Bayesian nonparametric set construction for robust optimization

This paper presents a Bayesian nonparametric, data-driven, nonconvex uncertainty set construction for robust optimization. First, a basic uncertainty set is constructed from a union of posterior predictive ellipsoids for the Dirichlet process Gaussian mixture. The robustification of linear optimization problems using this set is proven to be a tractable second order cone problem with probabilistic feasibility guarantees. Noting that this basic set is typically overly conservative, a scaled version of the set is obtained via stochastic bisection search, and convergence guarantees to the least conservative scaled set for a particular probabilistic guarantee are provided. Experiments on synthetic linear programs and a Mobility on Demand system design problem demonstrate that the proposed set improves upon the robust optimal objective over simpler uncertainty sets, more accurately achieves the desired level of conservatism, and requires little design input from the user.