Nonparametric Estimation of Uncertainty Sets for Robust Optimization

We investigate a data-driven approach to constructing uncertainty sets for robust optimization problems, where the uncertain problem parameters are modeled as random variables whose joint probability distributioyn is not known. Relying only on independent samples drawn from this distribution, we provide a nonparametric method to estimate uncertainty sets whose probability mass is guaranteed to approximate a given target mass within a given tolerance with high confidence. The nonparametric estimators that we consider are also shown to obey distribution-free finite-sample performance bounds that imply their convergence in probability to the given target mass. In addition to being efficient to compute, the proposed estimators result in uncertainty sets that yield computationally tractable robust optimization problems for a large family of constraint functions.