What is a good direction vector set for the R2-based hypervolume contribution approximation

The hypervolume contribution is an important concept in hypervolume-based evolutionary multi-objective optimization algorithms. It describes the loss of the hypervolume when a solution is removed from the current population. Since its calculation is #P-hard in the number of objectives, its approximation is necessary for many-objective optimization problems. Recently, an R2-based hypervolume contribution approximation method was proposed. This method relies on a set of direction vectors for the approximation. However, the influence of different direction vector generation methods on the approximation quality has not been studied yet. This paper aims to investigate this issue. Five direction vector generation methods are investigated, including Das and Dennis's method (DAS), unit normal vector method (UNV), JAS method, maximally sparse selection method with DAS (MSS-D), and maximally sparse selection method with UNV (MSS-U). Experimental results suggest that the approximation quality strongly depends on the direction vector generation method. The JAS and UNV methods show the best performance whereas the DAS method shows the worst performance. The reasons behind the results are also analyzed.