{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T18:05:58Z","timestamp":1773770758962,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T00:00:00Z","timestamp":1773705600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T00:00:00Z","timestamp":1773705600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 12071342"],"award-info":[{"award-number":["No. 12071342"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003787","name":"Natural Science Foundation of Hebei Province","doi-asserted-by":"publisher","award":["No. A2020202030"],"award-info":[{"award-number":["No. A2020202030"]}],"id":[{"id":"10.13039\/501100003787","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Open ProjectProgram of Key Laboratory of Discrete Mathematics with Applications of Ministry of Education","award":["J20230701"],"award-info":[{"award-number":["J20230701"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10957-026-02964-w","type":"journal-article","created":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T14:56:07Z","timestamp":1773759367000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Properties of Gradient-Based Methods for Minimax Problems with Nonlinear Constraints"],"prefix":"10.1007","volume":"209","author":[{"given":"Qingying","family":"Hu","sequence":"first","affiliation":[]},{"given":"Bo","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9203-6929","authenticated-orcid":false,"given":"Mengwei","family":"Xu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,17]]},"reference":[{"key":"2964_CR1","unstructured":"Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press (2014)"},{"issue":"2","key":"2964_CR2","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s00498-023-00371-4","volume":"36","author":"R Chinchilla","year":"2024","unstructured":"Chinchilla, R., Yang, G., Hespanha, J.P.: Newton and interior-point methods for (constrained) nonconvex\u2013nonconcave minmax optimization with stability and instability guarantees. Math. Control Signals Systems 36(2), 381\u2013421 (2024)","journal-title":"Math. Control Signals Systems"},{"key":"2964_CR3","doi-asserted-by":"publisher","first-page":"2883","DOI":"10.1137\/22M1535243","volume":"34","author":"YH Dai","year":"2022","unstructured":"Dai, Y.H., Wang, J., Zhang, L.: Optimality conditions and numerical algorithms for a class of linearly constrained minimax optimization problems. SIAM J. Optim. 34, 2883\u20132916 (2022)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"2964_CR4","doi-asserted-by":"publisher","first-page":"296","DOI":"10.4208\/csiam-am.2020-0014","volume":"1","author":"YH Dai","year":"2020","unstructured":"Dai, Y.H., Zhang, L.: Optimality conditions for constrained minimax optimization. CSIAM Transactions on Applied Mathematics 1(2), 296\u2013315 (2020)","journal-title":"CSIAM Transactions on Applied Mathematics"},{"key":"2964_CR5","unstructured":"Diakonikolas, J., Daskalakis, C., Jordan, M.I.: Efficient methods for structured nonconvex-nonconcave min-max optimization. International Conference on Artificial Intelligence and Statistics. PMLR, 2746\u20132754 (2021)"},{"key":"2964_CR6","unstructured":"Doan, T.: Convergence rates of two-time-scale gradient descent-ascent dynamics for solving nonconvex min-max problems. Learning for Dynamics and Control Conference. PMLR, 192\u2013206 (2022)"},{"issue":"1","key":"2964_CR7","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10107-012-0528-9","volume":"141","author":"J Eckstein","year":"2013","unstructured":"Eckstein, J., Silva, P.J.S.: A practical relative error criterion for augmented Lagrangians. Math. Program. 141(1), 319\u2013348 (2013)","journal-title":"Math. Program."},{"issue":"1","key":"2964_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.mlwa.2020.100017","volume":"3","author":"C Finlay","year":"2021","unstructured":"Finlay, C., Oberman, A.M.: Scaleable input gradient regularization for adversarial robustness. Machine Learning with Applications 3(1), 100017 (2021)","journal-title":"Machine Learning with Applications"},{"key":"2964_CR9","unstructured":"Ghadimi, S., Wang, M.: Approximation methods for bilevel programming. arXiv preprint arXiv:1802.02246 (2018)"},{"issue":"2","key":"2964_CR10","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1007\/s10107-021-01720-4","volume":"198","author":"J Jiang","year":"2023","unstructured":"Jiang, J., Chen, X.: Pure characteristics demand models and distributionally robust mathematical programs with stochastic complementarity constraints. Math. Program. 