Mathematical Programming

, Volume 174, Issue 1–2, pp 525–552 | Cite as

Spectral risk measures: the risk quadrangle and optimal approximation

  • Drew P. KouriEmail author
Full Length Paper Series B


We develop 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, we introduce an optimal approximation of spectral risk measures using quadrature. We prove the consistency of this approximation and demonstrate our results through numerical examples.


Stochastic optimization Risk measures Regression Quadrature Average value-at-risk 

Mathematics Subject Classification

49J20 49J55 49K20 49K45 90C15 


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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2018

Authors and Affiliations

  1. 1.Optimization and Uncertainty Quantification, MS-1320Sandia National LaboratoriesAlbuquerqueUSA

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