Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization
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Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.
KeywordsChance constraints Probabilistic constraints PDE constrained optimization
Mathematics Subject Classifications (2010)90C15 49J20
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The authors express their gratitude to two anonymous referees whose very careful reading and critical comments led to a substantially improved presentation of this paper.
- 11.Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics (2011)Google Scholar
- 15.Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics Volume: 19 (2010)Google Scholar
- 19.Henrion, R.: Qualitative stability of convex programs with probabilistic constraints. In: Nguyen, V.H., Strodiot, J.-J., Tossings, P. (eds.) Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 481, pp. 164–180. Springer, Berlin (2000)Google Scholar
- 31.Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, MPS-SIAM series on optimization 9 (2009)Google Scholar