A Crowding Distance That Provably Solves the Difficulties of the NSGA-II in Many-Objective Optimization

TL;DR

This paper introduces a modified crowding distance for NSGA-II, called truthful crowding distance, which provably improves its performance in many-objective optimization problems, matching the efficiency of NSGA-III and SMS-EMOA.

Contribution

The paper proposes a new crowding distance variant that ensures better scalability of NSGA-II in many-objective optimization, backed by mathematical runtime analyses.

Findings

NSGA-II with truthful crowding distance solves many-objective problems in polynomial time.

It performs comparably to NSGA-III and SMS-EMOA in complex scenarios.

In bi-objective cases, it benefits from smaller population sizes and slightly better Pareto front approximation.

Abstract

Recent theoretical works have shown that the NSGA-II can have enormous difficulties to solve problems with more than two objectives. In contrast, algorithms like the NSGA-III or SMS-EMOA, differing from the NSGA-II only in the secondary selection criterion, provably perform well in these situations. To remedy this shortcoming of the NSGA-II, but at the same time keep the advantages of the widely accepted crowding distance, we use the insights of these previous work to define a variant of the crowding distance, called truthful crowding distance. Different from the classic crowding distance, it has for any number of objectives the desirable property that a small crowding distance value indicates that some other solution has a similar objective vector. Building on this property, we conduct mathematical runtime analyses for the NSGA-II with truthful crowding distance. We show that this algorithm can solve the many-objective versions of the OneMinMax, COCZ, LOTZ, and OJZJ$_k$ problems in the same (polynomial) asymptotic runtimes as the NSGA-III and the SMS-EMOA. This contrasts the exponential lower bounds previously shown for the classic NSGA-II. For the bi-objective versions of these problems, our NSGA-II has a similar performance as the classic NSGA-II, gaining however from smaller admissible population sizes. For the bi-objective OneMinMax problem, we also observe a (minimally) better performance in approximating the Pareto front. These results suggest that our truthful version of the NSGA-II has the same good performance as the classic NSGA-II in two objectives, but can resolve the drastic problems in more than two objectives.