How to Specify a Reference Point in Hypervolume Calculation for Fair Performance Comparison

点(几何) 多目标优化 集合(抽象数据类型) 数学优化 最大化 数学 算法 帕累托原理 分布(数学) 计算机科学 几何学 数学分析 程序设计语言
作者
Hisao Ishibuchi,Ryo Imada,Yu Setoguchi,Yusuke Nojima
出处
期刊:Evolutionary Computation [The MIT Press]
卷期号:26 (3): 411-440 被引量:154
标识
DOI:10.1162/evco_a_00226
摘要

The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.
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