Ranking Vectors by Means of the Dominance Degree Matrix

In multi-/many-objective evolutionary algorithms (MOEAs), there are varieties of vector ranking schemes, including nondominated sorting, dominance counting, and so on. Usually, these vector ranking schemes in the classical MOEAs are of high computational complexity. Thus, in recent years, many researchers put emphasis on the further improvement of the computational complexity of the vector ranking schemes. In this paper, we propose the dominance degree matrix for a set of vectors and design a fast method to construct this new data structure, which requires <inline-formula> <tex-math notation="LaTeX">${O}$ </tex-math></inline-formula>(<italic>mN</italic>log <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula>) comparisons on average. The dominance degree matrix is constructed based on the properties of Pareto domination, and it can convert the dominance comparison into counting the number of special element pairs. Based on the dominance degree matrix, we develop a new and efficient implementation of nondominated sorting called dominance degree approach for nondominated sorting (DDA-NS), which has an average time complexity of <inline-formula> <tex-math notation="LaTeX">${O}$ </tex-math></inline-formula>(<italic>mN</italic><inline-formula> <tex-math notation="LaTeX">$^{2}$ </tex-math></inline-formula>) but only requires <inline-formula> <tex-math notation="LaTeX">${O}$ </tex-math></inline-formula>(<italic>mN</italic>log <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula>) objective function value comparisons on average. Empirical results demonstrate that DDA-NS clearly outperforms six other representative approaches for nondominated sorting in most cases and DDA-NS performs well when dealing with large-size and many-objective populations. In addition, we also use the dominance degree matrix to form a new method for calculating the dominance strength for Strength Pareto Evolutionary Algorithm (SPEA)2, which greatly improves the efficiency of the naive calculation method in SPEA2. Experiments on benchmark problems show that the Nondominated Sorting Genetic Algorithm (NSGA)-II and NSGA-III framework embedding DDA-NS and the SPEA2 framework embedding the new method of calculating the dominance strength indeed achieve the improvement of the runtime.