Exact Calculation of Inverted Generational Distance

Inverted Generational Distance (IGD) is an important performance indicator in the field of multi-objective optimization (MOO). Although it has been widely used for decades, applying IGD for fair and accurate performance evaluation remains challenging, with the biggest obstacle being the selection of the reference set. IGD generally represents the distance between the solution set and the Pareto front (PF). Since the real PF is often an infinite set, even if it is known, it is difficult to apply it directly to the calculation of IGD. As a workaround, past research typically samples a finite set, i.e., the reference set, from the PF as an approximation, indirectly used in the IGD calculation. This inevitably introduces a systematic error, which we refer to as discretization error. In this paper, we prove an upper bound for the discretization error, demonstrating that if the reference set is sufficiently dense and uniformly distributed on the entire PF, the discretization error will converge to zero. Additionally, we propose a numerical method for the exact calculation of IGD and IGD+. When the analytical expression of the PF is known, this method allows for the direct calculation of IGD and IGD+ using the real PF, thus avoiding discretization error.