Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

We study statistical inference and robust solution methods for stochastic optimization problems. We first develop an generalized empirical likelihood framework for stochastic optimization. We show an empirical likelihood theory for Hadamard differentiable functionals with general $f$-divergences and give conditions under which $T(P) = \inf_{x\in\mathcal{X}} \mathbb{E}_{P}[l(x; \xi)]$ is Hadamard differentiable. Noting that the right endpoint of the generalized empirical likelihood confidence interval is a distributionally robust optimization problem with uncertainty regions given by $f$-divergences, we show various statistical properties of robust optimization. First, we give a statistically principled method of choosing the size of the uncertainty set to obtain a \texit{calibrated} one-sided confidence interval. Next, we prove an asymptotic expansion for the robust formulation, showing how robustification is a variance regularization. Finally, we show that robust solutions are consistent under (essentially) identical conditions as that required for sample average approximations.