A Convergence Indicator for Multi-Objective Optimisation Algorithms

ABSTRACT The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it’s considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread (∆), Averaged Hausdorff distance (∆ p ), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require to know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.

摘要
近年来，多目标优化算法实现了长足发展。因此，亟需建立相应方法对这类算法的优化结果开展对比分析。就此而言，性能指标发挥着关键作用。通常而言，评估这类算法时需考量其多项核心属性，包括容量、收敛性、多样性以及收敛-多样性平衡。目前已有诸多成熟的性能指标，例如世代距离（generational distance, GD）、反向世代距离（inverted generational distance, IGD）、超体积（hypervolume, HV）、分布广度（Spread, ∆）、平均豪斯多夫距离（Averaged Hausdorff distance, ∆_p）以及R2指标（R2-indicator）等。
本文提出了一种基于香农熵（Shannon entropy）传统公式的新型收敛性评估指标。该指标的核心特性包括：1）无需知晓真实帕累托集（Pareto set）；2）相较于超体积（HV）指标，其计算成本适中。