  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.
