Hypervolume

Method
FrontContainer
Exact Hypervolume
`dimension_type hypervolume() const`
`dimension_type hypervolume(key_type reference_point) const`
Monte-Carlo Hypervolume
`dimension_type hypervolume(size_t sample_size) const`
`dimension_type hypervolume(size_t sample_size, const key_type &reference_point) const`

Parameters

reference_point - point used as reference for the hypervolume calculation. When not provided, it defaults to the nadir() point.
sample_size - number of samples for the hypervolume estimate

Return value

Hypervolume indicator

Complexity

Exact hypervolume: $O (n^{m - 2} \log n)$
Monte-Carlo hypervolume approximation: $O (s m \log n)$ , where $s$ is the number of samples

Notes

Because the solutions in a front are incomparable, we need performance indicators to infer the quality of a front. Indicators can measure several front attributes, such as cardinality, convergence, distribution, and spread. Correlation indicators can also estimate the relationship between objectives in a set.

The most popular indicator of a front quality is its hypervolume, as it measures both convergence and distribution quality. The front hypervolume refers to the total hypervolume dominated by the front.

Hypervolume Approximation

When $m$ is large, the exact hypervolume calculation becomes impractical. Our benchmarks provide a reference on the impact of these approximations.

Example

Continuing from the previous example:

C++

std::cout << "Exact hypervolume: " << pf.hypervolume(pf.nadir()) << std::endl;
std::cout << "Hypervolume approximation (10000 samples): " << pf.hypervolume(10000, pf.nadir()) << std::endl;

Python

print('Exact hypervolume:', pf.hypervolume(pf.nadir()))
print('Hypervolume approximation (10000 samples):', pf.hypervolume(10000, pf.nadir()))

Output

Exact hypervolume: 55.4029
Hypervolume approximation (10000 samples): 54.4734