Generating Well-Spaced Points on a Unit Simplex for Evolutionary Many-Objective Optimization

Most evolutionary many-objective optimization (EMaO) algorithms start with a description of a number of the 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 (HV), deviation in triangularized simplices, distance of the closest point pair, and variance of the geometric means to nearest neighbors in up to 15-D spaces. We show that an iterative improvement based on Riesz s-energy is able to effectively find an arbitrary number of well-spaced 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.