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Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

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Abstract

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.

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Acknowledgments

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.

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Correspondence to R. Henrion.

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This research was partially carried out in the framework of MATHEON supported by the Einstein Foundation Berlin within the ECMath project SE13 as well as within project B04 of the Sonderforschungsbereich / Transregio 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks funded by Deutsche Forschungsgemeinschaft.

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Farshbaf-Shaker, M.H., Henrion, R. & Hömberg, D. Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization. Set-Valued Var. Anal 26, 821–841 (2018). https://doi.org/10.1007/s11228-017-0452-5

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  • DOI: https://doi.org/10.1007/s11228-017-0452-5

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