Approximation of Stochastic Programming Problems

Summary

In Stochastic Programming, the aim is often the optimization of a criterion function that can be written as an integral or mean functional with respect to a probability measure $\mathbb{P}$. When this functional cannot be computed in closed form, it is customary to approximate it through an empirical mean functional based on a random Monte Carlo sample. Several improved methods have been proposed, using quasi-Monte Carlo samples, quadrature rules, etc. In this paper, we propose a result on the epigraphical approximation of an integral functional through an approximate one. This result allows us to deal with Monte Carlo, quasi-Monte Carlo and quadrature methods. We propose an application to the epi-convergence of stochastic programs approximated through the empirical measure based on an asymptotically mean stationary (ams) sequence. Because of the large scope of applications of ams measures in Applied Probability, this result turns out to be relevant for approximation of stochastic programs through real data.