Learning to Approximate: Auto Direction Vector Set Generation for Hypervolume Contribution Approximation

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> <tex-math notation="LaTeX">$R_{2}^{\text {HVC}}$ </tex-math></inline-formula> indicator) is proposed to approximate the hypervolume contribution. The <inline-formula> <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> <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> <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> <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.