The NSGA-II is one of the most prominent algorithms to solve multi-objective optimization problems. Despite numerous successful applications, several studies have shown that the NSGA-II is less effective for larger numbers of objectives. In this work, we use mathematical runtime analyses to rigorously demonstrate and quantify this phenomenon. We show that even on the simple $m$-objective generalization of the discrete OneMinMax benchmark, where every solution is Pareto optimal, the NSGA-II also with large population sizes cannot compute the full Pareto front (objective vectors of all Pareto optima) in sub-exponential time when the number of objectives is at least three. The reason for this unexpected behavior lies in the fact that in the computation of the crowding distance, the different objectives are regarded independently. This is not a problem for two objectives, where any sorting of a pair-wise incomparable set of solutions according to one objective is also such a sorting according to the other objective (in the inverse order).

翻译：NSGA-II是解决多目标优化问题最著名的算法之一。尽管已有大量成功应用，但多项研究表明该算法在处理较多目标时效果较差。本研究通过数学运行时间分析，严格论证并量化了这一现象。我们证明，即使在离散OneMinMax基准的$m$目标泛化问题（其中所有解均为帕累托最优）上，当目标数至少为三时，即使采用较大种群规模，NSGA-II也无法在次指数时间内计算出完整帕累托前沿（所有帕累托最优解的目标向量）。这一反常行为的根源在于拥挤距离计算中不同目标被独立处理。对于两个目标问题，这种处理方式不构成问题——任何按某一目标对成对不可比较解集进行排序的结果，必然也是按另一目标（逆序）的排序结果。