Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted generational distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a real-world optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties, such as Pareto compliance, immunity to dominated/duplicate solutions, and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.

中文翻译：

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基于网格的多/多目标优化的倒代距离


评估帕累托前沿（PF）近似的性能是进化多目标优化领域的一个关键问题。倒代距离（IGD）已被广泛接受作为评估 PF 近似综合质量的性能指标。然而，当面对现实世界的优化问题时，IGD 通常变得不可行，因为它需要先验地知道真实的 PF。此外，IGD 的时间复杂度随着解/参考集的大小呈二次方增长。为了解决上述问题，提出了一种基于网格的 IGD（Grid-IGD）来估计 PF 近似的收敛性和多样性，以实现多目标/多目标优化。在 Grid-IGD 中，基于网格环境中所有近似值的代表性非支配解，通过估计所讨论问题的 PF 来生成一组参考点。为了降低时间复杂度，Grid-IGD 在每个参考点的近似中仅考虑网格邻域内最接近的解。 Grid-IGD 还具有其他理想的特性，例如帕累托合规性、对主导/重复解决方案的免疫力以及无需标准化。在实验研究中，Grid-IGD 在由五个多目标优化器获得的人工和真实 PF 近似值上进行了验证。网格规范对 Grid-IGD 行为的影响也从理论上和实验上进行了详细讨论。