{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T23:39:23Z","timestamp":1769729963711,"version":"3.49.0"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G081222N"],"award-info":[{"award-number":["G081222N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G033822N"],"award-info":[{"award-number":["G033822N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G0A0920N"],"award-info":[{"award-number":["G0A0920N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["C14\/24\/103"],"award-info":[{"award-number":["C14\/24\/103"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s10589-025-00730-8","type":"journal-article","created":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T08:26:39Z","timestamp":1758529599000},"page":"617-649","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global convergence analysis of the power proximal point and augmented Lagrangian method"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6828-1385","authenticated-orcid":false,"given":"Konstantinos A.","family":"Oikonomidis","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Bodard","sequence":"additional","affiliation":[]},{"given":"Emanuel","family":"Laude","sequence":"additional","affiliation":[]},{"given":"Panagiotis","family":"Patrinos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,22]]},"reference":[{"key":"730_CR1","doi-asserted-by":"crossref","unstructured":"Ahookhosh, M.,\u00a0Nesterov, Y.: High-order methods beyond the classical complexity bounds, II: inexact high-order proximal-point methods with segment search (2024). URL http:\/\/arxiv.org\/abs\/2109.12303","DOI":"10.1007\/s10107-024-02067-2"},{"issue":"1","key":"730_CR2","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s10107-023-02041-4","volume":"208","author":"M Ahookhosh","year":"2024","unstructured":"Ahookhosh, M., Nesterov, Y.: High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods. Math. Program. 208(1), 365\u2013407 (2024)","journal-title":"Math. Program."},{"issue":"1","key":"730_CR3","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/BF01681332","volume":"9","author":"DP Bertsekas","year":"1975","unstructured":"Bertsekas, D.P.: Necessary and sufficient conditions for a penalty method to be exact. Math. Program. 9(1), 87\u201399 (1975)","journal-title":"Math. Program."},{"key":"730_CR4","volume-title":"Constrained Optimization and Lagrange Multiplier Methods","author":"DP Bertsekas","year":"2014","unstructured":"Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (2014)"},{"key":"730_CR5","unstructured":"Bodard, A.,\u00a0Oikonomidis, K.,\u00a0Laude, E.,\u00a0Patrinos, P.: The inexact power augmented lagrangian method for constrained nonconvex optimization, (2024). URL https:\/\/arxiv.org\/abs\/2410.20153"},{"key":"730_CR6","volume-title":"Perturbation Analysis of Optimization Problems","author":"JF Bonnans","year":"2013","unstructured":"Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2013)"},{"issue":"4","key":"730_CR7","doi-asserted-by":"publisher","first-page":"1324","DOI":"10.1287\/moor.2020.1099","volume":"46","author":"JV Burke","year":"2021","unstructured":"Burke, J.V., Tim, H., Nguyen, Q.V.: A study of convex convex-composite functions via infimal convolution with applications. Math. Oper. Res. 46(4), 1324\u20131348 (2021)","journal-title":"Math. Oper. Res."},{"issue":"5","key":"730_CR8","doi-asserted-by":"publisher","first-page":"1190","DOI":"10.1137\/0916069","volume":"16","author":"RH Byrd","year":"1995","unstructured":"Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190\u20131208 (1995)","journal-title":"SIAM J. Sci. Comput."},{"key":"730_CR9","unstructured":"Doikov, N., Nesterov, Y.E.: Inexact tensor methods with dynamic accuracies. In ICML, pages 2577\u20132586 (2020)"},{"key":"730_CR10","doi-asserted-by":"crossref","unstructured":"Bourkhissi, L.El.,\u00a0Necoara, I.: Convergence analysis of linearized $$\\ell _q$$ penalty methods for nonconvex optimization with nonlinear equality constraints (2025). https:\/\/arxiv.org\/abs\/2503.08522","DOI":"10.1007\/s10589-025-00737-1"},{"issue":"2","key":"730_CR11","first-page":"1","volume":"18","author":"B Grimmer","year":"2023","unstructured":"Grimmer, B.: On optimal universal first-order methods for minimizing heterogeneous sums. Optim. Lett. 18(2), 1\u201319 (2023)","journal-title":"Optim. Lett."