{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T03:22:25Z","timestamp":1780370545662,"version":"3.54.1"},"reference-count":88,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T00:00:00Z","timestamp":1757635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["124041500008-1"],"award-info":[{"award-number":["124041500008-1"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s10898-025-01534-0","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T01:35:54Z","timestamp":1757640954000},"page":"359-411","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence analysis of primal-dual augmented Lagrangian methods and duality theory"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0677-0797","authenticated-orcid":false,"given":"M.V.","family":"Dolgopolik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"1534_CR1","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, New York (1982)"},{"key":"1534_CR2","volume-title":"Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems","author":"M Fortin","year":"1983","unstructured":"Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam (1983)"},{"key":"1534_CR3","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718614","volume-title":"Lagrange Multiplier Approach to Variational Problems and Applications","author":"K Ito","year":"2008","unstructured":"Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)"},{"key":"1534_CR4","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. SIAM, Philadelphia (2014)"},{"key":"1534_CR5","doi-asserted-by":"crossref","first-page":"2579","DOI":"10.1007\/s11831-022-09878-6","volume":"30","author":"E Burman","year":"2023","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: The augmented Lagrangian method as a framework for stabilised methods in computational mechanics. Arch. Comput. Methods Eng. 30, 2579\u20132604 (2023)","journal-title":"Arch. Comput. Methods Eng."},{"key":"1534_CR6","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1287\/moor.28.3.533.16395","volume":"28","author":"XX Huang","year":"2003","unstructured":"Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533\u2013552 (2003)","journal-title":"Math. Oper. Res."},{"key":"1534_CR7","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s10898-004-5695-7","volume":"31","author":"XX Huang","year":"2005","unstructured":"Huang, X.X., Yang, X.Q.: Further study on augmented Lagrangian duality theory. J. Glob. Optim. 31, 193\u2013210 (2005)","journal-title":"J. Glob. Optim."},{"key":"1534_CR8","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1137\/050647621","volume":"18","author":"RS Burachik","year":"2007","unstructured":"Burachik, R.S., Rubinov, A.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413\u2013436 (2007)","journal-title":"SIAM J. Optim."},{"key":"1534_CR9","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1080\/02331930802355416","volume":"57","author":"Q Liu","year":"2008","unstructured":"Liu, Q., Yang, X.: Zero duality and saddle points of a class of augmented Lagrangian functions in constrained non-convex optimization. Optim. 57, 655\u2013667 (2008)","journal-title":"Optim."},{"key":"1534_CR10","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10957-008-9455-6","volume":"140","author":"YY Zhou","year":"2009","unstructured":"Zhou, Y.Y., Yang, X.Q.: Duality and penalization in optimization via an augmented Lagrangian function with applications. J. Optim. Theory Appl. 140, 171\u2013188 (2009)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR11","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1287\/moor.2013.0591","volume":"38","author":"CY Wang","year":"2012","unstructured":"Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear augmented Lagrangian and duality theory. Math. Oper. Res. 38, 740\u2013760 (2012)","journal-title":"Math. Oper. Res."},{"key":"1534_CR12","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s00186-020-00708-8","volume":"92","author":"GD Yalcin","year":"2020","unstructured":"Yalcin, G.D., Kasimbeyli, R.: On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization. Math. Methods Oper. Res. 92, 199\u2013228 (2020)","journal-title":"Math. Methods Oper. Res."},{"key":"1534_CR13","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s10957-010-9711-4","volume":"147","author":"RS Burachik","year":"2010","unstructured":"Burachik, R.S., Iusem, A.N., Melo, J.G.: Duality and exact penalization for general augmented Lagrangians. J. Optim. Theory Appl. 147, 125\u2013140 (2010)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR14","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10898-011-9688-z","volume":"52","author":"YY Zhou","year":"2012","unstructured":"Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian functions for constrained optimization problems. J. Glob. Optim. 