Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty

Uncertain programs have been developed to deal with optimization problems including inexact data, i.e., uncertainty. A deterministic approach called robust optimization is commonly applied to solve these problems. Recently, Calafiore and Campi have proposed a randomized approach based on sampling of constraints, where the number of samples is determined so that only small portion of original constraints is violated at the randomized solution. Our main concern is not only the probability of violation, but also the degree of violation i.e., the worst-case violation. We derive an upper bound of the worst-case violation for the sampled convex programs and consider the relation between the probability of violation and worst-case violation. The probability of violation and the degree of violation are simultaneously bounded by small values, when the number of random samples is sufficiently large. Our method is applicable to not only a bounded uncertainty set but also an unbounded one such as Gaussian uncertain variables.