We show ways to leverage parametric information and Monte Carlo to obtain certified feasible solutions for chance-constrained problems under limited data. Our approach makes use of distributionally robust optimization (DRO) that translates the data size requirement in scenario optimization into a Monte Carlo size requirement drawn from a generating distribution. We show that, while the optimal choice for such distribution is in a sense, the baseline from a distance-based DRO, it is not necessarily so in the parametric case, and leads us to procedures that improve upon these basic choices.
joint work with Henry Lam
Optimization formulations to handle data-driven decision-making under uncertain constraints, such as robust optimization, often encounter a statistical trade-off between feasibility and optimality that potentially leads to over-conservative solutions. We exploit the intrinsic low dimensionality of the solution sets possibly output from these formulations to enhance this trade-off. For several common paradigms of data-driven optimization, we demonstrate the dimension-free performance of our strategy in obtaining solutions that are, in a sense, both feasible and asymptotically optimal.
joint work with Henry Lam
A bottleneck in analyzing extreme events is that, by their own nature, tail data are often scarce. Conventional approaches fit data using justified parametric distributions, but often encounter difficulties in confining both estimation biases and variances. We discuss approaches using distributionally robust optimization as a nonparametric alternative that, through a different conservativeness-variance tradeoff, can mitigate some of the statistical challenges in estimating tails. We explain the statistical connections and comparisons of our framework with conventional extreme value theory.