In this paper, we study distributional robust optimization approaches for a one stage stochastic minimization problem, where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of worst possible distribution from that set. We consider the case when the distributional set is constructed through samples and investigate asymptotic convergence of optimal values and optimal solutions as sample size increases. The analysis provides a unified framework for asymptotic convergence of some data-driven problems and extends the classical asymptotic convergence analysis in stochastic programming. The discussion is extended to a stochastic Nash equilibrium problem where each player takes a robust action on the basis of their subjective expected objective value.
School of Engineering and Mathematical Sciences, City University of London, May, 2013.