198(2), 1449\u20131484 (2023)","journal-title":"Math. Program."},{"key":"2964_CR11","unstructured":"Jin, C., Netrapalli, P., Jordan, M.: What is local optimality in nonconvex-nonconcave minimax optimization? International Conference on Machine Learning. PMLR, 4880\u20134889 (2020)"},{"key":"2964_CR12","unstructured":"Lin, T., Jin, C., Jordan, M.I.: Near-optimal algorithms for minimax optimization. Conference on Learning Theory. PMLR, 2738\u20132779 (2020)"},{"key":"2964_CR13","unstructured":"Liu, S., Lu, S., Chen, X., Feng, Y., Xu, K., Al-Dujaili, A., Hong, M., O\u2019Reilly, U.M.: Min-max optimization without gradients: Convergence and applications to black-box evasion and poisoning attacks. International Conference on Machine Learning. PMLR, 6282\u20136293 (2020)"},{"issue":"169","key":"2964_CR14","first-page":"1","volume":"22","author":"M Liu","year":"2021","unstructured":"Liu, M., Rafique, H., Lin, Q., Yang, T.: First-order convergence theory for weakly-convex-weakly-concave min-max problems. J. Mach. Learn. Res. 22(169), 1\u201334 (2021)","journal-title":"J. Mach. Learn. Res."},{"issue":"12","key":"2964_CR15","doi-asserted-by":"publisher","first-page":"2721","DOI":"10.1080\/02331934.2025.2468412","volume":"74","author":"X Liu","year":"2025","unstructured":"Liu, X., Xu, M., Zhang, L.: Second-order optimality conditions for bi-local solutions of bilevel programs. Optimization 74(12), 2721\u20132743 (2025)","journal-title":"Optimization"},{"key":"2964_CR16","unstructured":"Liu, M., Yuan, Z., Ying, Y., Yang, T.: Stochastic AUC maximization with deep neural networks. International Conference on Learning Representations, (2020)"},{"key":"2964_CR17","doi-asserted-by":"publisher","first-page":"3676","DOI":"10.1109\/TSP.2020.2986363","volume":"68","author":"S Lu","year":"2020","unstructured":"Lu, S., Tsaknakis, I., Hong, M., Chen, Y.: Hybrid block successive approximation for one-sided non-convex min-max problems: algorithms and applications. IEEE Trans. Signal Process. 68, 3676\u20133691 (2020)","journal-title":"IEEE Trans. Signal Process."},{"issue":"2","key":"2964_CR18","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":"2964_CR19","unstructured":"Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. International Conference on Learning Representations (2018)"},{"issue":"1","key":"2964_CR20","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1137\/18M1231298","volume":"2","author":"E Mazumdar","year":"2020","unstructured":"Mazumdar, E., Ratliff, L.J., Sastry, S.S.: On gradient-based learning in continuous games. SIAM Journal on Mathematics of Data Science 2(1), 103\u2013131 (2020)","journal-title":"SIAM Journal on Mathematics of Data Science"},{"issue":"1","key":"2964_CR21","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1137\/S1052623403425629","volume":"15","author":"A Nemirovski","year":"2004","unstructured":"Nemirovski, A.: Prox-method with rate of convergence O (1\/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15(1), 229\u2013251 (2004)","journal-title":"SIAM J. Optim."},{"key":"2964_CR22","unstructured":"Neumann, J.V., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1953)"},{"issue":"2","key":"2964_CR23","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10107-006-0034-z","volume":"109","author":"Y Nesterov","year":"2007","unstructured":"Nesterov, Y.: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109(2), 319\u2013344 (2007)","journal-title":"Math. Program."},{"key":"2964_CR24","first-page":"14934","volume":"32","author":"M Nouiehed","year":"2019","unstructured":"Nouiehed, M., Sanjabi, M., Huang, T., Lee, J.D., Razaviyayn, M.: Solving a class of non-convex min-max games using iterative first order methods. Adv. Neural. Inf. Process. Syst. 32, 14934\u201314942 (2019)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"issue":"1","key":"2964_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-019-01420-0","volume":"185","author":"Y Ouyang","year":"2021","unstructured":"Ouyang, Y., Xu, Y.: Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems. Math. Program. 185(1), 1\u201335 (2021)","journal-title":"Math. Program."