52, 95\u2013108 (2012)","journal-title":"J. Glob. Optim."},{"key":"1534_CR15","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s10898-011-9784-0","volume":"54","author":"J Zhou","year":"2012","unstructured":"Zhou, J., Xiu, N., Wang, C.: Saddle point and exact penalty representation for generalized proximal Lagrangians. J. Glob. Optim. 54, 669\u2013687 (2012)","journal-title":"J. Glob. Optim."},{"key":"1534_CR16","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1080\/02331934.2013.830117","volume":"64","author":"RS Burachik","year":"2015","unstructured":"Burachik, R.S., Iusem, A.N., Melo, J.G.: The exact penalty map for nonsmooth and nonconvex optimization. Optim. 64, 717\u2013738 (2015)","journal-title":"Optim."},{"key":"1534_CR17","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s10107-016-1012-8","volume":"161","author":"MJ Feizollahi","year":"2017","unstructured":"Feizollahi, M.J., Ahmed, S., Sun, A.: Exact augmented Lagrangian duality for mixed integer linear programming. Math. Program. 161, 365\u2013387 (2017)","journal-title":"Math. Program."},{"key":"1534_CR18","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1137\/030602770","volume":"15","author":"XL Sun","year":"2005","unstructured":"Sun, X.L., Li, D., McKinnon, K.I.M.: On saddle points of augmented Lagrangians for constrained nonconvex optimization. SIAM J. Optim. 15, 1128\u20131146 (2005)","journal-title":"SIAM J. Optim."},{"key":"1534_CR19","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s10898-011-9731-0","volume":"53","author":"HX Wu","year":"2012","unstructured":"Wu, H.X., Luo, H.Z.: Saddle points of general augmented Lagrangians for constrained nonconvex optimization. J. Glob. Optim. 53, 683\u2013697 (2012)","journal-title":"J. Glob. Optim."},{"key":"1534_CR20","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1007\/s10898-013-0093-7","volume":"59","author":"HX Wu","year":"2014","unstructured":"Wu, H.X., Luo, H.Z., Yang, J.F.: Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming. J. Glob. Optim. 59, 695\u2013727 (2014)","journal-title":"J. Glob. Optim."},{"key":"1534_CR21","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s10898-014-0252-5","volume":"62","author":"J Zhou","year":"2015","unstructured":"Zhou, J., Chen, J.S.: On the existence of saddle points for nonlinear second-order cone programming problems. J. Glob. Optim. 62, 459\u2013480 (2015)","journal-title":"J. Glob. Optim."},{"key":"1534_CR22","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s10957-014-0634-3","volume":"165","author":"H Luo","year":"2015","unstructured":"Luo, H., Wu, H., Liu, J.: On saddle points in semidefinite optimization via separation scheme. J. Optim. Theory Appl. 165, 113\u2013150 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR23","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s10898-016-0456-y","volume":"68","author":"C Wang","year":"2017","unstructured":"Wang, C., Liu, Q., Qu, B.: Global saddle points of nonlinear augmented Lagrangian functions. J. Glob. Optim. 68, 125\u2013146 (2017)","journal-title":"J. Glob. Optim."},{"key":"1534_CR24","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10898-017-0603-0","volume":"71","author":"MV Dolgopolik","year":"2018","unstructured":"Dolgopolik, M.V.: Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property. J. Glob. Optim. 71, 237\u2013296 (2018)","journal-title":"J. Glob. Optim."},{"key":"1534_CR25","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1287\/moor.1040.0103","volume":"29","author":"A Shapiro","year":"2004","unstructured":"Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479\u2013491 (2004)","journal-title":"Math. Oper. Res."},{"key":"1534_CR26","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s10107-007-0115-7","volume":"116","author":"JJ R\u00fcckmann","year":"2009","unstructured":"R\u00fcckmann, J.J., Shapiro, A.: Augmented Lagrangians in semi-infinite programming. Math. Program. 116, 499\u2013512 (2009)","journal-title":"Math. Program."},{"key":"1534_CR27","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10898-013-0046-1","volume":"58","author":"YY Zhou","year":"2014","unstructured":"Zhou, Y.Y., Zhou, J.C., Yang, X.Q.: Existence of augmented Lagrange multipliers for cone constrained optimization problems. J. Glob. Optim. 58, 243\u2013260 (2014)","journal-title":"J. Glob. Optim."