{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:27:29Z","timestamp":1791505649679,"version":"4.3.4"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>This paper is meant to prove the consistency of the scenario approach \u00e0 la Calafiore, Campi, and Garatti with convex constraints. Scenario convex problems are usually stated in two equivalent forms: first, as the minimum of a linear function over the intersection of a finite random sample of independent and identically distributed convex sets, or second, as the min-max of a finite random sample of independent and identically distributed convex functions. The paper shows that, under fairly general assumptions, as the size of the sample increases the minimum attained by the solution of a problem of the first kind converges almost surely to the minimum attained by a suitably defined \u201cessential\u201d robust problem (or diverges if such a robust problem is infeasible), and that the minimum attained by the solution of a scenario problem of the second kind converges almost surely to the minimum of the pointwise essential supremum taken over all the possible convex functions (or diverges if such essential supremum takes the only value $+\\infty$). In both cases, if the solution of the essential problem exists and is unique, the solution of the scenario problem converges to it almost surely.<\/jats:p>","DOI":"10.1137\/16m109819x","type":"journal-article","created":{"date-parts":[[2018,1,11]],"date-time":"2018-01-11T13:32:37Z","timestamp":1515677557000},"page":"135-162","source":"Crossref","is-referenced-by-count":18,"title":["Consistency of the Scenario Approach"],"prefix":"10.1137","volume":"28","author":[{"given":"Federico Alessandro","family":"Ramponi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,1,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.23.4.769"},{"key":"atypb2","unstructured":"D. P. Bertsekas,\n                      Convex Optimization Theory\n                      , Athena Scientific, Belmont, MA, 2009."},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"G. C. Calafiore and M. C. Campi,\n                      The scenario approach to robust control design\n                      , IEEE Trans. Automat. Control, (2006), pp. 742-753.","DOI":"10.1109\/TAC.2006.875041"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/07069821X"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-010-9754-6"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"M. C. Campi and S. Garatti,\n                      Wait-and-judge scenario optimization\n                      , Math. Program., Sprnger, New York, 2016,https:\/\/doi.org\/10.1007\/s10107-016-1056-9.","DOI":"10.1007\/s10107-016-1056-9"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.arcontrol.2009.07.001"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"M. C. Campi, S. Garatti, and F. A. Ramponi,\n                      Non-convex scenario optimization with application to system identification\n                      , in Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, 2015.","DOI":"10.1109\/CDC.2015.7402845"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/130928546"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-011-9923-2"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623499363220"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. Lucchetti,\n                      Convexity and Well-Posed Problems\n                      , Springer, New York, 2006.","DOI":"10.1007\/0-387-31082-7"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2014.2330702"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01584661"},{"key":"atypb15","unstructured":"R. T. 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Shiryaev,\n                      Probability\n                      , Springer, New York, 1996.","DOI":"10.1007\/978-1-4757-2539-1"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M109819X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:14:57Z","timestamp":1787336097000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M109819X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M109819X"],"URL":"https:\/\/doi.org\/10.1137\/16m109819x","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}