},{"key":"2964_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/0-387-24273-2_1","volume":"4","author":"S Puntanen","year":"2005","unstructured":"Puntanen, S., Styan, G.P.H.: Historical introduction: Issai Schur and the early development of the Schur complement. The Schur Complement and Its Applications 4, 1\u201316 (2005)","journal-title":"The Schur Complement and Its Applications"},{"issue":"3","key":"2964_CR27","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1080\/10556788.2021.1895152","volume":"37","author":"H Rafique","year":"2022","unstructured":"Rafique, H., Liu, M., Lin, Q., Yang, T.: Weakly-convex\u2013concave min\u2013max optimization: provable algorithms and applications in machine learning. Optimization Methods and Software 37(3), 1087\u20131121 (2022)","journal-title":"Optimization Methods and Software"},{"issue":"1","key":"2964_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01585500","volume":"7","author":"SM Robinson","year":"1974","unstructured":"Robinson, S.M.: Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms. Math. Program. 7(1), 1\u201316 (1974)","journal-title":"Math. Program."},{"key":"2964_CR29","unstructured":"Robinson, S.M.: An Implicit-Function Theorem for generalized variational inequalties. Technical summary report no.1672, Mathematics Research Center, University of Wisconsin-Madison (1976)"},{"key":"2964_CR30","unstructured":"Stackelberg H.V.: Market Structure and Equilibrium. Springer (2010)"},{"key":"2964_CR31","first-page":"12680","volume":"32","author":"KK Thekumparampil","year":"2019","unstructured":"Thekumparampil, K.K., Jain, P., Netrapalli, P., Oh, S.: Efficient algorithms for smooth minimax optimization. Adv. Neural. Inf. Process. Syst. 32, 12680\u201312691 (2019)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"issue":"4","key":"2964_CR32","doi-asserted-by":"publisher","first-page":"2675","DOI":"10.1137\/21M1462428","volume":"33","author":"I Tsaknakis","year":"2023","unstructured":"Tsaknakis, I., Hong, M., Zhang, S.: Minimax problems with coupled linear constraints: Computational complexity and duality. SIAM J. Optim. 33(4), 2675\u20132702 (2023)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"2964_CR33","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s10107-022-01919-z","volume":"201","author":"Z Xu","year":"2023","unstructured":"Xu, Z., Zhang, H., Xu, Y., Lan, G.: A unified single-loop alternating gradient projection algorithm for nonconvex-concave and convex-nonconcave minimax problems. Math. Program. 201(1), 635\u2013706 (2023)","journal-title":"Math. Program."},{"key":"2964_CR34","unstructured":"Zhang, H., Wang, J., Xu, Z., Dai, Y.H.: Primal dual alternating proximal gradient algorithms for nonsmooth nonconvex minimax problems with coupled linear constraints. arXiv preprint arXiv:2212.04672 (2022)"},{"key":"2964_CR35","first-page":"7377","volume":"33","author":"J Zhang","year":"2020","unstructured":"Zhang, J., Xiao, P., Sun, R., Luo, Z.: A single-loop smoothed gradient descent-ascent algorithm for nonconvex-concave min-max problems. Adv. Neural. Inf. Process. Syst. 33, 7377\u20137389 (2020)","journal-title":"Adv. Neural. Inf. Process. Syst."},{"key":"2964_CR36","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/978-3-030-60990-0_12","volume":"325","author":"K Zhang","year":"2021","unstructured":"Zhang, K., Yang, Z., Ba\u015far, T.: Multi-agent reinforcement learning: A selective overview of theories and algorithms. Handbook of Reinforcement Learning and Control 325, 321\u2013384 (2021)","journal-title":"Handbook of Reinforcement Learning and Control"},{"key":"2964_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1142\/S0217595920400011","volume":"37","author":"XY Zhao","year":"2020","unstructured":"Zhao, X.Y., Chen, L.: The linear and asymptotically superlinear convergence rates of the augmented lagangian method with a practical relative error criterion. Asia-Pacific J. Oper. Res. 37, 1\u201316 (2020)","journal-title":"Asia-Pacific J. Oper. Res."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02964-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-026-02964-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02964-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T14:56:16Z","timestamp":1773759376000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-026-02964-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,17]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["2964"],"URL":"https:\/\/doi.org\/10.1007\/s10957-026-02964-w","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,17]]},"assertion":[{"value":"12 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 March 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"13"}}