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为广泛的多目标进化算法（MOEA）。然而，与几种同样经过数学分析的简单MOEA不同，目前尚无针对NSGA-II的相关研究。本文表明，对NSGA-II进行数学运行时间分析同样可行。具体结果包括：我们证明，当种群规模比帕累托前沿大小大四倍时，采用两种经典变异算子与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。然而，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：在指数次迭代中，种群总会缺失帕累托前沿的恒定比例。我们的实验证实了上述发现。