A Stochastic Approximation Method for Simulation-Based Quantile Optimization

Abstract

We present a gradient-based algorithm for solving a class of simulation optimization problems in which the objective function is the quantile of a simulation output random variable. In contrast with existing quantile (quantile derivative) estimation techniques, which aim to eliminate the estimator bias by gradually increasing the simulation sample size, our algorithm incorporates a novel recursive procedure that only requires a single simulation sample at each step to simultaneously obtain quantile and quantile derivative estimators that are asymptotically unbiased. We show that these estimators, when coupled with the standard gradient descent method, lead to a multi-time-scale stochastic approximation type of algorithm that converges to an optimal quantile value with probability one. In our numerical experiments, the proposed algorithm is applied to optimal investment portfolio problems, resulting in new solutions that complement those obtained under the classical Markowitz mean-variance framework.

Keywords: Quantile sensitivities, stochastic approximation, simulation optimization