Three Objectives Degrade the Convergence Ability of Dominance-Based Multi-objective Evolutionary Algorithms

In the evolutionary multi-objective optimization (EMO) community, it is well known that the convergence ability of dominance-based multi-objective evolutionary algorithms (MOEAs) is severely deteriorated on many-objective problems with more than three objectives. In this paper, we clearly demonstrate that the convergence ability of NSGA-II deteriorates even in the case of three objectives. Our experimental results on multi-objective knapsack and traveling salesman problems with 2–6 objectives show that NSGA-II starts to deteriorate the quality of the current population after a number of generations even when it is applied to three-objective problems. Surprisingly, NSGA-III also shows a similar performance deterioration. We analyze the search behavior of NSGA-II, NSGA-III, three versions of MOEA/D, and SMS-EMOA. Then, we explain the reason for the performance deterioration of NSGA-II and NSGA-III, which exists in the environmental selection mechanism of each algorithm. Another interesting observation is that NSGA-II has the best or second best performance (next to MOEA/D with the weighted sum) among the examined algorithms on many-objective problems in early generations before it starts to show performance deterioration.