{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T15:52:34Z","timestamp":1768837954380,"version":"3.49.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,1,13]],"date-time":"2020-01-13T00:00:00Z","timestamp":1578873600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,1,13]],"date-time":"2020-01-13T00:00:00Z","timestamp":1578873600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2020,6]]},"DOI":"10.1007\/s10107-019-01462-4","type":"journal-article","created":{"date-parts":[[2020,1,13]],"date-time":"2020-01-13T06:02:41Z","timestamp":1578895361000},"page":"453-472","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Minimizing buffered probability of exceedance by progressive hedging"],"prefix":"10.1007","volume":"181","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1703-5069","authenticated-orcid":false,"given":"R. Tyrrell","family":"Rockafellar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stan","family":"Uryasev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,1,13]]},"reference":[{"key":"1462_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/s11069-016-2324-y","author":"RA Davis","year":"2016","unstructured":"Davis, R.A., Uryasev, S.: Analysis of tropical storm damage using buffered probability of exceedance. Nat. Hazards (2016). https:\/\/doi.org\/10.1007\/s11069-016-2324-y","journal-title":"Nat. Hazards"},{"key":"1462_CR2","volume-title":"Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and Its Applications","author":"DP Kouri","year":"2018","unstructured":"Kouri, D.P., Shapiro, A.: Optimization of PDEs with uncertain inputs. In: Antil, H., Kouri, D., Lacasse, M.D., Ridzal, D. (eds.) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and Its Applications, vol. 163. Springer, New York, NY (2018)"},{"key":"1462_CR3","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1007\/s11590-018-1300-8","volume":"13","author":"DP Kouri","year":"2019","unstructured":"Kouri, D.P.: Higher-moment buffered probability. Optm. Lett. 13, 657\u2013672 (2019)","journal-title":"Optm. Lett."},{"key":"1462_CR4","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1137\/15M1042644","volume":"28","author":"A Mafusalov","year":"2018","unstructured":"Mafusalov, A., Uryasev, S.: Buffered probability of exceedance: mathematical properties and optimization. SIAM. J. Optim. 28, 1077\u20131103 (2018)","journal-title":"SIAM. J. Optim."},{"key":"1462_CR5","doi-asserted-by":"publisher","first-page":"826","DOI":"10.1016\/j.ejor.2018.01.021","volume":"270","author":"A Mafusalov","year":"2018","unstructured":"Mafusalov, A., Shapiro, A., Uryasev, S.: Estimation and asymptotics for buffered probability of exceedance. Eur. J. Oper. Res. 270, 826\u2013836 (2018)","journal-title":"Eur. J. Oper. Res."},{"key":"1462_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-018-1312-2","author":"M Norton","year":"2018","unstructured":"Norton, M., Uryasev, S.: Maximization of AUC and Buffered AUC in binary classification. Math. Program. (2018). https:\/\/doi.org\/10.1007\/s10107-018-1312-2","journal-title":"Math. Program."},{"key":"1462_CR7","first-page":"1","volume":"18","author":"M Norton","year":"2017","unstructured":"Norton, M., Mafusalov, A., Uryasev, S.: Soft margin support vector classification as buffered probability minimization. J. Mach. Learn. Res. 18, 1\u201343 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"1462_CR8","doi-asserted-by":"publisher","first-page":"1726","DOI":"10.1137\/16M1095913","volume":"28","author":"M Norton","year":"2018","unstructured":"Norton, M., Mafusalov, A., Uryasev, S.: Cardinality of upper average and application to network optimization. SIAM J. Optim. 28(2), 1726\u20131750 (2018)","journal-title":"SIAM J. Optim."},{"key":"1462_CR9","unstructured":"Norton, M., Khokhlov, V., Uryasev, S.: Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation. Ann. Oper. Res. arXiv:1811.11301 (submitted for publication)"},{"key":"1462_CR10","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1462_CR11","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1007\/s11228-017-0437-4","volume":"26","author":"RT Rockafellar","year":"2017","unstructured":"Rockafellar, R.T.: Solving stochastic programming problems with risk measures by progessive hedging. Set-valued Variational Anal. 26, 759\u2013768 (2017)","journal-title":"Set-valued Variational Anal."},{"key":"1462_CR12","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.ress.2010.01.001","volume":"95","author":"RT Rockafellar","year":"2010","unstructured":"Rockafellar, R.T., Royset, J.O.: On buffered failure probability in design and optimization of structures. J. Reliab. Eng. Syst. Saf. 95, 499\u2013510 (2010)","journal-title":"J. Reliab. Eng. Syst. Saf."},{"key":"1462_CR13","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T., Royset, J.O.: Superquantiles and their applications to risk, random variables, and regression. In: Tutorials in Operations Research, pp. 151\u2013167. INFORMS (2013)","DOI":"10.1287\/educ.2013.0111"},{"key":"1462_CR14","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1016\/j.ejor.2013.10.046","volume":"234","author":"RT Rockafellar","year":"2014","unstructured":"Rockafellar, R.T., Royset, J.O., Miranda, S.J.: Superquantile regression with applications to buffered reliability, uncertainty quantification and conditional value-at-risk. Eur. J. Oper. Res. 234, 140\u2013154 (2014)","journal-title":"Eur. J. Oper. Res."},{"issue":"2","key":"1462_CR15","doi-asserted-by":"publisher","first-page":"04015005","DOI":"10.1061\/AJRUA6.0000816","volume":"1","author":"RT Rockafellar","year":"2015","unstructured":"Rockafellar, R.T., Royset, J.O.: Engineering decisions under risk averseness. J. Risk Uncertain. Eng. Syst. Part A Civ. Eng. 1(2), 04015005 (2015). https:\/\/doi.org\/10.1061\/AJRUA6.0000816","journal-title":"J. Risk Uncertain. Eng. Syst. Part A Civ. Eng."