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A new R2 indicator for better hypervolume approximation

In this paper, a new R2 indicator is proposed for better hypervolume approximation. First the fact that the original R2 indicator is not a good approximation for the hypervolume is illustrated by examples. Then the new R2 indicator is derived based on the Divergence theorem and Riemann sum approximation. The difference between the original R2 and the new R2 is only the added exponential in the new R2 where the exponential is the same as the dimensionality of the objective space (i.e., the number of objectives). The new R2, the original R2 and some other R2 variants are compared through comprehensive numerical studies on different solution sets under different scenarios. The results show the superiority of the proposed new R2 indicator over other R2 variants for the hypervolume approximation, where the new R2 indicator achieves the best linear relation with the true hypervolume.