Due to the more complicated population dynamics of the NSGA-II, none of the existing runtime guarantees for this algorithm is accompanied by a non-trivial lower bound. Via a first mathematical understanding of the population dynamics of the NSGA-II, that is, by estimating the expected number of individuals having a certain objective value, we prove that the NSGA-II with suitable population size needs $\Omega(Nn\log n)$ function evaluations to find the Pareto front of the OneMinMax problem and $\Omega(Nn^k)$ evaluations on the OneJumpZeroJump problem with jump size $k$. These bounds are asymptotically tight (that is, they match previously shown upper bounds) and show that the NSGA-II here does not even in terms of the parallel runtime (number of iterations) profit from larger population sizes. For the OneJumpZeroJump problem and when the same sorting is used for the computation of the crowding distance contributions of the two objectives, we even obtain a runtime estimate that is tight including the leading constant.

翻译：由于NSGA-II的种群动力学更为复杂，该算法现有的运行时保证均未附带非平凡的下界。通过首次对NSGA-II种群动力学的数学理解——即估计具有特定目标值的个体期望数量，我们证明：在合适的种群规模下，NSGA-II求解OneMinMax问题的帕累托前沿需要$\Omega(Nn\log n)$次函数评估，而求解跳跃规模为$k$的OneJumpZeroJump问题需要$\Omega(Nn^k)$次函数评估。这些下界是渐近紧的（即与先前证明的上界匹配），并表明此处的NSGA-II即使在并行运行时（迭代次数）也无法从更大的种群规模中获益。对于OneJumpZeroJump问题，当采用相同的排序方式计算两个目标的拥挤距离贡献时，我们甚至得到了包含首项常数的紧致运行时估计。