{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:35:13Z","timestamp":1787340913101,"version":"build-2736575974"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2025,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as the Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem, where the associated operator consists of an approximation operator satisfying a [Formula: see text]-order Lipschitz bound with respect to the original mapping, and the gradient of a [Formula: see text]-order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a [Formula: see text]-order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is a broad scheme, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds.<\/jats:p>","DOI":"10.1137\/23m1585258","type":"journal-article","created":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T03:02:26Z","timestamp":1751511746000},"page":"1469-1497","source":"Crossref","is-referenced-by-count":1,"title":["An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities"],"prefix":"10.1137","volume":"35","author":[{"given":"Kevin","family":"Huang","sequence":"first","affiliation":[{"name":"Institute of Industrial Engineering, National Taiwan University, Taipei, 10617 Taiwan."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-1676","authenticated-orcid":true,"given":"Shuzhong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,7,3]]},"reference":[{"key":"ref1","unstructured":"D. Adil, B. Bullins, A. Jambulapati, and S. Sachdeva, Optimal Methods for Higher-Order Smooth Monotone Variational Inequalities, preprint, arXiv:2205.06167, 2022."},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"ref3","series-title":"Math. Appl. 560","volume-title":"Handbook of Means and Their Inequalities","author":"Bullen P. S.","year":"2013"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/21M1395764"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-020-01606-x"},{"key":"ref6","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems","author":"Facchinei F.","year":"2003"},{"key":"ref7","first-page":"21858","volume-title":"in Advances in Neural Information Processing Systems 35","author":"Gorbunov E.","year":"2022"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-022-01819-6"},{"key":"ref9","unstructured":"K. Huang and S. Zhang, A Unifying Framework of Accelerated First-Order Approach to Strongly Monotone Variational Inequalities, preprint, arXiv:2103.15270, 2021."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/19M1286025"},{"key":"ref11","unstructured":"R. Jiang and A. Mokhtari, Generalized Optimistic Methods for Convex-Concave Saddle Point Problems, preprint, arXiv:2202.09674, 2022."},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01643-0"},{"key":"ref13","volume-title":"Logistic Regression","author":"Kleinbaum D. G.","year":"2002"},{"key":"ref14","first-page":"747","volume":"12","author":"Korpelevich G. M.","year":"1976","journal-title":"Matecon"},{"key":"ref15","first-page":"22588","author":"Lee S.","year":"2021","journal-title":"in Advances in Neural Information Processing Systems 34"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-020-01617-9"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-024-02075-2"},{"key":"ref18","first-page":"154","volume":"4","author":"Martinet B.","year":"1970","journal-title":"Rev. Fran\u00e7aise Informat. Recherche Op\u00e9rationnelle"},{"key":"ref19","first-page":"1497","volume-title":"International Conference on Artificial Intelligence and Statistics","author":"Mokhtari A.","year":"2019"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/19M127375X"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/090753127"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1137\/11083085X"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623403425629"},{"key":"ref24","series-title":"Wiley-Intersci. Ser. Discrete Math.","volume-title":"Problem Complexity and Method Efficiency in Optimization","author":"Nemirovski A.","year":"1983"},{"key":"ref25","first-page":"543","volume":"269","author":"Nesterov Y.","year":"1983","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref26","doi-asserted-by":"crossref","unstructured":"Y. Nesterov, Cubic Regularization of Newton\u2019s Method for Convex Problems with Constraints, CORE Discussion Paper No. 2006\/39, 2006.","DOI":"10.2139\/ssrn.921825"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-006-0034-z"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-019-01449-1"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-006-0706-8"},{"key":"ref30","author":"Nesterov Y.","year":"2006","journal-title":"CORE Discussion Paper 2006\/107"},{"key":"ref31","first-page":"357","volume":"14","author":"Ostroukhov P.","year":"2022","journal-title":"\u041a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u044b\u0435 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435"},{"key":"ref32","first-page":"17420","volume-title":"International Conference on Machine Learning","author":"Park J.","year":"2022"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(64)90137-5"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/BF01141092"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1137\/0314056"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575559"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581276"},{"key":"ref38","unstructured":"Q. Tran-Dinh and Y. Luo, Halpern-Type Accelerated and Splitting Algorithms for Monotone Inclusions, preprint, arXiv:2110.08150, 2021."},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)00094-H"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998338806"},{"key":"ref41","unstructured":"T. Yoon and E. K. Ryu, Accelerated algorithms for smooth convex-concave minimax problems with \\({O}(1\/k^2)\\) rate on squared gradient norm, in Proceedings of the 38th International Conference on Machine Learning, PMLR 139, 2021, pp. 12098\u201312109."},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/22M1504597"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01660-z"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1585258","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:55:43Z","timestamp":1787338543000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1585258"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,3]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9,30]]}},"alternative-id":["10.1137\/23M1585258"],"URL":"https:\/\/doi.org\/10.1137\/23m1585258","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,3]]}}}