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Fix Typo in Hypervolume formula #1

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See also Hypervolume-slide for the correct formula

The element $r_2^{(i-1)}$ isn't even defined, because $r$ is no list.

See also Hypervolume-slide for the correct formula
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BaumiCoder commented Dec 6, 2024

In the changes of this Pull Request I assumed the formula from the slides and the python code below the formula here is correct, but it seems they are wrong. That would make this Pull Request obsolete.

The first python code (without the normalization) uses this formula:
$H(V, r) = (r_1 - v_1^{(1)})(r_2 - v_2^{(1)}) + \sum_{i=2}^m (r_1 - v_1^{(i)})(v_2^{(i-1)} - v_2^{(i)})$
And the implementations with normalization (in both parts of the chapter) use this one:
$H(V, r) = (r_1 - v_1^{(1)})(r_2 - v_2^{(1)}) + \sum_{i=2}^m (v_1^{(i-1)} - v_1^{(i)})(r_2 - v_2^{(i)})$

Maybe because the functions with normalization are the "complete" ones, which are used more often, the bug in the "toy" implementation without the normalization remained undiscovered until now.

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