Comparison of Hypervolume, IGD and IGD~+ from the Viewpoint of Optimal Distributions of Solutions

摘要

Hypervolume (HV) and inverted generational distance (IGD) have been frequently used as performance indicators to evaluate the quality of solution sets obtained by evolutionary multiobjective optimization (EMO) algorithms. They have also been used in indicator-based EMO algorithms. In some studies on many-objective problems, only the IGD indicator was used due to a large computation load of HV calculation. However, the IGD indicator is not Pareto compliant. This means that a better solution set in terms of the Pareto dominance relation can be evaluated as being worse. Recently the IGD plus (IGD~+) indicator has been proposed as a weakly Pareto compliant version of IGD. In this paper, we compare these three indicators from the viewpoint of optimal distributions of solutions. More specifically, we visually demonstrate similarities and differences among the three indicators by numerically calculating near-optimal distributions of solutions to optimize each indicator for some test problems. Our numerical analysis shows that IGD~+ is more similar to HV than IGD whereas the formulations of IGD and IGD~+ are almost the same.