High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

TL;DR

This paper develops high-order filtered finite difference schemes for the Hamilton-Jacobi equation arising from nondominated sorting, ensuring stability, convergence, and improved accuracy in smooth regions, with applications in multi-objective optimization.

Contribution

It introduces a novel filtered scheme approach that combines high-order and monotone schemes for the Hamilton-Jacobi equation in nondominated sorting, guaranteeing stability and convergence.

Findings

Filtered schemes are stable and converge to the viscosity solution.

Numerical simulations demonstrate the schemes' improved accuracy.

The approach is applicable to multi-objective optimization problems.

Abstract

We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes.