A direct verification argument for the Hamilton-Jacobi equation continuum limit of nondominated sorting

Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. The original proof relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this proof is new in the homogenization literature, and may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle.