Stochastic Programming Using Expected Value Bounds

In this article, we address the problem of minimizing an expected value with stochastic constraints, known in the literature as stochastic programming. Our approach is based on computing and optimizing bounds for the expected value that are obtained by solving a deterministic optimization problem that uses the probability density function (pdf) to penalize unlikely values for the random variables. The suboptimal solution obtained through this approach has performances guarantees with respect to the optimal one, while satisfying stochastic and deterministic constraints. We illustrate this approach in the context of the following three different classes of optimization problems: finite horizon optimal stochastic control, with state or output feedback; parameter estimation with latent variables; and nonlinear Bayesian experiment design. By the means of several numerical examples, we show that our suboptimal solution achieves results similar to those obtained with Monte Carlo methods with a fraction of the computational burden, highlighting the usefulness of this approach in real-time optimization problems.