  We show that non-dominated sorting of a sequence of i.i.d. random variables
in Euclidean space has a continuum limit that corresponds to solving a
Hamilton-Jacobi equation involving the probability density function of the
random variables. Non-dominated sorting is a fundamental problem in
multi-objective optimization, and is equivalent to finding the canonical
antichain partition and to problems involving the longest chain among Euclidean
points. As an application of this result, we show that non-dominated sorting is
asymptotically stable under random perturbations in the data. We give a
numerical scheme for computing the viscosity solution of this Hamilton-Jacobi
equation and present some numerical simulations for various density functions.
