{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T05:02:30Z","timestamp":1764997350834,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2021,2,28]],"date-time":"2021-02-28T00:00:00Z","timestamp":1614470400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,2,28]],"date-time":"2021-02-28T00:00:00Z","timestamp":1614470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SU 963\/1-1","SU 963\/2-1"],"award-info":[{"award-number":["SU 963\/1-1","SU 963\/2-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best H\u00f6lder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.<\/jats:p>","DOI":"10.1007\/s11590-021-01707-2","type":"journal-article","created":{"date-parts":[[2021,2,28]],"date-time":"2021-02-28T09:02:30Z","timestamp":1614502950000},"page":"2733-2756","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["On quantitative stability in infinite-dimensional optimization under uncertainty"],"prefix":"10.1007","volume":"15","author":[{"given":"M.","family":"Hoffhues","sequence":"first","affiliation":[]},{"given":"W.","family":"R\u00f6misch","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2473-4984","authenticated-orcid":false,"given":"T. M.","family":"Surowiec","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,28]]},"reference":[{"key":"1707_CR1","doi-asserted-by":"publisher","unstructured":"Alt, H.W.: Linear functional analysis. Universitext: an application-oriented introduction. Springer, London Ltd, London. Translated from the German edition by Robert N\u00fcrnberg (2016). https:\/\/doi.org\/10.1007\/978-1-4471-7280-2","DOI":"10.1007\/978-1-4471-7280-2"},{"key":"1707_CR2","series-title":"IMA Vol. Math. Appl.","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-1-4939-8636-1_1","volume-title":"Frontiers in PDE-constrained optimization","author":"H Antil","year":"2018","unstructured":"Antil, H., Leykekhman, D.: A brief introduction to PDE-constrained optimization. Frontiers in PDE-constrained optimization. IMA Vol. Math. Appl., vol. 163, pp. 3\u201340. Springer, New York (2018)"},{"key":"1707_CR3","series-title":"MPS\/SIAM Series on Optimization","volume-title":"Variational analysis in Sobolev and BV spaces","author":"H Attouch","year":"2006","unstructured":"Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces. MPS\/SIAM Series on Optimization, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2006)"},{"issue":"1","key":"1707_CR4","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1051\/m2an\/2016045","volume":"51","author":"M Bachmayr","year":"2017","unstructured":"Bachmayr, M., Cohen, A., Migliorati, G.: Sparse polynomial approximation of parametric elliptic PDES. PART 1: Affine coefficients. ESAIM: M2AN 51(1), 321\u2013339 (2017). https:\/\/doi.org\/10.1051\/m2an\/2016045","journal-title":"ESAIM: M2AN"},{"issue":"1","key":"1707_CR5","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1142\/S0219530511001728","volume":"9","author":"A Cohen","year":"2011","unstructured":"Cohen, A., Devore, R., Schwab, C.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE\u2019s. Anal. Appl. (Singapore) 9(1), 11\u201347 (2011). https:\/\/doi.org\/10.1142\/S0219530511001728","journal-title":"Anal. Appl. (Singapore)"},{"issue":"4","key":"1707_CR6","doi-asserted-by":"publisher","first-page":"1183","DOI":"10.1214\/12-AIHP489","volume":"49","author":"S Dereich","year":"2013","unstructured":"Dereich, S., Scheutzow, M., Schottstedt, R.: Constructive quantization: approximation by empirical measures. Ann. Inst. Henri Poincar\u00e9 Probab. Stat. 49(4), 1183\u20131203 (2013). https:\/\/doi.org\/10.1214\/12-AIHP489","journal-title":"Ann. Inst. Henri Poincar\u00e9 Probab. Stat."},{"key":"1707_CR7","series-title":"Springer Series in Operations Research and Financial Engineering","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-1037-3","volume-title":"Implicit functions and solution mappings: a view from variational analysis","author":"AL Dontchev","year":"2014","unstructured":"Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings: a view from variational analysis. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2014)","edition":"2"},{"key":"1707_CR8","series-title":"The Wadsworth & Brooks\/Cole Mathematics Series","volume-title":"Real analysis and probability","author":"RM Dudley","year":"1989","unstructured":"Dudley, R.M.: Real analysis and probability. The Wadsworth & Brooks\/Cole Mathematics Series. Wadsworth & Brooks\/Cole Advanced Books & Software, Pacific Grove, CA (1989)"},{"key":"1707_CR9","unstructured":"Dunford, N., Schwartz, J.T.: Linear operators. Part I. Wiley Classics Library. General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication. Wiley, New York (1988)"},{"key":"1707_CR10","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/0022-247X(73)90022-X","volume":"44","author":"RS Falk","year":"1973","unstructured":"Falk, R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28\u201347 (1973). https:\/\/doi.org\/10.1016\/0022-247X(73)90022-X","journal-title":"J. Math. Anal. Appl."},{"issue":"3\u20134","key":"1707_CR11","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1007\/s00440-014-0583-7","volume":"162","author":"N Fournier","year":"2015","unstructured":"Fournier, N., Guillin, A.: On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162(3\u20134), 707\u2013738 (2015). https:\/\/doi.org\/10.1007\/s00440-014-0583-7","journal-title":"Probab. Theory Related Fields"},{"key":"1707_CR12","doi-asserted-by":"crossref","unstructured":"Gajewski, H., Gr\u00f6ger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrb\u00fccher und Monographien. II, Abteilung, Mathematische Monographien, Band 38. Akademie-Verlag, Berlin (1974)","DOI":"10.1002\/mana.19750672207"},{"key":"1707_CR13","unstructured":"Hille, E., Phillips, R.S.: Functional analysis and semi-groups. vol. 31, Rev. ed. