{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T07:37:36Z","timestamp":1773819456703,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T00:00:00Z","timestamp":1604448000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T00:00:00Z","timestamp":1604448000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871276"],"award-info":[{"award-number":["11871276"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771405"],"award-info":[{"award-number":["11771405"]}],"id":[{"id":"10.13039\/501100001809","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":[[2021,1]]},"DOI":"10.1007\/s10589-020-00234-7","type":"journal-article","created":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T20:02:32Z","timestamp":1604520152000},"page":"205-238","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems"],"prefix":"10.1007","volume":"78","author":[{"given":"Yannan","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1036-6327","authenticated-orcid":false,"given":"Hailin","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huifu","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,11,4]]},"reference":[{"key":"234_CR1","volume-title":"Measure Theory and Probability Theory","author":"KB Athreya","year":"2006","unstructured":"Athreya, K.B., Lahiri, S.N.: Measure Theory and Probability Theory. Springer, New York (2006)"},{"key":"234_CR2","doi-asserted-by":"publisher","first-page":"580","DOI":"10.1287\/moor.1100.0445","volume":"35","author":"D Bertsimas","year":"2010","unstructured":"Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35, 580\u2013602 (2010)","journal-title":"Math. Oper. Res."},{"key":"234_CR3","unstructured":"Bertsimas, D., Parys, B. V.: Bootstrap robust prescriptive analytics. arXiv:1711.09974 (2017)"},{"key":"234_CR4","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","volume":"40","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging. Vis. 40, 120\u2013145 (2011)","journal-title":"J. Math. Imaging. Vis."},{"key":"234_CR5","first-page":"592","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, 592\u2013612 (2010)","journal-title":"Oper. Res."},{"key":"234_CR6","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1137\/09076934X","volume":"3","author":"E Esser","year":"2010","unstructured":"Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for Convex Optimization in Imaging Science. SIAM J. Imaging Sci. 3, 1015\u20131046 (2010)","journal-title":"SIAM J. Imaging Sci."},{"key":"234_CR7","first-page":"42","volume":"39","author":"K Fan","year":"1953","unstructured":"Fan, K.: Minimax theorems. Izv. Nats. Akad. Nauk Armen. Mekh 39, 42\u201347 (1953)","journal-title":"Izv. Nats. Akad. Nauk Armen. Mekh"},{"key":"234_CR8","unstructured":"Gao, R., Kleywegt, A.: Distributionally robust stochastic optimization with Wasserstein distance. arXiv:1604.02199 (2016)"},{"key":"234_CR9","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1111\/j.1751-5823.2002.tb00178.x","volume":"70","author":"AL Gibbs","year":"2002","unstructured":"Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70, 419\u2013435 (2002)","journal-title":"Int. Stat. Rev."},{"key":"234_CR10","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1287\/opre.1090.0795","volume":"58","author":"J Goh","year":"2010","unstructured":"Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58, 902\u2013917 (2010)","journal-title":"Oper. Res."},{"key":"234_CR11","unstructured":"Goldstein, T., Li, M., Yuan, X., Esser, E., Baraniuk, R.: Adaptive primal-dual hybrid gradient methods for saddle-point problems. arXiv:1305.0546 (2013)"},{"key":"234_CR12","doi-asserted-by":"publisher","first-page":"784","DOI":"10.1137\/15M1036592","volume":"27","author":"S Guo","year":"2017","unstructured":"Guo, S., Xu, H., Zhang, L.: Convergence analysis for mathematical programs with distributionally robust chance constraint. SIAM J. Optim. 27, 784\u2013816 (2017)","journal-title":"SIAM J. Optim."},{"key":"234_CR13","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s10107-018-1307-z","volume":"174","author":"S Guo","year":"2019","unstructured":"Guo, S., Xu, H.: Distributionally robust shortfall risk optimization model and its approximation. Math. Program. 174, 473\u2013498 (2019)","journal-title":"Math. Program."},{"key":"234_CR14","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1287\/opre.2017.1698","volume":"66","author":"GA Hanasusanto","year":"2018","unstructured":"Hanasusanto, G.A., Kuhn, D.: Conic programming reformulations of two-stage distributionally robust linear programs over Wasserstein balls. Oper. Res. 66, 849\u2013869 (2018)","journal-title":"Oper. Res."},{"key":"234_CR15","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s10851-017-0709-5","volume":"58","author":"B He","year":"2017","unstructured":"He, B., Ma, F., Yuan, X.: An algorithm framework of generalized primal-dual hybrid gradient methods for saddle point problems. J. Math. Imaging. Vis. 58, 279\u2013293 (2017)","journal-title":"J. Math. Imaging. Vis."},{"key":"234_CR16","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1137\/100814494","volume":"5","author":"B He","year":"2012","unstructured":"He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119\u2013149 (2012)","journal-title":"SIAM J. Imaging Sci."