Module hypervolume 

Hypervolume indicator computation for multi-objective optimization.

Provides exact and approximate algorithms for computing the hypervolume dominated by a Pareto front with respect to a reference point.

Algorithms

Function	Algorithm	Complexity
hypervolume_2d	Sweep-line on sorted front	O(n log n)
hypervolume_3d	Slice-and-sweep	O(n² log n)
hypervolume_wfg	WFG recursive algorithm	O(n^(d-1))
hypervolume_contribution_wfg	Remove-and-recompute	O(n^d)
exclusive_hypervolume	Per-solution contributions	O(n^d)

References

• While, L., Hingston, P., Barone, L., & Huband, S. (2006). A faster algorithm for calculating hypervolume. IEEE Transactions on Evolutionary Computation, 10(1), 29-38.
• Emmerich, M., Beume, N., & Naujoks, B. (2005). An EMO algorithm using the hypervolume measure as selection criterion.
• Bradstreet, L., While, L., & Barone, L. (2007). A fast many-objective hypervolume algorithm. ISPA, 3-10.

Functions

exclusive_hypervolume

Compute the exclusive hypervolume contribution of every solution in the front.

hypervolume_2d

Compute the hypervolume dominated by a 2-D Pareto front.

hypervolume_3d

Compute the exact 3-D hypervolume using a slice-and-sweep approach.

hypervolume_contribution_wfg

Compute the hypervolume contribution of solution idx to the total hypervolume.

hypervolume_wfg

Compute the exact hypervolume using the WFG (While-Hingston-Barone-Huband) recursive algorithm for arbitrary dimensionality.