},{"key":"730_CR12","doi-asserted-by":"crossref","unstructured":"Hermans, B.,\u00a0Themelis, A.,\u00a0Patrinos, P.: QPALM: A proximal augmented lagrangian method for nonconvex quadratic programs (2022). http:\/\/arxiv.org\/abs\/2010.02653","DOI":"10.1007\/s12532-022-00218-0"},{"issue":"5","key":"730_CR13","doi-asserted-by":"publisher","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(5), 303\u2013320 (1969)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"730_CR14","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1007\/s10589-023-00476-1","volume":"85","author":"AF Izmailov","year":"2023","unstructured":"Izmailov, A.F., Solodov, M.V.: Convergence rate estimates for penalty methods revisited. Comput. Optim. Appl. 85(3), 973\u2013992 (2023)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"730_CR15","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/20M136147X","volume":"33","author":"W Kong","year":"2023","unstructured":"Kong, W., Melo, J.G., Monteiro, R.D.: Iteration complexity of an inner accelerated inexact proximal augmented lagrangian method based on the classical lagrangian function. SIAM J. Optim. 33(1), 181\u2013210 (2023)","journal-title":"SIAM J. Optim."},{"key":"730_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-025-02204-5","author":"E Laude","year":"2025","unstructured":"Laude, E., Patrinos, P.: Anisotropic proximal gradient. Math. Program. (2025). https:\/\/doi.org\/10.1007\/s10107-025-02204-5","journal-title":"Math. Program."},{"issue":"4","key":"730_CR17","doi-asserted-by":"publisher","first-page":"2721","DOI":"10.1137\/21M1465913","volume":"33","author":"E Laude","year":"2023","unstructured":"Laude, E., Themelis, A., Patrinos, P.: Dualities for non-euclidean smoothness and strong convexity under the light of generalized conjugacy. SIAM J. Optim. 33(4), 2721\u20132749 (2023)","journal-title":"SIAM J. Optim."},{"key":"730_CR18","doi-asserted-by":"crossref","unstructured":"Li, T.,\u00a0Lan, G.: A simple uniformly optimal method without line search for convex optimization (2024). https:\/\/arxiv.org\/abs\/2310.10082","DOI":"10.1007\/s10107-025-02250-z"},{"issue":"2","key":"730_CR19","doi-asserted-by":"publisher","first-page":"632","DOI":"10.1287\/moor.2018.0939","volume":"44","author":"Y-F Liu","year":"2019","unstructured":"Liu, Y.-F., Liu, X., Ma, S.: On the nonergodic convergence rate of an inexact augmented lagrangian framework for composite convex programming. Math. Oper. Res. 44(2), 632\u2013650 (2019)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"730_CR20","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1137\/0322019","volume":"22","author":"FJ Luque","year":"1984","unstructured":"Luque, F.J.: Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control. Optim. 22(2), 277\u2013293 (1984)","journal-title":"SIAM J. Control. Optim."},{"key":"730_CR21","unstructured":"Luque, J.: Nonlinear proximal point algorithms. PhD thesis, Massachusetts Institute of Technology, Department of Civil Engineering (1984)"},{"key":"730_CR22","unstructured":"Luque, J.: The nonlinear proximal point algorithm. Massachusetts Institute of Technology. Laboratory for Information and Decision Systems, Technical report (1986)"},{"key":"730_CR23","unstructured":"Luque, J.: The nonlinear poximal point algorithm and multiplier methods. Massachusetts Institute of Technology. Laboratory for Information and Decision Systems, Technical report (1986)"},{"key":"730_CR24","doi-asserted-by":"crossref","unstructured":"Luque, J.: A nonlinear proximal point algorithm. In: 26th IEEE Conference on Decision and Control, volume\u00a026, pp. 816\u2013817. IEEE (1987)","DOI":"10.1109\/CDC.1987.272506"},{"key":"730_CR25","doi-asserted-by":"crossref","unstructured":"Mohammadisiahroudi, M.,\u00a0Fakhimi, R.,\u00a0Augustino, B.,\u00a0Terlaky, T.: Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments (2024). http:\/\/arxiv.org\/abs\/2302.00711","DOI":"10.1080\/10556788.2024.2308677"},{"issue":"2","key":"730_CR26","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1080\/10556788.2017.1380642","volume":"34","author":"I Necoara","year":"2019","unstructured":"Necoara, I., Patrascu, A., Glineur, F.: Complexity of first-order inexact lagrangian and penalty methods for conic convex programming. Optim. Methods Softw. 34(2), 305\u2013335 (2019)","journal-title":"Optim. Methods Softw."