A Newton Method for Hausdorff Approximations of the Pareto Front within Multi-objective Evolutionary Algorithms

TL;DR

This paper introduces a set-based Newton method for Hausdorff approximations of the Pareto front, enhancing multi-objective evolutionary algorithms by improving approximation quality with a novel post-processing approach.

Contribution

It proposes a new Newton method for Hausdorff approximations, generalizes previous Newton steps for constrained problems, and integrates a data-driven reference set strategy within evolutionary algorithms.

Findings

The Newton method improves Pareto front approximations.

The approach enhances existing evolutionary algorithms.

Post-processing with the Newton method yields better results on benchmarks.

Abstract

A common goal in evolutionary multi-objective optimization is to find suitable finite-size approximations of the Pareto front of a given multi-objective optimization problem. While many multi-objective evolutionary algorithms have proven to be very efficient in finding good Pareto front approximations, they may need quite a few resources or may even fail to obtain optimal or nearly approximations. Hereby, optimality is implicitly defined by the chosen performance indicator. In this work, we propose a set-based Newton method for Hausdorff approximations of the Pareto front to be used within multi-objective evolutionary algorithms. To this end, we first generalize the previously proposed Newton step for the performance indicator for the treatment of constrained problems for general reference sets. To approximate the target Pareto front, we propose a particular strategy for generating the reference set that utilizes the data gathered by the evolutionary algorithm during its run. Finally, we show the benefit of the Newton method as a post-processing step on several benchmark test functions and different base evolutionary algorithms.