On the Impact of Crossover in Many-Objective Optimization: A Runtime Analysis of NSGA-III

TL;DR

This paper provides a theoretical runtime analysis of NSGA-III, showing that incorporating crossover significantly improves optimization speed on many-objective problems, supported by bounds on specific benchmark functions.

Contribution

It offers the first theoretical analysis demonstrating the benefits of crossover in NSGA-III for many-objective optimization, with bounds on specific functions.

Findings

NSGA-III with crossover outperforms without crossover on m-Objective m-OJZJ.

Crossover leads to asymptotically faster optimization for large parameter regimes.

Lower bounds established for runtime without crossover on 4-OJZJ.

Abstract

In recent years, a theoretical understanding has rapidly advanced regarding how popular multi-objective evolutionary algorithms (MOEAs) can optimize many-objective problems. However, the benefits of using crossover in many-objective optimization are theoretically not understood, except for specifically designed benchmark functions tuned to particular crossover operators, and still lag significantly behind its practical use. In this paper, we build upon this line of research and present a theoretical runtime analysis of the widely used NSGA-III algorithm on the classical $m$-objective $m$-OneJumpZeroJump function ($m$-OJZJ for short). Our results demonstrate that NSGA-III with crossover optimizes $m$-OJZJ asymptotically faster than NSGA-III without crossover for any number $m$ of objectives for huge parameter regimes. We complement our analysis by providing a lower runtime bound on $4$-OJZJ when crossover is turned off.