},{"key":"234_CR17","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1287\/opre.2018.1729","volume":"66","author":"R Jiang","year":"2018","unstructured":"Jiang, R., Guan, Y.: Risk-averse two-stage stochastic program with distributional ambiguity. Oper. Res. 66, 1390\u20131405 (2018)","journal-title":"Oper. Res."},{"key":"234_CR18","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1287\/moor.2018.0939","volume":"44","author":"Y Liu","year":"2019","unstructured":"Liu, Y., Pichler, A., Xu, H.: Discrete approximation and quantification in distributionally robust optimization. Math. Oper. Res. 44, 19\u201337 (2019)","journal-title":"Math. Oper. Res."},{"key":"234_CR19","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1016\/j.orl.2017.10.001","volume":"45","author":"Y Liu","year":"2017","unstructured":"Liu, Y., Yuan, X., Zeng, S., Zhang, J.: Primal-dual hybrid gradient method for distributionally robust optimization problems. Oper. Res. Lett. 45, 625\u2013630 (2017)","journal-title":"Oper. Res. Lett."},{"key":"234_CR20","unstructured":"Liu, Y., Yuan, X., Zhang, J.: Quantitative stability analysis of stochastic programs with distributionally robust second order dominance constraints, manuscript (2017)"},{"key":"234_CR21","unstructured":"Love, D., Bayrakcan, G.: Phi-divergence constrained ambiguous stochastic programs for data-driven optimization, available on researchgate.net (2016)"},{"key":"234_CR22","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10107-017-1172-1","volume":"171","author":"P Mohajerin Esfahani","year":"2018","unstructured":"Mohajerin Esfahani, P., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. 171, 115\u2013166 (2018)","journal-title":"Math. Program."},{"key":"234_CR23","volume-title":"Statistical Inference Based on Divergence Measures","author":"L Pardo","year":"2005","unstructured":"Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman and Hall\/CRC, Boca Raton (2005)"},{"key":"234_CR24","volume-title":"Stochastic Optimization Methods in Finance and Energy, vol. 163 of International Series in Operations Research & Management Science","author":"GC Pflug","year":"2011","unstructured":"Pflug, G.C., Pichler, A.: Approximations for probability distributions and stochastic optimization problems. In: Bertocchi, M., Consigli, G., Dempster, M.A.H. (eds.) Stochastic Optimization Methods in Finance and Energy, vol. 163 of International Series in Operations Research & Management Science. Springer, New York (2011)"},{"key":"234_CR25","volume-title":"Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering","author":"GC Pflug","year":"2014","unstructured":"Pflug, G.C., Pichler, A.: Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)"},{"key":"234_CR26","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1080\/14697680701455410","volume":"7","author":"GC Pflug","year":"2007","unstructured":"Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Financ. 7, 435\u2013442 (2007)","journal-title":"Quant. Financ."},{"key":"234_CR27","doi-asserted-by":"crossref","unstructured":"Pichler, A., Xu, H.: Quantitative stability analysis for minimax distributionally robust risk optimization, To appear in Math. Program. (2018)","DOI":"10.1007\/s10107-018-1347-4"},{"key":"234_CR28","volume-title":"Probability Metrics and the Stability of Stochastic Models","author":"ST Rachev","year":"1991","unstructured":"Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, West Sussex (1991)"},{"key":"234_CR29","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1287\/moor.16.1.119","volume":"16","author":"R Rockafellar","year":"1991","unstructured":"Rockafellar, R., Wets, R.J.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119\u2013147 (1991)","journal-title":"Math. Oper. Res."},{"key":"234_CR30","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/s10107-018-1251-y","volume":"174","author":"R Rockafellar","year":"2019","unstructured":"Rockafellar, R., Sun, J.: Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174, 453\u2013471 (2019)","journal-title":"Math. Program."},{"key":"234_CR31","volume-title":"Stochastic Programming. Handbook in OR & MS","author":"W R\u00f6misch","year":"2003","unstructured":"R\u00f6misch, W.: Stability of stochastic programming problems. In: Rusczy\u0144ski, A., Shapiro, A. (eds.) Stochastic Programming. Handbook in OR & MS, vol. 10. North-Holland Publishing Company, Amsterdam (2003)"},{"key":"234_CR32","volume-title":"Stochastic Programming, Handbook in OR & MS","author":"A Ruszczy\u0144ski","year":"2003","unstructured":"Ruszczy\u0144ski, A.: Decomposition method. In: Rusczy\u0144ski, A., Shapiro, A. (eds.) Stochastic Programming, Handbook in OR & MS, vol. 10. North-Holland Publishing Company, Amsterdam (2003)"},{"key":"234_CR33","first-page":"333","volume":"79","author":"A Ruszczy\u0144ski","year":"1997","unstructured":"Ruszczy\u0144ski, A.: Decomposition methods in stochastic programming. Math. Program. 79, 333\u2013353 (1997)","journal-title":"Math. Program."},{"key":"234_CR34","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/s10107-017-1224-6","volume":"173","author":"H Rahimian","year":"2019","unstructured":"Rahimian, H., Bayraksan, G., Homem-de-Mello, T.: Identifying effective scenarios in distributionally robust stochastic programs with total variation distance. Math. Program. 