Exact Calculation of Inverted Generational Distance
本文证明了倒置世代距离（IGD）中参考集离散化误差的上界，并提出一种当帕累托前沿解析表达式已知时精确计算IGD和IGD+的数值方法，避免离散化误差。
Inverted generational distance (IGD) is an important performance indicator in the field of multiobjective 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 article, 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.