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

TL;DR

This paper presents a new direct verification proof for the continuum limit of nondominated sorting, a combinatorial algorithm, connecting it to the Hamilton-Jacobi equation without relying on variational principles.

Contribution

It introduces a novel proof method in homogenization theory that bypasses the variational approach, potentially applicable to other stochastic homogenization problems.

Findings

Established a direct verification proof for the Hamilton-Jacobi limit

Avoided the use of variational principles in the proof

Potentially generalized to other stochastic homogenization problems

Abstract

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.