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 LeadingOnesTrailingZeroes 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.

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，与通过数学手段分析的几个简单的 MOEA 相比，迄今为止还没有针对 NSGA-II 的此类研究。在这项工作中，我们证明数学运行时分析对于 NSGA-II 也是可行的。作为特定结果，我们证明，当总体规模比 Pareto 前沿规模大四倍时，具有两个经典变异算子和四种不同选择父母的方式的 NSGA-II 满足与 SEMO 和 GSEMO 相同的渐近运行时保证基于OneMinMax和leadingOnesTrailingZeroes的算法基准。然而，如果总体大小仅等于 Pareto 前沿的大小，则 NSGA-II 无法有效计算完整的 Pareto 前沿：对于指数次数的迭代，总体将始终错过 Pareto 前沿的常数部分。我们的实验证实了上述发现。