{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T19:44:32Z","timestamp":1791920672579,"version":"4.3.4"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["SNL 014150709"],"award-info":[{"award-number":["SNL 014150709"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-12-1-0427"],"award-info":[{"award-number":["FA9550-12-1-0427"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-11-1-0258"],"award-info":[{"award-number":["FA9550-11-1-0258"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006602","name":"Air Force Research Laboratory","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100006602","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>This paper studies a probabilistic characteristic called buffered probability of exceedance (bPOE). It is a function of a random variable and a real-valued threshold. By definition, bPOE is the probability of a tail such that the average of this tail equals the threshold. This characteristic is an extension of the so-called buffered failure probability and it is equal to one minus inverse of the conditional value-at-risk (CVaR). bPOE is a quasi-convex function of the random variable w.r.t. the regular addition operation and a concave function w.r.t. the mixture operation; it is a monotonic function of the random variable; it is a strictly decreasing function of the threshold on the interval between the expectation and the essential supremum. The multiplicative inverse of the bPOE is a convex function of the threshold, and a piecewise-linear function in the case of discretely distributed random variable. The paper provides efficient calculation formulas for bPOE. Minimization of bPOE is reduced to a convex program for a convex feasible region and to linear programming for a polyhedral feasible region and discretely distributed random variables. A family of bPOE minimization problems and corresponding CVaR minimization problems share the same set of optimal solutions.<\/jats:p>","DOI":"10.1137\/15m1042644","type":"journal-article","created":{"date-parts":[[2018,4,10]],"date-time":"2018-04-10T14:48:58Z","timestamp":1523371738000},"page":"1077-1103","source":"Crossref","is-referenced-by-count":52,"title":["Buffered Probability of Exceedance: Mathematical Properties and Optimization"],"prefix":"10.1137","volume":"28","author":[{"given":"Alexander","family":"Mafusalov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stan","family":"Uryasev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,4,10]]},"reference":[{"key":"atypb1","first-page":"1","author":"Davis J. R.","year":"2015","journal-title":"Nat. Hazards"},{"key":"atypb2","unstructured":"E. Delage, D. Kuhn, and W. Wiesemann,\n                      ``Dice''-sion Making under Uncertainty: When Can a Random Decision Reduce Risk?\n                      , 2016,http:\/\/www.optimization-online.org\/DB_FILE\/2016\/08\/5582.pdf."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2011.11.044"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623402420528"},{"key":"atypb5","first-page":"4","volume":"57","author":"Dentcheva D.","year":"2004","journal-title":"C. R. Acad. Bulgare Sci."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s007800200072"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"H. Fo\u0308llmer and A. Schied,\n                      Stochastic Finance: An Introduction in Discrete Time\n                      , Walter de Gruyter, Berlin, 2011.","DOI":"10.1515\/9783110218053"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-011-9752-8"},{"key":"atypb9","unstructured":"K. Miettinen,\n                      Nonlinear Multiobjective Optimization\n                      , Vol. 12, Springer, New York, 2012."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/050622328"},{"key":"atypb11","unstructured":"M. Norton, A. Mafusalov, and S. Uryasev,\n                      Cardinality of Upper Average and Application to Network Optimization\n                      , Research report 2015-1, ISE Department, University of Florida, 2015."},{"key":"atypb12","first-page":"1","volume":"18","author":"Norton M.","year":"2017","journal-title":"J. Mach. Learn. Res."},{"key":"atypb13","unstructured":"M. Norton and S. Uryasev,\n                      Maximization of AUC and Buffered AUC in Classification\n                      , Research report 2014-2, ISE Department, University of Florida, 2014."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623400375075"},{"key":"atypb15","unstructured":"I. Olkin and A. W. Marshall,\n                      Inequalities: Theory of Majorization and its Applications\n                      , Vol. 143, Academic Press, New York, 2016."},{"key":"atypb16","unstructured":"K. Pavlikov and S. Uryasev,\n                      CVaR Distance Between Univariate Probability Distributions and Approximation Problems\n                      , Research report 2015-6, ISE Department, University of Florida, 2015."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"R. T. Rockafellar,\n                      Conjugate Duality and Optimization\n                      , SIAM, Philadelphia, 1974.","DOI":"10.1137\/1.9781611970524"},{"key":"atypb18","unstructured":"R. T. Rockafellar,\n                      Convex Analysis\n                      , Princeton Landmarks Math., Princeton University Press, Princeton, NJ, 1997."},{"key":"atypb19","unstructured":"R. T. Rockafellar,\n                      Safeguarding strategies in risky optimization\n                      , in Proceedings of the International Workshop on Engineering Risk Control and Optimization, Gainesville, FL, 2009."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2010.01.001"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0801-1"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4266(02)00271-6"},{"key":"atypb23","first-page":"18","author":"Rockafellar R. T.","year":"2013","journal-title":"Surv. Oper. Res. Manag. Sci."},{"key":"atypb24","first-page":"92","volume":"117","author":"Ryff J. V.","year":"1965","journal-title":"Trans. Amer. Math. Soc."},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"M. Shaked and J. G. Shanthikumar,\n                      Stochastic Orders\n                      , Springer, New York, 2007.","DOI":"10.1007\/978-0-387-34675-5"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-016-2354-6"},{"key":"atypb27","unstructured":"S. Uryasev,\n                      Buffered Probability of Exceedance and Buffered Service Level: Definitions and Properties\n                      , Research report 2014-3, ISE Department, University of Florida, 2014."},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"M. Zabarankin and S. Uryasev,\n                      Statistical Decision Problems\n                      , Springer, New York, 2014.","DOI":"10.1007\/978-1-4614-8471-4"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1042644","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:02:53Z","timestamp":1787338973000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1042644"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/15M1042644"],"URL":"https:\/\/doi.org\/10.1137\/15m1042644","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}