},{"issue":"5","key":"730_CR27","doi-asserted-by":"publisher","first-page":"3109","DOI":"10.1137\/120897547","volume":"52","author":"V Nedelcu","year":"2014","unstructured":"Nedelcu, V., Necoara, I., Tran-Dinh, Q.: Computational complexity of inexact gradient augmented lagrangian methods: application to constrained mpc. SIAM J. Control. Optim. 52(5), 3109\u20133134 (2014)","journal-title":"SIAM J. Control. Optim."},{"key":"730_CR28","volume-title":"Introductory Lectures on Convex Optimization: A Basic Course","author":"Y Nesterov","year":"2003","unstructured":"Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer, Berlin (2003)"},{"issue":"1\u20132","key":"730_CR29","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s10107-014-0790-0","volume":"152","author":"Y Nesterov","year":"2015","unstructured":"Nesterov, Y.: Universal gradient methods for convex optimization problems. Math. Program. 152(1\u20132), 381\u2013404 (2015)","journal-title":"Math. Program."},{"key":"730_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01727-x","author":"Y Nesterov","year":"2021","unstructured":"Nesterov, Y.: Inexact accelerated high-order proximal-point methods. Math. Program. (2021). https:\/\/doi.org\/10.1007\/s10107-021-01727-x","journal-title":"Math. Program."},{"key":"730_CR31","volume-title":"Numerical Optimization. Springer Series in Operations Research","author":"J Nocedal","year":"2006","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research, 2nd edn. Springer, New York (2006)","edition":"2"},{"key":"730_CR32","unstructured":"Oikonomidis, K.,\u00a0Laude, E.,\u00a0Latafat, P.,\u00a0Themelis, A.,\u00a0Patrinos, P.: Adaptive proximal gradient methods are universal without approximation. In: International Conference on Machine Learning, pages 38663\u201338682. PMLR (2024)"},{"key":"730_CR33","first-page":"283","volume-title":"Optimization","author":"MJ Powell","year":"1969","unstructured":"Powell, M.J.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283\u2013298. Academic Press, New York (1969)"},{"issue":"2","key":"730_CR34","first-page":"268","volume":"12","author":"RT Rockafellar","year":"1974","unstructured":"Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Optim. 12(2), 268\u2013285 (1974)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"730_CR35","doi-asserted-by":"publisher","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":"730_CR36","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.: Variational Analysis. Springer, New York (1998)"},{"key":"730_CR37","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s10107-019-01425-9","volume":"185","author":"Y Xu","year":"2021","unstructured":"Xu, Y.: Iteration complexity of inexact augmented lagrangian methods for constrained convex programming. Math. Program. 185, 199\u2013244 (2021)","journal-title":"Math. Program."},{"key":"730_CR38","first-page":"5667","volume":"33","author":"J Yang","year":"2020","unstructured":"Yang, J., Zhang, S., Kiyavash, N., He, N.: A catalyst framework for minimax optimization. Adv. Neural. Inf. Process. Syst. 33, 5667\u20135678 (2020)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"issue":"4","key":"730_CR39","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1145\/279232.279236","volume":"23","author":"C Zhu","year":"1997","unstructured":"Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: Algorithm 778: L-BFGS-b: Fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. 23(4), 550\u2013560 (1997). https:\/\/doi.org\/10.1145\/279232.279236","journal-title":"ACM Trans. Math. Softw."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00730-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00730-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00730-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T12:01:52Z","timestamp":1769688112000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00730-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,22]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["730"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00730-8","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,22]]},"assertion":[{"value":"7 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 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 authors declare that they have no conflict of interest regarding this research.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}