{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:46:17Z","timestamp":1740123977654,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T00:00:00Z","timestamp":1722297600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T00:00:00Z","timestamp":1722297600000},"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":["12201084"],"award-info":[{"award-number":["12201084"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2023M730417"],"award-info":[{"award-number":["2023M730417"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s10957-024-02497-0","type":"journal-article","created":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T08:49:17Z","timestamp":1722934157000},"page":"227-255","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Distributionally Robust Variational Inequalities: Relaxation, Quantification and Discretization"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3985-8701","authenticated-orcid":false,"given":"Jie","family":"Jiang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,30]]},"reference":[{"key":"2497_CR1","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316962","volume-title":"Convergence of Probability Measures","author":"P Billingsley","year":"1999","unstructured":"Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)","edition":"2"},{"issue":"4","key":"2497_CR2","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1287\/moor.1050.0160","volume":"30","author":"X Chen","year":"2005","unstructured":"Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022\u20131038 (2005)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"2497_CR3","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10107-017-1132-9","volume":"165","author":"X Chen","year":"2017","unstructured":"Chen, X., Pong, T.K., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165(1), 71\u2013111 (2017)","journal-title":"Math. Program."},{"issue":"1","key":"2497_CR4","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1137\/17M1162822","volume":"29","author":"X Chen","year":"2019","unstructured":"Chen, X., Shapiro, A., Sun, H.: Convergence analysis of sample average approximation of two-stage stochastic generalized equations. SIAM J. Optim. 29(1), 135\u2013161 (2019)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"2497_CR5","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s10107-018-1266-4","volume":"177","author":"X Chen","year":"2019","unstructured":"Chen, X., Sun, H., Huifu, X.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. Math. Program. 177(1\u20132), 255\u2013289 (2019)","journal-title":"Math. Program."},{"issue":"2","key":"2497_CR6","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1137\/110825248","volume":"22","author":"X Chen","year":"2012","unstructured":"Chen, X., Wets, R.J.-B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649\u2013673 (2012)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"2497_CR7","doi-asserted-by":"crossref","first-page":"1396","DOI":"10.1137\/17M1120063","volume":"28","author":"Z Chen","year":"2018","unstructured":"Chen, Z., Jiang, J.: Stability analysis of optimization problems with $$k$$th order stochastic and distributionally robust dominance constraints induced by full random recourse. SIAM J. Optim. 28(2), 1396\u20131419 (2018)","journal-title":"SIAM J. Optim."},{"key":"2497_CR8","unstructured":"Claus, M.: Advancing Stability Analysis of Mean-risk Stochastic Programs: Bilevel and Two-stage Models. Ph.D. thesis, Universit\u00e4tsbibliothek Duisburg-Essen (2016)"},{"key":"2497_CR9","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719000","volume-title":"The Linear Complementarity Problem","author":"RW Cottle","year":"2009","unstructured":"Cottle, R.W., Pang, J.-S., Stone Richard, E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)"},{"issue":"3","key":"2497_CR10","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1287\/opre.1090.0741","volume":"58","author":"E Delage","year":"2010","unstructured":"Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595\u2013612 (2010)","journal-title":"Oper. Res."},{"key":"2497_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0084195","volume-title":"Well-posed Optimization Problems","author":"L Dontchev Assen","year":"1993","unstructured":"Dontchev Assen, L., Tullio, Z.: Well-posed Optimization Problems. Springer, Berlin (1993)"},{"key":"2497_CR12","volume-title":"Finite-dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2007","unstructured":"Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2007)"},{"issue":"1","key":"2497_CR13","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s10107-019-01435-7","volume":"185","author":"L Guo","year":"2021","unstructured":"Guo, L., Chen, X.: Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation. Math. Program. 185(1), 455\u2013485 (2021)","journal-title":"Math. Program."},{"key":"2497_CR14","doi-asserted-by":"crossref","unstructured":"G\u00fcl, G., Yonca\u00a0\u00d6zge, A., Robinson Stephen, M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84(2) (1999)","DOI":"10.1007\/s101070050024"},{"issue":"4","key":"2497_CR15","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1080\/10556788.2023.2167995","volume":"38","author":"A Hori","year":"2023","unstructured":"Hori, A., Yamakawa, Y., Yamashita, N.: Distributionally robust expected residual minimization for stochastic variational inequality problems. Optim. Methods Softw. 