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进化多目标算法。NSGA-II适用于处理少量目标（如2个和3个目标），而NSGA-III则专为应对大量目标问题而设计。在最近的一项突破性研究中，Wietheger与Doerr（IJCAI 2023）首次针对3目标OneMinMax问题开展了NSGA-III的运行时间分析，证明这一前沿算法可被严谨分析。我们推进了这一新兴研究方向，首次对NSGA-III在流行多目标基准问题mLOTZ、mOMM和mCOCZ上进行了运行时间分析，其中目标数量$m$为任意常数。我们的分析为算法关键参数（参考点数量与种群规模）的设置提供了依据，从而确保算法性能得到保障。我们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。据我们所知，这是首次针对超过3个目标的NSGA-III运行时间分析。