Optimization-based Estimation of Expected Values with Application to Stochastic Programming

This paper constructs bounds on the expected value of a scalar function of a random vector. The bounds are obtained using an optimization method, which can be computed efficiently using state-of-the-art solvers, and do not require integration or sampling the random vector. This optimization based approach is especially useful in stochastic programming, where the criteria to be minimized takes the form of an expected value. In particular, we minimize the bounds to solve problems of discrete time finite horizon open-loop control with stochastic perturbations and also uncertainty in the system's parameters. We illustrate this application with two numerical examples.