38(4), 756\u2013780 (2023)","journal-title":"Optim. Methods Softw."},{"issue":"6","key":"2497_CR16","doi-asserted-by":"crossref","first-page":"1462","DOI":"10.1109\/TAC.2008.925853","volume":"53","author":"H Jiang","year":"2008","unstructured":"Jiang, H., Huifu, X.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53(6), 1462\u20131475 (2008)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"2497_CR17","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1137\/22M1482238","volume":"5","author":"J Jiang","year":"2023","unstructured":"Jiang, J., Chen, X.: Optimality conditions for nonsmooth nonconvex-nonconcave min-max problems and generative adversarial networks. SIAM J. Math. Data Sci. 5(3), 693\u2013722 (2023)","journal-title":"SIAM J. Math. Data Sci."},{"issue":"2","key":"2497_CR18","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s10589-020-00185-z","volume":"76","author":"J Jiang","year":"2020","unstructured":"Jiang, J., Chen, X., Chen, Z.: Quantitative analysis for a class of two-stage stochastic linear variational inequality problems. Comput. Optim. Appl. 76(2), 431\u2013460 (2020)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"2497_CR19","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1007\/s10957-021-01905-z","volume":"190","author":"J Jiang","year":"2021","unstructured":"Jiang, J., Li, S.: Regularized sample average approximation approach for two-stage stochastic variational inequalities. J. Optim. Theory Appl. 190(2), 650\u2013671 (2021)","journal-title":"J. Optim. Theory Appl."},{"issue":"9","key":"2497_CR20","doi-asserted-by":"crossref","first-page":"2591","DOI":"10.1007\/s11590-021-01838-6","volume":"16","author":"J Jiang","year":"2022","unstructured":"Jiang, J., Li, S.: Statistical robustness of two-stage stochastic variational inequalities. Optim. Lett. 16(9), 2591\u20132605 (2022)","journal-title":"Optim. Lett."},{"issue":"6","key":"2497_CR21","doi-asserted-by":"crossref","first-page":"813","DOI":"10.4208\/jcm.1906-m2019-0025","volume":"37","author":"J Jiang","year":"2019","unstructured":"Jiang, J., Shi, Y., Wang, X., Chen, X.: Regularized two-stage stochastic variational inequalities for Cournot\u2013Nash equilibrium under uncertainty. J. Comput. Math. 37(6), 813\u2013842 (2019)","journal-title":"J. Comput. Math."},{"issue":"1","key":"2497_CR22","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11075-021-01110-z","volume":"89","author":"J Jiang","year":"2022","unstructured":"Jiang, J., Sun, H., Zhou, B.: Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage. Numer. Algorithms 89(1), 167\u2013194 (2022)","journal-title":"Numer. Algorithms"},{"issue":"3","key":"2497_CR23","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s10589-019-00120-x","volume":"74","author":"A Kannan","year":"2019","unstructured":"Kannan, A., Shanbhag, U.V.: Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants. Comput. Optim. Appl. 74(3), 779\u2013820 (2019)","journal-title":"Comput. Optim. Appl."},{"key":"2497_CR24","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719451","volume-title":"An Introduction to Variational Inequalities and Their Applications","author":"D Kinderlehrer","year":"2000","unstructured":"Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)"},{"issue":"1","key":"2497_CR25","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.jmva.2011.06.005","volume":"103","author":"V Kr\u00e4tschmer","year":"2012","unstructured":"Kr\u00e4tschmer, V., Schied, A., Z\u00e4hle, H.: Qualitative and infinitesimal robustness of tail-dependent statistical functionals. J. Multivar. Anal. 103(1), 35\u201347 (2012)","journal-title":"J. Multivar. Anal."},{"issue":"1","key":"2497_CR26","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1007\/s10957-020-01686-x","volume":"186","author":"M Li","year":"2020","unstructured":"Li, M., Zhang, C.: Two-stage stochastic variational inequality arising from stochastic programming. J. Optim. Theory Appl. 186(1), 324\u2013343 (2020)","journal-title":"J. Optim. Theory Appl."},{"key":"2497_CR27","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.apm.2021.09.033","volume":"102","author":"M Li","year":"2022","unstructured":"Li, M., Zhang, C., Ding, M., Lv, R.: A two-stage stochastic variational inequality model for storage and dynamic distribution of medical supplies in epidemic management. Appl. Math. Model. 102, 35\u201361 (2022)","journal-title":"Appl. Math. Model."},{"issue":"1","key":"2497_CR28","first-page":"19","volume":"44","author":"Y Liu","year":"2019","unstructured":"Liu, Y., Pichler, A., Huifu, X.: Discrete approximation and quantification in distributionally robust optimization. Math. Oper. Res. 44(1), 19\u201337 (2019)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"2497_CR29","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1137\/1101016","volume":"1","author":"YV Prokhorov","year":"1956","unstructured":"Prokhorov, Y.V.: Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1(2), 157\u2013214 (1956)","journal-title":"Theory Probab. Appl."},{"key":"2497_CR30","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-4869-3","volume-title":"The Methods of Distances in the Theory of Probability and Statistics","author":"ST Rachev","year":"2013","unstructured":"Rachev, S.T., Klebanov, L., Stoyanov Stoyan, V., Fabozzi, F.: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York (2013)"},{"issue":"1\u20132","key":"2497_CR31","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s10107-018-1251-y","volume":"174","author":"RT Rockafellar","year":"2019","unstructured":"Rockafellar, R.T., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174(1\u20132), 453\u2013471 (2019)","journal-title":"Math. Program."