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.

翻译：近年来，关于主流多目标进化算法如何优化多目标问题的理论研究取得了快速进展。然而，除针对特定交叉算子专门设计的基准函数外，交叉操作在多目标优化中的优势在理论上尚未被充分理解，其理论研究仍显著滞后于实际应用。本文延续这一研究方向，对经典$m$目标$m$-OneJumpZeroJump函数（简称$m$-OJZJ）上广泛使用的NSGA-III算法进行了理论运行时间分析。结果表明：在广泛参数区间内，对于任意目标数$m$，使用交叉的NSGA-III优化$m$-OJZJ的渐进速度均优于不使用交叉的NSGA-III。我们通过提供禁用交叉时$4$-OJZJ的下界运行时间，进一步补充验证了这一分析结果。