},{"key":"1534_CR28","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/s10957-017-1091-6","volume":"173","author":"RS Burachik","year":"2017","unstructured":"Burachik, R.S., Yang, X.Q., Zhou, Y.Y.: Existence of augmented Lagrange multipliers for semi-infinite programming problems. J. Optim. Theory Appl. 173, 471\u2013503 (2017)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR29","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10107-017-1122-y","volume":"166","author":"MV Dolgopolik","year":"2017","unstructured":"Dolgopolik, M.V.: Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle. Math. Program. 166, 297\u2013326 (2017)","journal-title":"Math. Program."},{"key":"1534_CR30","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s10898-005-3270-5","volume":"34","author":"RS Burachik","year":"2006","unstructured":"Burachik, R.S., Gasimov, R.N., Ismayilova, N.A., Kaya, C.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Glob. Optim. 34, 55\u201378 (2006)","journal-title":"J. Glob. Optim."},{"key":"1534_CR31","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s10898-009-9429-8","volume":"46","author":"RS Burachik","year":"2010","unstructured":"Burachik, R.S., Iusem, A.N., Melo, J.G.: A primal dual modified subgradient algorithm with sharp Lagrangian. J. Glob. Optim. 46, 55\u201378 (2010)","journal-title":"J. Glob. Optim."},{"key":"1534_CR32","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1080\/10556788.2018.1496431","volume":"34","author":"AM Bagirov","year":"2019","unstructured":"Bagirov, A.M., Ozturk, G., Kasimbeyli, R.: A shapr augmented Lagrangian-based method for constrained non-convex optimization. Optim. Methods Softw. 34, 462\u2013488 (2019)","journal-title":"Optim. Methods Softw."},{"key":"1534_CR33","unstructured":"Burachik, R.S., Liu, X.: An inexact deflected subgradient algorithm in infinite dimensional spaces. arXiv: 2302.02072 (2023)"},{"key":"1534_CR34","unstructured":"Burachik, R.S., Kaya, C.Y., Liu, X.: A primal-dual algorithm as applied to optimal control problems. arXiv: 2304.03465 (2023)"},{"key":"1534_CR35","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1080\/02331934.2010.527971","volume":"60","author":"RS Burachik","year":"2011","unstructured":"Burachik, R.S.: On primal convergence for augmented Lagrangian duality. Optim. 60, 979\u2013990 (2011)","journal-title":"Optim."},{"key":"1534_CR36","doi-asserted-by":"crossref","unstructured":"Liu, Y.J., Zhang, L.W.: Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal.: Theory, Methods, Appl. 67, 1359\u2013 1373 ( 2007)","DOI":"10.1016\/j.na.2006.07.022"},{"key":"1534_CR37","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s10957-008-9390-6","volume":"139","author":"YJ Liu","year":"2008","unstructured":"Liu, Y.J., Zhang, L.W.: Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. J. Optim. Theory Appl. 139, 557\u2013575 (2008)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR38","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10107-007-0105-9","volume":"114","author":"D Sun","year":"2008","unstructured":"Sun, D., Sun, J., Zhang, L.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349\u2013391 (2008)","journal-title":"Math. Program."},{"key":"1534_CR39","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1007\/s10898-011-9779-x","volume":"54","author":"HZ Luo","year":"2012","unstructured":"Luo, H.Z., Wu, H.X., Chen, G.T.: On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. J. Glob. Optim. 54, 599\u2013618 (2012)","journal-title":"J. Glob. Optim."},{"key":"1534_CR40","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s10589-013-9568-1","volume":"56","author":"H Wu","year":"2013","unstructured":"Wu, H., Luo, H., Ding, X., Chen, G.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming. Comput. Optim. Appl. 56, 531\u2013558 (2013)","journal-title":"Comput. Optim. Appl."},{"key":"1534_CR41","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10589-009-9279-9","volume":"49","author":"L Zhang","year":"2011","unstructured":"Zhang, L., Gu, J., Xiao, X.: A class of nonlinear Lagrangians for nonconvex second order cone programming. Comput. Optim. Appl. 49, 61\u201399 (2011)","journal-title":"Comput. Optim. Appl."},{"key":"1534_CR42","first-page":"149","volume":"15","author":"Y Li","year":"2009","unstructured":"Li, Y., Zhang, L.: A new nonlinear Lagrangian method for nonconvex semidefinite programming. J. Appl. Anal. 15, 149\u2013172 (2009)","journal-title":"J. Appl. Anal."},{"key":"1534_CR43","first-page":"1209","volume":"18","author":"HZ Luo","year":"2007","unstructured":"Luo, H.Z., Sun, X.L., Li, D.: On the convergence of augmented Lagrangian methods for constrained global optimization. SIAM J. Optim. 