Uncertainty in Sample Average Approximation

We consider stochastic optimization problems in which the input probability distribution is not fully known, and can only be observed through data. Common procedures handle such problems by optimizing an empirical counterpart, namely via using an empirical distribution of the input. The optimal solutions obtained through such procedures are hence subject to uncertainty of the data. In this paper, we explore techniques to quantify this uncertainty that have potentially good finite-sample performance. We consider three approaches: the empirical likelihood method, nonparametric Bayesian approach, and the bootstrap approach. They are designed to approximate the confidence intervals or posterior distributions of the optimal values or the optimality gaps. We present computational procedures for each of the approaches and discuss their relative benefits. A numerical example on conditional value-at-risk is used to demonstrate these methods.