Spectral risk measures: the risk quadrangle and optimal approximation
Countless science and engineering decisions are formulated as optimization problems, many of which are riddled with uncertainty. For such problems, it is critical that the optimal solutions not only account for, but are resilient to this uncertainty. Risk measures provide a convenient utility for quantifying the hazard associated with random objectives and constraints. The risk quadrangle provides a rigorous connection between statistical estimation and risk-averse optimization by furnishing fundamental relationships connecting regular measures of risk, deviation, error and regret. The author develops a general risk quadrangle that gives rise to a large class of spectral risk measures. The statistic of this new risk quadrangle is the average value-at-risk at a specific confidence level. As such, this risk quadrangle generates a continuum of error measures that can be used for superquantile regression. For risk-averse optimization, an optimal approximation of spectral risk measures is introduced using quadrature. The consistency of this approximation is proved. Numerical examples are presented with a problem of optimal contaminant mitigation.
scientific article; zbMATH DE number 7047601
scientific article