},{"key":"2497_CR32","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R.T., Wets Roger, J.-B.: Variational Analysis, vol. 317. Springer, New York (2009)"},{"key":"2497_CR33","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T., Wets, Roger JB.: Stochastic variational inequalities: single-stage to multistage. Math. Program. 165(1), 331\u2013360 (2017)","DOI":"10.1007\/s10107-016-0995-5"},{"key":"2497_CR34","first-page":"483","volume":"10","author":"W R\u00f6misch","year":"2003","unstructured":"R\u00f6misch, W.: Stability of stochastic programming problems. Handb. Oper. Res. Management Sci. 10, 483\u2013554 (2003)","journal-title":"Handb. Oper. Res. Management Sci."},{"key":"2497_CR35","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611973433","volume-title":"Lectures on Stochastic Programming: Modeling and Theory","author":"A Shapiro","year":"2014","unstructured":"Shapiro, A., Dentcheva, D., Ruszczy\u0144ski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2014)"},{"issue":"1","key":"2497_CR36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40305-019-00267-8","volume":"9","author":"H Sun","year":"2021","unstructured":"Sun, H., Chen, X.: Two-stage stochastic variational inequalities: theory, algorithms and applications. J. Oper. Res. Soc. China 9(1), 1\u201332 (2021)","journal-title":"J. Oper. Res. Soc. China"},{"issue":"1","key":"2497_CR37","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s10107-022-01889-2","volume":"200","author":"H Sun","year":"2023","unstructured":"Sun, H., Shapiro, A., Chen, X.: Distributionally robust stochastic variational inequalities. Math. Program. 200(1), 279\u2013317 (2023)","journal-title":"Math. Program."},{"issue":"2","key":"2497_CR38","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1287\/moor.2015.0732","volume":"41","author":"H Sun","year":"2016","unstructured":"Sun, H., Huifu, X.: Convergence analysis for distributionally robust optimization and equilibrium problems. Math. Oper. Res. 41(2), 377\u2013401 (2016)","journal-title":"Math. Oper. Res."},{"key":"2497_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2009","unstructured":"Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2009)"},{"issue":"01","key":"2497_CR40","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1142\/S0217595910002569","volume":"27","author":"H Xu","year":"2010","unstructured":"Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pac. J. Oper. Res. 27(01), 103\u2013119 (2010)","journal-title":"Asia-Pac. J. Oper. Res."},{"issue":"1","key":"2497_CR41","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1080\/02331939508844060","volume":"33","author":"J-J Ye","year":"1995","unstructured":"Ye, J.-J., Zhu, D.-L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9\u201327 (1995)","journal-title":"Optimization"},{"issue":"3","key":"2497_CR42","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1137\/15M1038529","volume":"26","author":"J Zhang","year":"2016","unstructured":"Zhang, J., Huifu, X., Zhang, L.: Quantitative stability analysis for distributionally robust optimization with moment constraints. SIAM J. Optim. 26(3), 1855\u20131882 (2016)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"2497_CR43","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s10107-017-1116-9","volume":"165","author":"J Zhang","year":"2017","unstructured":"Zhang, J., Huifu, X., Zhang, L.: Quantitative stability analysis of stochastic quasi-variational inequality problems and applications. Math. Program. 165(1), 433\u2013470 (2017)","journal-title":"Math. Program."},{"issue":"3","key":"2497_CR44","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1137\/17M1151067","volume":"29","author":"M Zhang","year":"2019","unstructured":"Zhang, M., Sun, J., Honglei, X.: Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms. SIAM J. Optim. 29(3), 1799\u20131818 (2019)","journal-title":"SIAM J. Optim."},{"key":"2497_CR45","unstructured":"Zhang, Y.: Distributionally Robust Stochastic Variational Inequalities and Applications. Ph.D. thesis, The Hong Kong Polytechnic University, Hong Kong (2012)"},{"issue":"3","key":"2497_CR46","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1080\/10556788.2016.1277997","volume":"32","author":"L Zhu","year":"2017","unstructured":"Zhu, L., Bo, Yu., Liyan, X.: The distributionally robust complementarity problem. Optim. Methods Softw. 32(3), 650\u2013668 (2017)","journal-title":"Optim. Methods Softw."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02497-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02497-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02497-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T18:05:01Z","timestamp":1730484301000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02497-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,30]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2497"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02497-0","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2024,7,30]]},"assertion":[{"value":"7 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}