Illustrating the Efficiency of Popular Evolutionary Multi-Objective Algorithms Using Runtime Analysis

TL;DR

This paper demonstrates that popular evolutionary multi-objective algorithms like NSGA-II, NSGA-III, and SMS-EMOA significantly outperform the simple GSEMO algorithm on a specially designed problem, showcasing their efficiency through rigorous runtime analysis.

Contribution

The paper introduces a new problem, OneTrapZeroTrap, and proves that popular EMO algorithms have a polynomial runtime advantage over GSEMO on this problem, highlighting their superior efficiency.

Findings

GSEMO requires at least n^n evaluations to solve OneTrapZeroTrap.

NSGA-II, NSGA-III, and SMS-EMOA solve it in O(n log n) evaluations.

Diversity mechanisms are crucial for the efficiency of these algorithms.

Abstract

Runtime analysis has recently been applied to popular evolutionary multi-objective (EMO) algorithms like NSGA-II in order to establish a rigorous theoretical foundation. However, most analyses showed that these algorithms have the same performance guarantee as the simple (G)SEMO algorithm. To our knowledge, there are no runtime analyses showing an advantage of a popular EMO algorithm over the simple algorithm for deterministic problems. We propose such a problem and use it to showcase the superiority of popular EMO algorithms over (G)SEMO: OneTrapZeroTrap is a straightforward generalization of the well-known Trap function to two objectives. We prove that, while GSEMO requires at least $n^n$ expected fitness evaluations to optimise OneTrapZeroTrap, popular EMO algorithms NSGA-II, NSGA-III and SMS-EMOA, all enhanced with a mild diversity mechanism of avoiding genotype duplication, only require $O(n \log n)$ expected fitness evaluations. Our analysis reveals the importance of the key components in each of these sophisticated algorithms and contributes to a better understanding of their capabilities.