Another Difficulty of Inverted Triangular Pareto Fronts for Decomposition-Based Multi-Objective Algorithms

A set of uniformly sampled weight vectors from a unit simplex has been frequently used in decomposition-based multi-objective algorithms. The number of the generated weight vectors is controlled by a user-defined parameter H. In the literature, good results are often reported on test problems with triangular Pareto fronts since the shape of the Pareto fronts is consistent with the distribution of the weight vectors. However, when a problem has an inverted triangular Pareto front, well-distributed solutions over the entire Pareto front are not obtained due to the inconsistency between the Pareto front shape and the weight vector distribution. In this paper, we demonstrate that the specification of H has an unexpected large effect on the performance of decomposition-based multi-objective algorithms when the test problems have inverted triangular Pareto fronts. We clearly explain why their performance is sensitive to the specification of H in an unexpected manner (e.g., H = 3 is bad but H = 4 is good for three-objective problems whereas H = 3 is good but H = 4 is bad for four-objective problems). After these discussions, we suggest a simple weight vector specification method for inverted triangular Pareto fronts.