18, 1209\u20131230 (2007)","journal-title":"SIAM J. Optim."},{"key":"1534_CR44","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1080\/10556780802124648","volume":"23","author":"H Luo","year":"2008","unstructured":"Luo, H., Sun, X., Wu, H.: Convergence properties of augmented Lagrangian methods for constrained global optimization. Optim. Methods Softw. 23, 763\u2013778 (2008)","journal-title":"Optim. Methods Softw."},{"key":"1534_CR45","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s10898-008-9347-1","volume":"44","author":"CY Wang","year":"2009","unstructured":"Wang, C.Y., Li, D.: Unified theory of augmented Lagrangian methods for constrained global optimization. J. Glob. Optim. 44, 433\u2013458 (2009)","journal-title":"J. Glob. Optim."},{"key":"1534_CR46","doi-asserted-by":"crossref","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":"1534_CR47","first-page":"61","volume":"3","author":"MV Dolgopolik","year":"2021","unstructured":"Dolgopolik, M.V.: Minimax exactness and global saddle points of nonlinear augmented Lagrangians. J. Appl. Numer. Optim. 3, 61\u201383 (2021)","journal-title":"J. Appl. Numer. Optim."},{"key":"1534_CR48","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1534_CR49","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1020261001771","volume":"24","author":"RN Gasimov","year":"2002","unstructured":"Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Glob. Optim. 24, 187\u2013203 (2002)","journal-title":"J. Glob. Optim."},{"key":"1534_CR50","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":"1534_CR51","first-page":"283","volume-title":"Optimization","author":"MJD Powell","year":"1969","unstructured":"Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283\u2013298. Academic Press, London (1969)"},{"key":"1534_CR52","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1137\/0313045","volume":"12","author":"OL Mangasarian","year":"1975","unstructured":"Mangasarian, O.L.: Unconstrained Lagrangians in nonlinear programming. SIAM J. Control 12, 772\u2013791 (1975)","journal-title":"SIAM J. Control"},{"key":"1534_CR53","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9172-0","volume-title":"Lagrange-type Functions in Constrained Nonconvex Optimization","author":"AM Rubinov","year":"2003","unstructured":"Rubinov, A.M., Yang, X.Q.: Lagrange-type Functions in Constrained Nonconvex Optimization. Kluwer Academic Publishers, Boston (2003)"},{"key":"1534_CR54","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/BF00934777","volume":"12","author":"RT Rockafellar","year":"1973","unstructured":"Rockafellar, R.T.: The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl. 12, 555\u2013562 (1973)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR55","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1137\/0312021","volume":"12","author":"RT Rockafellar","year":"1974","unstructured":"Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control. Optim. 12, 268\u2013285 (1974)","journal-title":"SIAM J. Control. Optim."},{"key":"1534_CR56","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/1035044","volume":"35","author":"RT Rockafellar","year":"1993","unstructured":"Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183\u2013238 (1993)","journal-title":"SIAM Rev."},{"key":"1534_CR57","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/BF02192031","volume":"88","author":"KC Kiwiel","year":"1996","unstructured":"Kiwiel, K.C.: On the twice differentiable cubic augmented Lagrangian. J. Optim. Theory Appl. 88, 233\u2013236 (1996)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR58","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01580598","volume":"60","author":"P Tseng","year":"1993","unstructured":"Tseng, P., Bertsekas, D.P.: On the convergence of the exponential multiplier method for convex programming. Math. Program. 60, 1\u201319 (1993)","journal-title":"Math. Program."},{"key":"1534_CR59","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1023\/A:1010938423538","volume":"101","author":"RA Polyak","year":"2001","unstructured":"Polyak, R.A.: Log-sigmoid multipliers method in constrained optimization. Ann. Oper. Res. 101, 427\u2013460 (2001)","journal-title":"Ann. Oper. Res."},{"key":"1534_CR60","doi-asserted-by":"crossref","unstructured":"Polyak, R.A.: Nonlinear rescaling vs. smoothing technique in convex optimization. Math. Program. 