{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:37:36Z","timestamp":1787341056879,"version":"build-2736575974"},"reference-count":42,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["YIP FA- 9550-14-1-0059"],"award-info":[{"award-number":["YIP FA- 9550-14-1-0059"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-1453934"],"award-info":[{"award-number":["CMMI-1453934"]}],"id":[{"id":"10.13039\/100000001","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>A large class of stochastic programs involve optimizing an expectation taken with respect to an underlying distribution that is unknown in practice. One popular approach to addressing the distributional uncertainty, known as the distributionally robust optimization, is to hedge against the worst case over an uncertainty set of candidate distributions. However, it has been observed that inappropriate construction of the uncertainty set can sometimes result in overconservative solutions. To explore the middle ground between optimistically ignoring the distributional uncertainty and pessimistically fixating on the worst-case scenario, we propose a Bayesian risk optimization (BRO) framework for parametric underlying distributions, which is to optimize a risk functional applied to the posterior distribution of an unknown distribution parameter. Of our particular interest are four risk functionals: mean, mean-variance, value-at-risk, and conditional value-at-risk. To unravel the implication of BRO, we establish the consistency of objective functions and optimal solutions, as well as the asymptotic normality of objective functions and optimal values. More importantly, our analysis reveals a hidden interpretation: the objectives of BRO can be approximately viewed as a weighted sum of posterior mean objective and the (squared) half-width of the true objective's confidence interval.<\/jats:p>","DOI":"10.1137\/16m1101933","type":"journal-article","created":{"date-parts":[[2018,5,8]],"date-time":"2018-05-08T11:53:02Z","timestamp":1525780382000},"page":"1588-1612","source":"Crossref","is-referenced-by-count":31,"title":["A Bayesian Risk Approach to Data-driven Stochastic Optimization: Formulations and Asymptotics"],"prefix":"10.1137","volume":"28","author":[{"given":"Di","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helin","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5399-6508","authenticated-orcid":true,"given":"Enlu","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,5,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00068"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"G. Bayraksan and D. K. Love,\n                      Data-driven stochastic programming using phi-divergences\n                      , in The Operations Research Revolution, Tutor. Oper. Res., INFORMS, Catonsville, MD, 2015,https:\/\/doi.org\/10.1287\/educ.2015.0134.","DOI":"10.1287\/educ.2015.0134"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1120.1641"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1287\/moor.23.4.769"},{"key":"atypb5","unstructured":"D. Bertsimas, V. Gupta, and N. Kallus,\n                      Data-Driven Robust Optimization\n                      , preprint, arXiv:1401.0212, 2013."},{"key":"atypb6","first-page":"469","author":"Bertsimas D.","year":"2000","journal-title":"MA"},{"key":"atypb7","unstructured":"P. Billingsley,\n                      Convergence of Probability Measures\n                      , John Wiley & Sons, New York, 2013."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1214\/15-AOS1336"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1090.0741"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10463-016-0559-8"},{"key":"atypb11","first-page":"23","author":"Doob J. L.","year":"1949","journal-title":"Paris"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. Durrett,\n                      Probability: Theory and Examples\n                      , Cambridge University Press, Cambridge, 2010.","DOI":"10.1017\/CBO9780511779398"},{"key":"atypb13","unstructured":"P. M. Esfahani and D. Kuhn,\n                      Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations\n                      , preprint, arXiv:1505.05116, 2015."},{"key":"atypb14","unstructured":"R. Gao and A. J. Kleywegt,\n                      Distributionally Robust Stochastic Optimization with Wasserstein Distance\n                      , preprint, arXiv:1604.02199, 2016."},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"J.y. Gotoh, M. J. Kim, and A. Lim,\n                      Robust Empirical Optimization Is Almost the Same as Mean-Variance Optimization\n                      , 2015; available athttps:\/\/ssrn.com\/abstract=2827400.","DOI":"10.2139\/ssrn.2827400"},{"key":"atypb16","unstructured":"F. Hans and A. Schied,\n                      Stochastic Finance: An Introduction in Discrete Time\n                      , de Gruyter Stud. Math. 27, de Gruyter, Berlin, 2002."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"J.B. Hiriart-Urruty and C. Lemare\u0301chal,\n                      Fundamentals of Convex Analysis\n                      , Grundlehren Text Ed., Springer, Berlin, Heidelberg, 2001.","DOI":"10.1007\/978-3-642-56468-0"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0929-7"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1214\/009053604000000526"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623499363220"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2015.0776"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1060.0353"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1214\/12-AOS1004"},{"key":"atypb24","first-page":"33","volume":"18","author":"Rockafellar R. T.","year":"2013","journal-title":"Surv. Oper. Res. Manag. Sci."},{"key":"atypb25","unstructured":"H. L. Royden and P. Fitzpatrick,\n                      Real Analysis\n                      , 3rd ed., Macmillan, New York, 1988."},{"key":"atypb26","unstructured":"W. Rudin,\n                      Principles of Mathematical Analysis\n                      , 3rd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1964."},{"key":"atypb27","unstructured":"H. Scarf, K. Arrow, and S. Karlin,\n                      A min-max solution of an inventory problem\n                      , in Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Redwood City, CA, 1958, pp. 201-209."},{"key":"atypb28","unstructured":"M. J. Schervish,\n                      Theory of Statistics\n                      , Springer Ser. 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Bruckner,\n                      Elementary Real Analysis\n                      , 2nd ed., 2008; available athttp:\/\/classicalrealanalysis.info\/com\/Elementary-Real-Analysis.php."},{"key":"atypb36","unstructured":"A. W. Van der Vaart,\n                      Asymptotic Statistics\n                      , Camb. Ser. Stat. Probab. Math. 3, Cambridge University Press, Cambridge, 2000."},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1007\/s10287-015-0240-3"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2014.1314"},{"key":"atypb39","first-page":"423","volume":"91","author":"Z\u030ca\u0301c\u030ckova\u0301 J.","year":"1966","journal-title":"C\u030cas. Pe\u030cst. Mat."},{"key":"atypb40","first-page":"3714","author":"Zhou E.","year":"2015","journal-title":"NJ"},{"key":"atypb41","unstructured":"H. Zhu, J. Hale, and E. Zhou,\n                      Simulation optimization of risk measures with adaptive risk levels\n                      , preprint, arXiv:1608.00663, 2016."},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0494-7"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1101933","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:03:39Z","timestamp":1787339019000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1101933"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1101933"],"URL":"https:\/\/doi.org\/10.1137\/16m1101933","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}