Stopping criteria automatically determine when to stop an evolutionary algorithm, so as not to waste function evaluations on a stagnant population. Although stopping criteria play an important role in real-world applications, they have attracted little attention in the evolutionary multi-objective optimization (EMO) community. In fact, new stopping criteria for EMO have been rarely developed in recent years. One reason for the stagnation in developing stopping criteria for EMO is a lack of effective benchmarking methodologies. To address this issue, this paper proposes (i) a performance measure of stopping criteria for EMO and (ii) a file-based benchmarking approach. This paper also proposes (iii) a data representation method that effectively stores population states in text files. (i) The proposed measure represents the performance of stopping criteria as a single scalar value, making comparison easy. (ii) The proposed file-based approach not only simplifies the benchmarking process but also facilitates reproducibility. (iii) The proposed data representation method addresses the issue of file size in (ii). We demonstrate the effectiveness of our three contributions (i)--(iii) by benchmarking five representative stopping criteria for EMO.

翻译：停止准则能够自动确定进化算法的终止时机，从而避免在停滞种群上浪费函数评估次数。尽管停止准则在实际应用中具有重要作用，但在进化多目标优化（EMO）领域却鲜受关注。事实上，近年来针对EMO的新停止准则开发进展缓慢。究其原因，在于缺乏有效的基准测试方法论。为解决此问题，本文提出：(i) 针对EMO停止准则的性能度量方法；(ii) 基于文件的基准测试范式；以及(iii) 能有效存储种群状态的数据表征方法。(i) 所提度量方法将停止准则性能转化为单一标量值，便于比较；(ii) 所提基于文件的策略不仅简化了基准测试流程，还可促进结果复现；(iii) 所提数据表征方法解决了(ii)中存在的文件体积问题。我们通过测试五种代表性EMO停止准则，验证了上述三项贡献(i)-(iii)的有效性。