92, 197\u2013 235 ( 2002)","DOI":"10.1007\/s101070100293"},{"key":"1534_CR61","doi-asserted-by":"crossref","first-page":"5611","DOI":"10.1016\/S0362-546X(01)00663-0","volume":"47","author":"D Li","year":"2001","unstructured":"Li, D., Sun, X.L.: Convexification and existence of saddle point in a pth-power reformulation for nonconvex constrained optimization. Nonlinear Anal. 47, 5611\u20135622 (2001)","journal-title":"Nonlinear Anal."},{"key":"1534_CR62","first-page":"1231","volume":"61","author":"HX Wu","year":"2012","unstructured":"Wu, H.X., Luo, H.Z.: A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization. Optim. 61, 1231\u20131245 (2012)","journal-title":"Optim."},{"key":"1534_CR63","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1111\/1475-3995.t01-1-00330","volume":"8","author":"AE Xavier","year":"2001","unstructured":"Xavier, A.E.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. Int. Trans. Oper. Res. 8, 659\u2013671 (2001)","journal-title":"Int. Trans. Oper. Res."},{"key":"1534_CR64","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1051\/ro\/2023153","volume":"57","author":"LM Ramirez","year":"2023","unstructured":"Ramirez, L.M., Maculan, N., Xavier, A.E., Xavier, V.L.: Dislocation hyperbolic augmented Lagrangian algorithm for nonconvex optimization. RAIRO Oper. Res. 57, 2941\u20132950 (2023)","journal-title":"RAIRO Oper. Res."},{"key":"1534_CR65","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF01586050","volume":"54","author":"R Polyak","year":"1992","unstructured":"Polyak, R.: Modified barrier functions: theory and methods. Math. Program. 54, 177\u2013222 (1992)","journal-title":"Math. Program."},{"key":"1534_CR66","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s12190-011-0504-1","volume":"38","author":"S He","year":"2012","unstructured":"He, S., Wu, L., Meng, H.: A nonlinear Lagrangian for constrained optimization problems. J. Appl. Math. Comput. 38, 669\u2013685 (2012)","journal-title":"J. Appl. Math. Comput."},{"key":"1534_CR67","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.A., Martinez, J.M.: Numerical comparison of augmented Lagragian algorithms for nonconvex problems. Comput. Optim. Appl. 31, 31\u201355 (2005)","journal-title":"Comput. Optim. Appl."},{"key":"1534_CR68","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1137\/S1052623400380365","volume":"12","author":"M Fukushima","year":"2001","unstructured":"Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436\u2013460 (2001)","journal-title":"SIAM J. Optim."},{"key":"1534_CR69","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1287\/moor.1070.0300","volume":"33","author":"D Sun","year":"2008","unstructured":"Sun, D., Sun, J.: L\u00f6wner\u2019s operator and spectral functions in euclidean jordan algebras. Math. Oper. Res. 33, 421\u2013445 (2008)","journal-title":"Math. Oper. Res."},{"key":"1534_CR70","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/0-387-24276-7_32","volume-title":"Variational Analysis and Applications","author":"XX Huang","year":"2005","unstructured":"Huang, X.X., Yang, X.Q., Teo, K.L.: Augmented Lagrangian and nonlinear semidefinite programs. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications, pp. 513\u2013529. Springer, Boston (2005)"},{"key":"1534_CR71","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s10957-006-9078-8","volume":"129","author":"J Sun","year":"2006","unstructured":"Sun, J., Zhang, L.W., Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 129, 437\u2013456 (2006)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR72","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1137\/080718206","volume":"20","author":"XY Zhao","year":"2010","unstructured":"Zhao, X.Y., Sun, D., Toh, K.-C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737\u20131765 (2010)","journal-title":"SIAM J. Optim."},{"key":"1534_CR73","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s12532-010-0017-1","volume":"2","author":"Z Wen","year":"2010","unstructured":"Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203\u2013230 (2010)","journal-title":"Math. Program. Comput."},{"key":"1534_CR74","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/naco.2011.1.1","volume":"1","author":"J Sun","year":"2011","unstructured":"Sun, J.: On methods for solving nonlinear semidefinite optimization problems. Numer. Algebra Control Optim. 1, 1\u201314 (2011)","journal-title":"Numer. Algebra Control Optim."},{"key":"1534_CR75","first-page":"24","volume":"58","author":"H Yamashita","year":"2015","unstructured":"Yamashita, H., Yabe, H.: A survey of numerical methods for nonlinear semidefinite programming. J. Oper. Res. Soc. Jpn. 58, 24\u201360 (2015)","journal-title":"J. Oper. Res. Soc. Jpn."