Prof. Domenico Quagliarella, Italian Aerospace Research Center (CIRA)
Risk measures and optimization under uncertainty
Many industrial optimization processes must take account of the stochastic nature of the system and processes to be designed or re-designed and have to consider the random variability of some of the parameters that describe them. Thus it is necessary to characterize the system that is being studied from various points of view related to the treatment of uncertainty. This talk 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 functiontional 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. Finally 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.