},{"key":"1462_CR16","doi-asserted-by":"publisher","unstructured":"Rockafellar, R.T., Royset, J.O.: Risk measures in engineering design under uncertainty. In: Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP), Vancouver (2015). https:\/\/doi.org\/10.14288\/1.0076159","DOI":"10.14288\/1.0076159"},{"key":"1462_CR17","doi-asserted-by":"publisher","unstructured":"Rockafellar, R.T., Royset, J.O., Harajli, M.M.: Importance sampling in the evaluation and optimization of buffered probability of failure. In: Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP), Vancouver (2015). https:\/\/doi.org\/10.14288\/1.0076214","DOI":"10.14288\/1.0076214"},{"key":"1462_CR18","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1137\/151003271","volume":"25","author":"RT Rockafellar","year":"2015","unstructured":"Rockafellar, R.T., Royset, J.O.: Measures of residual risk with connections to regression, risk tracking, surrogate models and ambiguity. SIAM J. Optim. 25, 1179\u20131208 (2015)","journal-title":"SIAM J. Optim."},{"key":"1462_CR19","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10479-016-2129-0","volume":"262","author":"RT Rockafellar","year":"2018","unstructured":"Rockafellar, R.T., Royset, J.O.: Superquantile\/CVaR risk measures: second-order theory. Ann. Oper. Res. 262, 3\u201329 (2018)","journal-title":"Ann. Oper. Res."},{"key":"1462_CR20","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/s10107-014-0801-1","volume":"128","author":"RT Rockafellar","year":"2014","unstructured":"Rockafellar, R.T., Royset, J.O.: Random variables, monotone relations and convex analysis. Math. Program. B 128, 297\u2013331 (2014)","journal-title":"Math. Program. B"},{"key":"1462_CR21","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/s10107-018-1251-y","volume":"174","author":"RT Rockafellar","year":"2018","unstructured":"Rockafellar, R.T., Sun, J.: Soving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. B 174, 453\u2013471 (2018)","journal-title":"Math. Program. B"},{"key":"1462_CR22","doi-asserted-by":"publisher","unstructured":"Rockafellar, R.T., Sun, J.: Solving Lagrangian variational inequalities with applications to stochastic programming. Math. Program. (2020). https:\/\/doi.org\/10.1007\/s10107-019-01458-0","DOI":"10.1007\/s10107-019-01458-0"},{"key":"1462_CR23","doi-asserted-by":"publisher","first-page":"21","DOI":"10.21314\/JOR.2000.038","volume":"2","author":"RT Rockafellar","year":"2000","unstructured":"Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21\u201343 (2000)","journal-title":"J. Risk"},{"key":"1462_CR24","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.1016\/S0378-4266(02)00271-6","volume":"26","author":"RT Rockafellar","year":"2002","unstructured":"Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443\u20131471 (2002)","journal-title":"J. Bank. Finance"},{"key":"1462_CR25","first-page":"33","volume":"18","author":"RT Rockafellar","year":"2013","unstructured":"Rockafellar, R.T., Uryasev, S.: The fundamental risk quadrangle in risk management, optimization and statistical esimation. Surv. Oper. Res. Manag. Sci. 18, 33\u201353 (2013)","journal-title":"Surv. Oper. Res. Manag. Sci."},{"key":"1462_CR26","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s00780-005-0165-8","volume":"10","author":"RT Rockafellar","year":"2006","unstructured":"Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Finance Stoch. 10, 51\u201374 (2006)","journal-title":"Finance Stoch."},{"key":"1462_CR27","doi-asserted-by":"publisher","first-page":"170","DOI":"10.1007\/BFb0120750","volume":"6","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Nonanticipativity and $${\\cal{L}}^1$$-martingales in stochastic optimization problems. Stoch. Syst. Model. Identif. Optim. Math. Programm. Study 6, 170\u2013187 (1976)","journal-title":"Stoch. Syst. Model. Identif. Optim. Math. Programm. Study"},{"key":"1462_CR28","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1287\/moor.16.1.119","volume":"16","author":"RT Rockafellar","year":"1991","unstructured":"Rockafellar, R.T., 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":"1462_CR29","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/s10107-016-0995-5","volume":"165","author":"RT Rockafellar","year":"2017","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. B 165, 291\u2013330 (2017)","journal-title":"Math. Program. B"},{"key":"1462_CR30","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1007\/s10479-016-2354-6","volume":"260","author":"D Shang","year":"2016","unstructured":"Shang, D., Kuzmenko, V., Uryasev, S.: Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk. Ann. Oper. Res. 260, 501\u2013514 (2016)","journal-title":"Ann. Oper. Res."},{"key":"1462_CR31","unstructured":"Uryasev, S.: Buffered probability of exceedance and buffered service level: definitions and properties, Research Report 2014-3. ISE Department, University of Florida (2014)"},{"key":"1462_CR32","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s10287-010-0125-4","volume":"8","author":"J-P Watson","year":"2010","unstructured":"Watson, J.-P., Woodruff, D.L.: Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems. Comput. Manag. Sci. 8, 355\u2013370 (2010)","journal-title":"Comput. Manag. Sci."},{"key":"1462_CR33","unstructured":"Zhitlukhin, M.: Monotone Sharpe ratios and related measures of investment performance (2018). arXiv:1809.10193"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-019-01462-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-019-01462-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-019-01462-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T21:08:34Z","timestamp":1610485714000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-019-01462-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,13]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["1462"],"URL":"https:\/\/doi.org\/10.1007\/s10107-019-01462-4","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,13]]},"assertion":[{"value":"31 March 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}