In this letter, a new hypervolume contribution approximation method is proposed which is formulated as an R2 indicator. The basic idea of the proposed method is to use different line segments only in the hypervolume contribution region for the hypervolume contribution approximation. Comparing with a traditional method which is based on the R2 indicator to approximate the hypervolume, the new method can directly approximate the hypervolume contribution and will utilize all the direction vectors only in the hypervolume contribution region. The new method, the traditional method, and the Monte Carlo sampling method together with two exact methods are compared through comprehensive experiments. Our results show the advantages of the new method over the other methods. Comparing with the other two approximation methods, the new method achieves the best performance for comparing hypervolume contributions of different solutions and identifying the solution with the smallest hypervolume contribution. Comparing with the exact methods, the new method is computationally efficient in high-dimensional spaces where the exact methods are impractical to use.

中文翻译：

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基于 R2 的超体积贡献近似

在这封信中，提出了一种新的超体积贡献近似方法，该方法被表述为 R2 指标。所提出方法的基本思想是仅在超体积贡献区域中使用不同的线段进行超体积贡献近似。与传统的基于R2指标逼近超体积的方法相比，新方法可以直接逼近超体积贡献，并且只利用超体积贡献区域内的所有方向向量。通过综合实验比较了新方法、传统方法、蒙特卡洛抽样方法和两种精确方法的比较。我们的结果显示了新方法相对于其他方法的优势。与其他两种近似方法相比，新方法在比较不同解决方案的超体积贡献和识别具有最小超体积贡献的解决方案方面取得了最佳性能。与精确方法相比，新方法在无法使用精确方法的高维空间中具有计算效率。