Simulation optimization of risk measures with adaptive risk levels

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.