Learning to Approximate: Auto Direction Vector Set Generation for Hypervolume Contribution Approximation
提出一种名为LtA的方法，自动从训练数据中学习方向向量集，用于改进R2HVC指标对超体积贡献的近似质量，实验表明优于现有方法。
Hypervolume contribution is an important concept in evolutionary multiobjective optimization (EMO). It involves hypervolume-based EMO algorithms and hypervolume subset selection algorithms. Its main drawback is that it is computationally expensive in high-dimensional spaces, which limits its applicability to many-objective optimization. Recently, an R2 indicator variant (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator) is proposed to approximate the hypervolume contribution. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator uses line segments along a number of direction vectors for hypervolume contribution approximation. It has been shown that different direction vector sets lead to different approximation qualities. In this article, we propose learning to approximate (LtA), a direction vector set generation method for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator. The direction vector set is automatically learned from training data. The learned direction vector set can then be used in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator to improve its approximation quality. The usefulness of the proposed LtA method is examined by comparing it with other commonly used direction vector set generation methods for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator. Experimental results suggest the superiority of LtA over the other methods for generating high-quality direction vector sets.