173, 393\u2013430 (2019)","journal-title":"Math. Program."},{"key":"234_CR35","volume-title":"Stochastic Programming, Handbook in OR & MS","author":"A Rusczy\u0144ski","year":"2003","unstructured":"Rusczy\u0144ski, A., Shapiro, A.: Stochastic Programming, Handbook in OR & MS, vol. 10. North-Holland Publishing Company, Amsterdam (2003)"},{"key":"234_CR36","first-page":"201","volume-title":"Studies in the Mathematical Theory of Inventory and Production","author":"H Scarf","year":"1958","unstructured":"Scarf, H.: A min\u2013max solution of an inventory problem. In: Arrow, K.S., Karlin, S., Scarf, H.E. (eds.) Studies in the Mathematical Theory of Inventory and Production, pp. 201\u2013209. Stanford University Press, Palo Alto (1958)"},{"key":"234_CR37","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1137\/S1052623403434012","volume":"14","author":"A Shapiro","year":"2004","unstructured":"Shapiro, A., Ahmed, S.: On a class of minimax stochastic programs. SIAM J. Optim. 14, 1237\u20131249 (2004)","journal-title":"SIAM J. Optim."},{"key":"234_CR38","volume-title":"Semi-Infinite Programming. Nonconvex Optimization and Its Applications","author":"A Shapiro","year":"2001","unstructured":"Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.A., L\u00f3pez, M.A. (eds.) Semi-Infinite Programming. Nonconvex Optimization and Its Applications, vol. 57. Springer, Boston (2001)"},{"key":"234_CR39","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/s10107-017-1131-x","volume":"168","author":"J Sun","year":"2018","unstructured":"Sun, J., Liao, L.Z., Rodrigues, B.: Quadratic two-stage stochastic optimization with coherent measures of risk. Math. Program. 168, 599\u2013613 (2018)","journal-title":"Math. Program."},{"key":"234_CR40","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1287\/moor.2015.0732","volume":"41","author":"H Sun","year":"2016","unstructured":"Sun, H., Xu, H.: Convergence analysis for distributionally robust optimization and equilibrium problems. Math. Oper. Res. 41, 377\u2013401 (2016)","journal-title":"Math. Oper. Res."},{"key":"234_CR41","doi-asserted-by":"publisher","first-page":"2047","DOI":"10.1137\/070696143","volume":"31","author":"P Weiss","year":"2009","unstructured":"Weiss, P., Blanc-Feraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31, 2047\u20132080 (2009)","journal-title":"SIAM J. Sci. Comput."},{"key":"234_CR42","doi-asserted-by":"publisher","first-page":"1358","DOI":"10.1287\/opre.2014.1314","volume":"62","author":"W Wiesemann","year":"2014","unstructured":"Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62, 1358\u2013376 (2014)","journal-title":"Oper. Res."},{"key":"234_CR43","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1007\/s10107-017-1143-6","volume":"169","author":"H Xu","year":"2018","unstructured":"Xu, H., Liu, Y., Sun, H.: Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods. Math. Program. 169, 489\u2013529 (2018)","journal-title":"Math. Program."},{"key":"234_CR44","doi-asserted-by":"publisher","first-page":"2922","DOI":"10.1137\/17M1158707","volume":"28","author":"Y Zhang","year":"2018","unstructured":"Zhang, Y., Jiang, R., Shen, S.: Ambiguous chance-constrained binary programs under mean-covariance information. SIAM J. Optim. 28, 2922\u20132944 (2018)","journal-title":"SIAM J. Optim."},{"key":"234_CR45","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s10915-010-9408-8","volume":"46","author":"X Zhang","year":"2010","unstructured":"Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20\u201346 (2010)","journal-title":"J. Sci. Comput."},{"key":"234_CR46","unstructured":"Zhao, C., Guan, Y.: Data-driven risk-averse two-stage stochastic program with \u03b6-structure probability metrics, available at Optimization Online (2015).  http:\/\/www.optimization-online.org\/DB_FILE\/2015\/07\/5014.pdf"},{"key":"234_CR47","first-page":"264","volume":"28","author":"VM Zolotarev","year":"1983","unstructured":"Zolotarev, V.M.: Probability metrics. Teoriya Veroyatnostei IEE Primeneniya 28, 264\u2013287 (1983)","journal-title":"Teoriya Veroyatnostei IEE Primeneniya"},{"key":"234_CR48","unstructured":"Zhang, Z., Ahmend, S., Lan, G.: Efficient algorithms for distributionally robust stochastic optimization with discrete scenario support. arXiv:1909.11216 (2019)"},{"key":"234_CR49","unstructured":"Zhu, M., Chan, T. F.: An efficient primal dual hybrid gradient algorithm for total variation image restoration. CAM Report 08-34, UCLA, Los Angeles, CA (2008)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00234-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-020-00234-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00234-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,11]],"date-time":"2021-01-11T13:36:07Z","timestamp":1610372167000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-020-00234-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,4]]},"references-count":49,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["234"],"URL":"https:\/\/doi.org\/10.1007\/s10589-020-00234-7","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,4]]},"assertion":[{"value":"29 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}