Pareto Fronts of Many-Objective Degenerate Test Problems


In general, an M-objective continuous optimization problem has an (M - 1)-dimensional Pareto front in the objective space. If its dimension is smaller than (M - 1), it is called a degenerate Pareto front. Deb-Thiele-Laumanns-Zitzler (DTLZ)5 and Walking Fish Group (WFG)3 have often been used as many-objective continuous test problems with degenerate Pareto fronts. However, it was noted that DTLZ5 has a nondegenerate part of the Pareto front. Constraints have been proposed to remove the nondegenerate part. In this letter, first we show that WFG3 also has a nondegenerate part. Then, we derive constraints to remove the nondegenerate part. Finally, we show that the existence of the nondegenerate part makes WFG3 an interesting test problem through computational experiments.