NSGA-II and NSGA-III are two of the most popular evolutionary multi-objective algorithms used in practice. While NSGA-II is used for few objectives such as 2 and 3, NSGA-III is designed to deal with a larger number of objectives. In a recent breakthrough, Wietheger and Doerr (IJCAI 2023) gave the first runtime analysis for NSGA-III on the 3-objective OneMinMax problem, showing that this state-of-the-art algorithm can be analyzed rigorously. We advance this new line of research by presenting the first runtime analyses of NSGA-III on the popular many-objective benchmark problems mLOTZ, mOMM, and mCOCZ, for an arbitrary constant number $m$ of objectives. Our analysis provides ways to set the important parameters of the algorithm: the number of reference points and the population size, so that a good performance can be guaranteed. We show how these parameters should be scaled with the problem dimension, the number of objectives and the fitness range. To our knowledge, these are the first runtime analyses for NSGA-III for more than 3 objectives.

翻译：NSGA-II和NSGA-III是实践中应用最广泛的两种多目标演化算法。NSGA-II适用于2至3个目标的低维优化问题，而NSGA-III则专门设计用于处理更多目标的情形。在近期一项突破性工作中，Wietheger与Doerr（IJCAI 2023）首次对NSGA-III在三维OneMinMax问题上进行了运行时间分析，验证了该前沿算法可被严格分析。我们通过呈现NSGA-III在经典多目标基准问题mLOTZ、mOMM及mCOCZ上的首次运行时间分析，进一步拓展了这一新兴研究方向，其中目标数$m$为任意常数。我们的分析提供了算法关键参数（参考点数量与种群规模）的设置方法，从而确保其优异性能。研究揭示了这些参数应如何随问题维度、目标数量及适应度范围进行缩放调整。据我们所知，这是NSGA-III在超过三维目标情形下首次得到的运行时间分析结果。