Overcome the Difficulties of NSGA-II via Truthful Crowding Distance with Theoretical Guarantees

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.