A rate result for simulation optimization with conditional value-at-risk constraints

We study a stochastic optimization problem that has its roots in financial portfolio design. The problem has a specified deterministic objective function and constraints on the conditional value-at-risk of the portfolio. Approximate optimal solutions to this problem are usually obtained by solving a sample-average approximation. We derive bounds on the gap in the objective value between the true optimal and an approximate solution so obtained. We show that under certain regularity conditions the approximate optimal value converges to the true optimal at the canonical rate O(n^-1/2), where n represents the sample size. The constants in the expression are explicitly defined.