Optimal Distribution of Solutions for Crowding Distance on Linear Pareto Fronts of Two-Objective Optimization Problems

Characteristics of an evolutionary multi-objective optimization (EMO) algorithm can be explained using its best solution set. For example, the best solution set for SMS-EMOA is the same as the optimal distribution of solutions for hypervolume maximization. For NSGA-III, if the Pareto front has intersection points with all reference lines, all of those intersection points are the best solution set. For MOEA/D, the best solution set is the set of the optimal solution of each sub-problem. Whereas these EMO algorithms can be analyzed in this manner, the best solution set for the most well-known and frequently-used EMO algorithm NSGA-II has not been discussed in the literature. This is because NSGA-II is not based on any clear criterion to be optimized (e.g., hypervolume maximization, distance minimization to the nearest reference line). As the first step toward the best solution set analysis for NSGA-II, we discuss the optimal distribution of solutions for the crowding distance under the simplest setting: the maximization of the minimum crowding distance on linear Pareto fronts of two-objective optimization problems. That is, we discuss the optimal distribution of solutions on a straight line. Our theoretical analysis shows that the uniformly distributed solutions are not the best solution set. However, it is also shown by computational experiments that the uniformly distributed solutions (except for the duplicated two extreme solutions at each edge of the Pareto front) are obtained from a modified NSGA-II with the ($\mu$ + 1) generation update scheme.