":"2","key":"1815_CR65","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"CH Zhang","year":"2010","unstructured":"Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"key":"1815_CR66","unstructured":"Zhang, J., Luo, Z.Q.: A global dual error bound and its application to the analysis of linearly constrained nonconvex optimization. ArXiv e-prints (2020)"},{"issue":"3","key":"1815_CR67","doi-asserted-by":"crossref","first-page":"2272","DOI":"10.1137\/19M1242276","volume":"30","author":"J Zhang","year":"2020","unstructured":"Zhang, J., Luo, Z.Q.: A proximal alternating direction method of multiplier for linearly constrained nonconvex minimization. SIAM J. Optim. 30(3), 2272\u20132302 (2020)","journal-title":"SIAM J. Optim."},{"key":"1815_CR68","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/j.acha.2019.06.004","volume":"48","author":"DX Zhou","year":"2020","unstructured":"Zhou, D.X.: Universality of deep convolutional neural networks. Appl. Comput. Harmonic Anal. 48, 787\u2013794 (2020)","journal-title":"Appl. Comput. Harmonic Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01815-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-022-01815-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-022-01815-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:17:52Z","timestamp":1651065472000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-022-01815-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,7]]},"references-count":68,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,5]]}},"alternative-id":["1815"],"URL":"https:\/\/doi.org\/10.1007\/s10915-022-01815-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,7]]},"assertion":[{"value":"20 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 December 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 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 declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"61"}}