Scalability of evolutionary algorithms refers to assessing how their performance changes as problem size increases. In the area of multi-objective optimisation, research on the scalability of multi-objective evolutionary algorithms (MOEAs) has predominantly focussed on continuous problems. However, multi-objective combinatorial optimisation problems (MOCOPs) differ from continuous ones. Their discrete and rigid structure often brings rugged landscape, numerous local optimal solutions and disjoint global optimal regions. This leads to different behaviour of MOEAs. For example, SEMO, a simple MOEA without mating selection and diversity maintenance mechanisms, has been shown to be highly competitive, and in many cases to outperform more sophisticated MOEAs on MOCOPs. Yet, it remains unclear whether such findings hold for large-scale cases. In this paper, we conduct an empirical investigation into the scalability of MOEAs on combinatorial problems, with problem size from 50 to 5,000. Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.

翻译：摘要：进化算法的可扩展性指的是衡量其性能如何随问题规模增大而变化。在多目标优化领域，关于多目标进化算法（MOEA）可扩展性的研究主要集中在连续问题上。然而，多目标组合优化问题（MOCOP）与连续问题存在差异。其离散且刚性的结构通常导致崎岖的搜索景观、大量局部最优解以及不连通的全局最优区域，这使得MOEA表现出不同的行为。例如，SEMO（一种缺乏交配选择和多样性维持机制的简单MOEA）已被证明在MOCOP上具有很强的竞争力，且在多种情况下优于更复杂的MOEA。然而，这些发现是否适用于大规模问题尚不清楚。本文针对组合问题，对问题规模从50到5,000的MOEA可扩展性进行了实证研究。结果表明，与NSGA-II、SMS-EMOA和MOEA/D等其他MOEA相比，SEMO的收敛速度随维度增加而下降。我们进一步证明，缺乏交叉操作是SEMO在大规模问题上性能不佳的主要原因，而将交叉引入SEMO通常可以显著加速收敛，尽管这会在帕累托前沿上分散解集时产生负面影响。