The NSGA-II is proven to encounter difficulties for more than two objectives, and the deduced reason is the crowding distance computed by regarding the different objectives independently. The recent theoretical efficiency of the NSGA-III and the SMS-EMOA also supports the deduced reason as both algorithms consider the dependencies of objectives in the second criterion after the non-dominated sorting but with complicated structure or difficult computation. However, there is still a question of whether a simple modification of the original crowding distance can help. This paper proposes such a variant, called truthful crowding distance. This variant inherits the simple structure of summing the component for each objective. For each objective, it first sorts the set of solutions in order of descending objective values, and uses the smallest normalized L1 distance between the current solution and solutions in the earlier positions of the sorted list as the component. Summing up all components gives the value of truthful crowding distance. We call this NSGA-II variant by NSGA-II-T that replaces the original crowding distance with the truthful one, and that sequentially updates the crowding distance value after each removal. We prove that the NSGA-II-T can efficiently cover the full Pareto front for many-objective mOneMinMax and mOJZJ, in contrast to the exponential runtime of the original NSGA-II. Besides, we also prove that it theoretically achieves a slightly better approximation of the Pareto front for OneMinMax than the original NSGA-II with sequential survival selection. Besides, it is the first NSGA-II variant with a simple structure that performs well for many objectives with theoretical guarantees.

翻译：NSGA-II已被证明在目标数超过两个时会遇到困难，推导出的原因在于其拥挤距离计算将不同目标视为独立处理。近期NSGA-III和SMS-EMOA的理论效率研究也支持这一推断，因为这两种算法在非支配排序后的第二准则中考虑了目标间的依赖关系，但具有复杂结构或计算困难。然而，是否可以通过简单修改原始拥挤距离来改善性能仍是一个悬而未决的问题。本文提出一种称为真实拥挤距离的变体。该变体继承了为每个目标分量求和的简单结构：针对每个目标，首先按目标值降序排列解集，然后使用当前解与排序列表中靠前解的最小归一化L1距离作为该目标分量。对所有分量求和即得到真实拥挤距离值。我们将采用真实拥挤距离替代原始距离、并在每次移除后顺序更新拥挤距离值的NSGA-II变体称为NSGA-II-T。我们证明，对于多目标mOneMinMax和mOJZJ问题，NSGA-II-T能够高效覆盖完整帕累托前沿，而原始NSGA-II需要指数级运行时间。此外，我们还证明在理论上，NSGA-II-T对OneMinMax问题帕累托前沿的逼近效果略优于采用顺序生存选择的原始NSGA-II。值得注意的是，这是首个具有简单结构且在多目标问题上表现优异、同时具备理论保证的NSGA-II变体。