Most evolutionary many-objective optimization (EMaO) algorithms start with a description of a number of predefined set of reference points on a unit simplex. So far, most studies have used the Das and Dennis’s structured approach for generating well-spaced reference points. Due to the highly structured nature of the procedure, this method cannot produce an arbitrary number of points, which is desired in an EMaO application. Although a layer-wise implementation has been suggested, EMO researchers always felt the need for a more generic approach. Motivated by earlier studies, we introduce a metric for defining well-spaced points on a unit simplex and propose a number of viable methods for generating such a set. We compare the proposed methods on a variety of performance metrics such as hypervolume, deviation in triangularized simplices, distance of the closest point pair, and variance of the geometric means to nearest neighbors in up to 15-dimensional spaces. We show that an iterative improvement based on Riesz s-Energy is able to effectively find an arbitrary number of wellspaced points even in higher-dimensional spaces. Reference points created using the proposed Riesz s-Energy method for a number of standard combinations of objectives and reference points as well as a source code written in Python are available publicly at https://www.egr.msu.edu/coinlab/blankjul/uniform.

中文翻译：

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为进化多目标优化在单位单纯形上生成间隔良好的点

大多数进化多目标优化 (EMaO) 算法开始于对单位单纯形上的许多预定义参考点集的描述。到目前为止，大多数研究都使用 Das 和 Dennis 的结构化方法来生成间隔良好的参考点。由于该过程的高度结构化性质，该方法不能产生任意数量的点，而这在 EMaO 应用程序中是需要的。尽管有人建议采用分层实施，但 EMO 研究人员始终认为需要一种更通用的方法。受早期研究的启发，我们引入了一个度量来定义单位单纯形上的间隔良好的点，并提出了许多生成这样一个集合的可行方法。我们在各种性能指标上比较了所提出的方法，例如超体积、三角化单纯形的偏差、最近点对的距离，以及几何平均值与最多 15 维空间中最近邻点的方差。我们表明，即使在高维空间中，基于 Riesz s-Energy 的迭代改进也能够有效地找到任意数量的间隔良好的点。使用提议的 Riesz s-Energy 方法为目标和参考点的许多标准组合创建的参考点以及用 Python 编写的源代码可在 https://www.egr.msu.edu/coinlab/blankjul 公开获得/制服。