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Value-at-Risk and Conditional Value-at-Risk in Optimization Under Uncertainty

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Uncertainty Management for Robust Industrial Design in Aeronautics

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 140))

Abstract

This work is related to the use of various risk measures in the context of robust- and reliability-based optimization. We start from the definition of risk measure and its formal setting, and then, we show how different risk functional definitions can lead to different approaches to the problem of optimization under uncertainty. In particular, the application of value-at-risk (VaR) and conditional value-at-risk (CVaR), also called quantiles and superquantiles, is here illustrated. These risk measures originated in the area of financial engineering, but they are very well and naturally suited to reliability-based design optimization problems and they represent a possible alternative to more traditional robust design approaches. We will then discuss the implementation of an efficient risk measure-based optimization algorithm based on the introduction of the weighted empirical cumulative distribution function (WECDF) and on the use of methods for changing the probability measure. Subsequently, we will discuss the problems related to the error in the estimation of the risk function and we will illustrate the “bootstrap” computational statistics technique to get an estimate of the standard error on VaR and CVaR. Finally, we will report some simple application examples of this approach to robust and reliability-based optimization.

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Authors and Affiliations

  1. Fluid Mechanics Department, C.I.R.A.—Italian Aerospace Research Centre, Via Maiorise, 81043, Capua, Italy

    Domenico Quagliarella

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  1. Domenico Quagliarella
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Correspondence to Domenico Quagliarella .

Editor information

Editors and Affiliations

  1. NUMECA International S.A., Brussels, Belgium

    Charles Hirsch

  2. NUMECA International S.A., Brussels, Belgium

    Dirk Wunsch

  3. Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Warsaw, Poland

    Jacek Szumbarski

  4. Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Warsaw, Poland

    Łukasz Łaniewski-Wołłk

  5. CIMNE International Centre for Numerical Methods in Engineering, Barcelona, Spain

    Jordi Pons-Prats

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Quagliarella, D. (2019). Value-at-Risk and Conditional Value-at-Risk in Optimization Under Uncertainty. In: Hirsch, C., Wunsch, D., Szumbarski, J., Łaniewski-Wołłk, Ł., Pons-Prats, J. (eds) Uncertainty Management for Robust Industrial Design in Aeronautics . Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-319-77767-2_34

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  • DOI: https://doi.org/10.1007/978-3-319-77767-2_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77766-5

  • Online ISBN: 978-3-319-77767-2

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