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。虽然它在双目标优化问题上表现明显良好，但经验研究表明，当应用于具有两个以上目标的问题时，其效果较差。最近的一项数学运行时间分析通过证明NGSA-II在指数次迭代中缺失了简单3目标OneMinMax问题帕累托前沿的常数因子，证实了这一观察结果。在本工作中，我们首次对NSGA-III进行了数学运行时间分析，NSGA-III是NSGA-II的改进版本，旨在更好地处理两个以上的目标。我们证明，具有足够多参考点（如该算法所建议的，比帕累托前沿大小多一个小的常数因子）的NSGA-III，在期望O(n log n)次迭代中计算出3目标OneMinMax基准的完整帕累托前沿。这一结果适用于所有种群规模（至少等于帕累托前沿大小）。它表明在该基准上NSGA-III相比NSGA-II具有显著优势。本文及先前关于NSGA-II工作中使用的数学论证表明，其他具有三个或更多目标的基准可能也会出现类似发现。