The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective OneMinMax problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points -- a small constant factor more than the size of the Pareto front, as suggested for this algorithm -- computes the complete Pareto front of the 3-objective OneMinMax benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives.

翻译：非支配排序遗传算法II（NSGA-II）是实际应用中最突出的多目标进化算法。尽管该算法在双目标优化问题上表现出色，但实证研究表明，当应用于具有三个以上目标的问题时，其效率有所下降。最近一项数学运行时间分析通过证明NSGA-II在指数级迭代次数后仍无法覆盖简单三目标OneMinMax问题帕累托前沿的常数因子，验证了这一观察结果。本文首次对NSGA-III（为更好处理三个以上目标而对NSGA-II进行的改进）进行了数学运行时间分析。我们证明，当NSGA-III采用足够多的参考点（如该算法建议的，比帕累托前沿规模大一个小的常数因子）时，能够在期望O(n log n)次迭代内计算出三目标OneMinMax基准问题的完整帕累托前沿。该结论对所有不小于帕累托前沿规模的种群规模均成立，表明在此基准问题上NSGA-III相较于NSGA-II具有显著优势。本文及先前关于NSGA-II的研究中使用的数学论证表明，对于其他具有三个及以上目标的基准问题，极可能观察到类似的结论。