},{"key":"1534_CR76","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"RA Horn","year":"1991","unstructured":"Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"1534_CR77","unstructured":"Stingl, M.: On the solution of nonlinear semidefinite programs by augmented Lagrangian methods. PhD thesis, Institute of Applied Mathematics II, Friedrech-Alexander University of Erlangen-Nuremberg, Erlangen, Germany (2006)"},{"key":"1534_CR78","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1080\/10556780701223970","volume":"22","author":"D Noll","year":"2007","unstructured":"Noll, D.: Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optim. Methods Softw. 22, 777\u2013802 (2007)","journal-title":"Optim. Methods Softw."},{"key":"1534_CR79","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1080\/02331934.2013.848861","volume":"63","author":"L Zhang","year":"2014","unstructured":"Zhang, L., Li, Y., Wu, J.: Nonlinear rescaling Lagrangians for nonconvex semidefinite programming. Optim. 63, 899\u2013920 (2014)","journal-title":"Optim."},{"key":"1534_CR80","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1287\/moor.27.4.775.295","volume":"27","author":"AM Rubinov","year":"2002","unstructured":"Rubinov, A.M., Huang, X.X., Yang, X.Q.: The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res. 27, 775\u2013791 (2002)","journal-title":"Math. Oper. Res."},{"key":"1534_CR81","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1394-9","volume-title":"Perturbation Analysis of Optimization Problems","author":"JF Bonnans","year":"2000","unstructured":"Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)"},{"key":"1534_CR82","first-page":"11","volume":"8","author":"AN Iusem","year":"1999","unstructured":"Iusem, A.N.: Augmented Lagrangian methods and proximal point methods for convex optimization. Investigaci\u00f3n Operativa 8, 11\u201349 (1999)","journal-title":"Investigaci\u00f3n Operativa"},{"key":"1534_CR83","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1007\/s10589-011-9396-0","volume":"51","author":"EG Birgin","year":"2012","unstructured":"Birgin, E.G., Martinez, J.M.: Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization. Comput. Optim. Appl. 51, 941\u2013965 (2012)","journal-title":"Comput. Optim. Appl."},{"key":"1534_CR84","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s10107-021-01703-5","volume":"196","author":"MCW Oliveira","year":"2022","unstructured":"Oliveira, M.C.W., Sagastiz\u00e1bal, C.: Revisiting augmented Lagrangian duals. Math. Program. 196, 235\u2013277 (2022)","journal-title":"Math. Program."},{"key":"1534_CR85","doi-asserted-by":"crossref","first-page":"381","DOI":"10.3934\/jimo.2007.3.381","volume":"3","author":"RS Burachik","year":"2007","unstructured":"Burachik, R.S., Kaya, C.Y.: An update rule and a convergence result for a penalty function method. J. Ind. Manag. Optim. 3, 381\u2013398 (2007)","journal-title":"J. Ind. Manag. Optim."},{"key":"1534_CR86","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10589-008-9168-7","volume":"45","author":"RS Burachik","year":"2008","unstructured":"Burachik, R.S., Kaya, C.Y., Mammadov, M.: An inexact modified subgradient algorithm for nonconvex optimization. Comput. Optim. Appl. 45, 1\u201334 (2008)","journal-title":"Comput. Optim. Appl."},{"key":"1534_CR87","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1007\/s10957-012-0158-7","volume":"157","author":"RS Burachik","year":"2013","unstructured":"Burachik, R.S., Iusem, A.N., Melo, J.G.: An inexact modified subgradient algorithm for primal-dual problems via augmented Lagrangians. J. Optim. Theory Appl. 157, 108\u2013131 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1534_CR88","doi-asserted-by":"crossref","unstructured":"Conn, A., N. I. M. Gould, Toint, P.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545\u2013 572 ( 1991)","DOI":"10.1137\/0728030"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01534-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-025-01534-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01534-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T09:50:00Z","timestamp":1761904200000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-025-01534-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,12]]},"references-count":88,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1534"],"URL":"https:\/\/doi.org\/10.1007\/s10898-025-01534-0","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,12]]},"assertion":[{"value":"20 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 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 author has no conflicts of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}]}}