Rates of convergence for the continuum limit of nondominated sorting

TL;DR

This paper provides quantitative error estimates for how quickly nondominated sorting of random points converges to its continuum limit described by a Hamilton-Jacobi equation, using advanced mathematical tools.

Contribution

It introduces new semiconvexity estimates and applies the maximum principle to establish convergence rates for nondominated sorting's continuum limit.

Findings

Quantitative error bounds for convergence to the continuum limit.

Use of viscosity solutions and maximum principle in the analysis.

New semiconvexity estimates for domains with corners.

Abstract

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order, and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton-Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities.