A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting

TL;DR

This paper establishes a continuum limit for non-dominated sorting in multi-objective optimization by linking it to a Hamilton-Jacobi equation, providing theoretical insights, numerical methods, and stability analysis.

Contribution

It introduces a novel continuum limit description of non-dominated sorting via a Hamilton-Jacobi equation and develops a numerical scheme for its solution.

Findings

Non-dominated sorting converges to a Hamilton-Jacobi PDE in the continuum limit.

The PDE solution provides a stable approximation under data perturbations.

Numerical simulations validate the theoretical framework.

Abstract

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.