An Ergodic Theorem for Random Lagrangians with an Application to Stochastic Programming

Abstract

We prove an ergodic theorem showing the almost sure epi/hypo-convergence of a sequence of random lagrangians to a limit lagrangian where the random lagrangians are generated by stationary sampling of a probability measure. We apply this theorem to stochastic programming and demonstrate that the outer set-limit of the sequence of the set of saddle points from the sampled problems is a subset of the set of saddle points of the true problem.