The non-dominated sorting genetic algorithm II (NSGA-II) is the most intensively used multi-objective evolutionary algorithm (MOEA) in real-world applications. However, in contrast to several simple MOEAs analyzed also via mathematical means, no such study exists for the NSGA-II so far. In this work, we show that mathematical runtime analyses are feasible also for the NSGA-II. As particular results, we prove that with a population size four times larger than the size of the Pareto front, the NSGA-II with two classic mutation operators and four different ways to select the parents satisfies the same asymptotic runtime guarantees as the SEMO and GSEMO algorithms on the basic OneMinMax and LeadingOnesTrailingZeros benchmarks. However, if the population size is only equal to the size of the Pareto front, then the NSGA-II cannot efficiently compute the full Pareto front: for an exponential number of iterations, the population will always miss a constant fraction of the Pareto front. Our experiments confirm the above findings.

翻译：非支配排序遗传算法II（NSGA-II）是实际应用中使用最广泛的多目标进化算法（MOEA）。然而，与一些已通过数学手段分析的简单MOEA相比，目前尚不存在针对NSGA-II的此类研究。本文证明，数学运行时间分析同样适用于NSGA-II。具体而言，我们证明：当种群规模为帕累托前沿大小的四倍时，采用两种经典变异算子及四种不同父代选择方式的NSGA-II，在基本OneMinMax和LeadingOnesTrailingZeros基准问题上，与SEMO和GSEMO算法具有相同的渐近运行时间保证。但是，若种群规模仅等于帕累托前沿大小，则NSGA-II无法高效计算完整帕累托前沿：在指数级迭代次数内，种群始终会缺失帕累托前沿的恒定比例部分。我们的实验证实了上述发现。