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1957)"},{"issue":"3","key":"1707_CR14","doi-asserted-by":"publisher","first-page":"865","DOI":"10.1137\/S1052623401383558","volume":"13","author":"M Hinterm\u00fcller","year":"2002","unstructured":"Hinterm\u00fcller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865\u2013888 (2002)","journal-title":"SIAM J. Optim."},{"key":"1707_CR15","series-title":"Mathematical Modelling: Theory and Applications","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-8839-1_3","volume-title":"Optimization with PDE constraints","author":"M Hinze","year":"2009","unstructured":"Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)"},{"issue":"1","key":"1707_CR16","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1137\/140954556","volume":"26","author":"DP Kouri","year":"2016","unstructured":"Kouri, D.P., Surowiec, T.M.: Risk-averse PDE-constrained optimization using the conditional value-at-risk. SIAM J. Optim. 26(1), 365\u2013396 (2016). https:\/\/doi.org\/10.1137\/140954556","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1707_CR17","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1137\/16M1086613","volume":"6","author":"DP Kouri","year":"2018","unstructured":"Kouri, D.P., Surowiec, T.M.: Existence and optimality conditions for risk-averse PDE-constrained optimization. SIAM\/ASA J. Uncertain. Quantif. 6(2), 787\u2013815 (2018). https:\/\/doi.org\/10.1137\/16M1086613","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"issue":"2","key":"1707_CR18","doi-asserted-by":"publisher","first-page":"429","DOI":"10.2307\/1428011","volume":"29","author":"A M\u00fcller","year":"1997","unstructured":"M\u00fcller, A.: Integral probability metrics and their generating classes of functions. Adv. Appl. Probab. 29(2), 429\u2013443 (1997)","journal-title":"Adv. Appl. Probab."},{"key":"1707_CR19","unstructured":"Rachev, S.T.: Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, Chichester (1991)"},{"issue":"4","key":"1707_CR20","doi-asserted-by":"publisher","first-page":"792","DOI":"10.1287\/moor.27.4.792.304","volume":"27","author":"ST Rachev","year":"2002","unstructured":"Rachev, S.T., R\u00f6misch, W.: Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27(4), 792\u2013818 (2002). https:\/\/doi.org\/10.1287\/moor.27.4.792.304","journal-title":"Math. Oper. Res."},{"key":"1707_CR21","series-title":"Springer Series in Statistics","doi-asserted-by":"publisher","DOI":"10.1007\/b98888","volume-title":"Functional data analysis","author":"JO Ramsay","year":"2005","unstructured":"Ramsay, J.O., Silverman, B.W.: Functional data analysis. Springer Series in Statistics, 2nd edn. Springer, New York (2005)","edition":"2"},{"key":"1707_CR22","doi-asserted-by":"publisher","unstructured":"R\u00f6misch, W.: Stability of stochastic programming problems. In: Stochastic programming, Handbooks Oper. Res. Management Sci., vol.\u00a010, pp. 483\u2013554. Elsevier, Amsterdam (2003). https:\/\/doi.org\/10.1016\/S0927-0507(03)10008-4","DOI":"10.1016\/S0927-0507(03)10008-4"},{"key":"1707_CR23","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1137\/1112027","volume":"12","author":"F Tops\u00f8e","year":"1967","unstructured":"Tops\u00f8e, F.: On the connection between P-continuity and P-uniformity in weak convergence. Probab. Theory Appl. 12, 281\u2013290 (1967)","journal-title":"Probab. Theory Appl."},{"issue":"3","key":"1707_CR24","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1137\/S1052623400371569","volume":"13","author":"M Ulbrich","year":"2002","unstructured":"Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13(3), 805\u2013841 (2002)","journal-title":"SIAM J. Optim."},{"key":"1707_CR25","doi-asserted-by":"publisher","unstructured":"Ulbrich, M.: Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces. MOS-SIAM Series on Optimization, vol.\u00a011. SIAM, MOS, Philadelphia, PA (2011). https:\/\/doi.org\/10.1137\/1.9781611970692","DOI":"10.1137\/1.9781611970692"},{"issue":"1","key":"1707_CR26","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1137\/070683416","volume":"47","author":"B Vexler","year":"2008","unstructured":"Vexler, B., Wollner, W.: Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47(1), 509\u2013534 (2008). https:\/\/doi.org\/10.1137\/070683416","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1707_CR27","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1137\/1128025","volume":"28","author":"VM Zolotarev","year":"1983","unstructured":"Zolotarev, V.M.: Probability metrics. Theory Probab. Theory Appl. 28(2), 278\u2013302 (1983)","journal-title":"Theory Probab. Theory Appl."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-021-01707-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-021-01707-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-021-01707-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T01:29:29Z","timestamp":1632446969000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-021-01707-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,28]]},"references-count":27,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["1707"],"URL":"https:\/\/doi.org\/10.1007\/s11590-021-01707-2","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"type":"print","value":"1862-4472"},{"type":"electronic","value":"1862-4480"}],"subject":[],"published":{"date-parts":[[2021,2,28]]},"assertion":[{"value":"22 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}