Uncertainty Quantification of Pareto Fronts

Uncertainty quantification of Pareto fronts introduces new challenges connected to probabilities in infinite dimensional spaces. Indeed, Pareto fronts are, in general, manifolds belonging to infinite dimensional spaces: for instance, a curve in bi-objective optimization or a surface in three objective optimization. This article examines the methods for the determination of means, standard deviations and confidence intervals of Pareto fronts. We show that a punctual mean is not adapted and that the use of chaos expansions may lead to difficulties. Then we propose an approach based on a variational characterization of the mean and we show that it is effective to generate statistics of Pareto fronts. Finally, we examine the use of expansions to generate large samples and evaluate probabilities connected to Pareto fronts.