Simulation optimization of conditional value-at-risk

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Conditional value-at-risk (CVaR) is a well-established tool for measuring risk. In this article, we consider solving CVaR optimization problems within a general simulation context. We derive an analytical expression for CVaR gradient and propose a simultaneous perturbation-type gradient estimator. This naturally results in a two-time-scale stochastic gradient algorithm for differentiable CVaR optimization. The algorithm is easily implementable and uses only three simulation evaluations at each iteration without requiring knowledge of the simulation model. We prove the almost sure local convergence of the algorithm and show that for the class of strongly convex problems, the mean absolute error of the sequence of solutions produced by the algorithm diminishes at a rate that is bounded from above by O ( k − 2 / 7 ) , where k is the number of iterations. Simulation experiments are also carried out to illustrate and evaluate the performance of the algorithm.