Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

We study statistical inference and robust solution methods for stochastic optimization prob-lems. We first develop an empirical likelihood framework for stochastic optimization. We showan empirical likelihood theory for Hadamard differentiable functionals with general f -divergencesand give conditions under which T (P ) = infx∈X EP [`(x; ξ)] is Hadamard differentiable. Notingthat the right endpoint of the generalized empirical likelihood confidence interval is a distribu-tionally robust optimization problem with uncertainty regions given by f -divergences, we showvarious statistical properties of robust optimization. First, we give a statistically principledmethod of choosing the size of the uncertainty set to obtain a calibrated one-sided confidenceinterval. Next, we give general conditions under which the robust solutions are consistent. Fi-nally, we prove an asymptotic expansion for the robust formulation, showing how robustificationregularizes the problem.