Journal of Global Optimization

, Volume 70, Issue 4, pp 783–809 | Cite as

Simulation optimization of risk measures with adaptive risk levels

  • Helin Zhu
  • Joshua Hale
  • Enlu Zhou


Optimizing risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a general loss distribution is usually difficult, because (1) the loss function might lack structural properties such as convexity or differentiability since it is often generated via black-box simulation of a stochastic system; (2) evaluation of risk measures often requires rare-event simulation, which is computationally expensive. In this paper, we study the extension of the recently proposed gradient-based adaptive stochastic search to the optimization of risk measures VaR and CVaR. Instead of optimizing VaR or CVaR at the target risk level directly, we incorporate an adaptive updating scheme on the risk level, by initializing the algorithm at a small risk level and adaptively increasing it until the target risk level is achieved while the algorithm converges at the same time. This enables us to adaptively reduce the number of samples required to estimate the risk measure at each iteration, and thus improving the overall efficiency of the algorithm.


Simulation optimization Risk measures Black-box simulation Adaptive risk level 



This work was supported by National Science Foundation under Grants CMMI-1413790 and CAREER CMMI-1453934, and Air Force Office of Scientific Research under Grant YIP FA-9550-14-1-0059.


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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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