Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

We study statistical inference and robust solution methods for stochastic optimization problems, focusing on giving calibrated and adaptive confidence intervals for optimal values and solutions for a range of stochastic problems. As part of this, we develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric $f$-divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide principled methods of choosing distributional uncertainty regions so as to provide calibrated one- and two-sided confidence intervals. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations.