{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:38:59Z","timestamp":1787341139804,"version":"build-2736575974"},"reference-count":48,"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":[[2014,1]]},"abstract":"<jats:p>We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one-stage and two-stage stochastic minimization problems, two-stage stochastic mathematical programs with equilibrium constraints, and stochastic programs with second order dominance constraints.<\/jats:p>","DOI":"10.1137\/120880434","type":"journal-article","created":{"date-parts":[[2014,3,20]],"date-time":"2014-03-20T11:35:47Z","timestamp":1395315347000},"page":"467-497","source":"Crossref","is-referenced-by-count":13,"title":["Quantitative Stability Analysis of Stochastic Generalized Equations"],"prefix":"10.1137","volume":"24","author":[{"given":"Yongchao","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Werner","family":"R\u00f6misch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huifu","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,3,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(88)90041-3"},{"key":"atypb2","unstructured":"J.P. Aubin and H. Frankowska,<i>Set-Valued Analysis<\/i> Birkhauser, Boston, 1990."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(65)90049-1"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"J. F. Bonnans and A. Shapiro,<i>Perturbation Analysis of Optimization Problems<\/i> Springer Ser. Oper. Res., Springer-Verlag, Berlin, 2000.","DOI":"10.1007\/978-1-4612-1394-9"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0180884-9"},{"key":"atypb6","unstructured":"J. M. Borwein and Q. J. Zhu,<i>Techniques of Variational Analysis<\/i> Springer, New York, 2005."},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"C. Castaing and M. Valadier,<i>Convex Analysis and Measurable Multifunctions<\/i> Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.","DOI":"10.1007\/BFb0087685"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1050.0160"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/110825248"},{"key":"atypb10","unstructured":"F. H. Clarke,<i>Optimization and Nonsmooth Analysis<\/i> Wiley, New York, 1983."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623402420528"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-003-0453-z"},{"key":"atypb13","first-page":"1","volume":"36","author":"Dentcheva D.","year":"2007","journal-title":"Control Cybernet."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-02-03088-X"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"A. L. Dontchev and R. T. Rockafellar<i>Implicit Functions and Solution Mappings: A View from Variational Analysis<\/i> Springer, New York, 2009.","DOI":"10.1007\/978-0-387-87821-8"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"F. Facchinei and J. S. Pang,<i>Finite-dimensional Variational Inequalities and Complementarity Problems<\/i> Springer, New York, 2003.","DOI":"10.1007\/b97543"},{"key":"atypb17","first-page":"617","author":"Hess C.","year":"2002","journal-title":"Amsterdam"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1985-0800254-X"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0586-z"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581199"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0121218"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-012-0585-0"},{"key":"atypb23","first-page":"609","volume":"21","author":"Liu Y.","year":"2011","journal-title":"SIAM J. Optim."},{"key":"atypb24","unstructured":"I. Molchanov,<i>Theory of Random Sets<\/i> in Probability and Its Applications, J. Gani et al., eds., Springer, New York, 2005."},{"key":"atypb25","unstructured":"F. No\u017ei\u010dka, J. Guddat, H. Hollatz, and B. Bank,<i>Theorie der Linearen Parametrischen Optimierung<\/i> Akademie-Verlag, Berlin, 1974."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"J. Outrata, M. Kocvara, and J. Zowe,<i>Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results<\/i> Kluwer Academic Publishers, Boston, 1998.","DOI":"10.1007\/978-1-4757-2825-5"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(85)90078-8"},{"key":"atypb28","unstructured":"S. T. Rachev,<i>Probability Metrics and the Stability of Stochastic Models<\/i> Wiley, Chichester, UK, 1991."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1287\/moor.27.4.792.304"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0506"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/100792644"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1.2.130"},{"key":"atypb33","first-page":"346","author":"Robinson S. M.","year":"1983","journal-title":"Berlin"},{"key":"atypb34","unstructured":"R. T. Rockafellar,<i>Convex Analysis<\/i> Princeton University Press, Princeton, NJ, 1970."},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"R. T. Rockafellar and R. J.B. Wets,<i>Variational Analysis<\/i> Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-642-02431-3"},{"key":"atypb36","first-page":"483","author":"R\u00f6misch W.","year":"2003","journal-title":"Amsterdam"},{"key":"atypb37","first-page":"353","author":"Shapiro A.","year":"2003","journal-title":"Amsterdam"},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"A. Shapiro, D. Dentcheva, and A. Rusczy\u0144ski,<i>Lecture Notes on Stochastic Programming: Modelling and Theory<\/i> MPS-SIAM Ser. Optim., Philadelphia, 2009.","DOI":"10.1137\/1.9780898718751"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1080\/02331930801954177"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0711-7"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/1112027"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1969.28.465"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/1016053"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1137\/040608544"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.03.021"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1142\/S0217595910002569"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(94)90120-1"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/1128025"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120880434","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:07:19Z","timestamp":1787339239000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120880434"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/120880434"],"URL":"https:\/\/doi.org\/10.1137\/120880434","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}