[FEATURE] Approximating Hypervolume / Hypervolume Contributions by polar coordinates #582
Description
Is your feature request related to a problem? Please describe.
The monte-carlo based hypervolume approximation in pagmo bf_fpras / bf_approx is kinda outdated by now. There exists alternatives that deploy some number-theoretic tricks and coordinate transformations that makes them particular robust for certain geometries (e.g. highly convex or concave point sets) in the sense that they select the "correct" least contributor with higher chance. Finding the least contributor is critical for many-objective optimization if the number of objectives goes high (let's say >8).
Describe the solution you'd like
It would be cool to see the algorithm from Deng and Zhang (see context) implemented in pagmo. The idea of this algorithm is to approximate the hypervolume by integrating the lengths of rays coming from from the reference point towards the front, using some sampling of polar coordinates/angles. The implementation might be moderately complex, but probably doable with some commitment.
Describe alternatives you've considered
To be fair, the approximation right now in pygmo is not bad and does its job. This is more a suggestion for a hypervolume related project, that can be put forward for interested individuals or groups. For this to be useful, many-objective optimization problems (>8 objectives) would need to be of practical concern. Otherwise, this is mostly of academic interest.
Additional context
Deng, J. and Zhang, Q. Approximating hypervolume and hypervolume contributions using polar coordinate. IEEE Transactions on Evolutionary Computation, 2019. [link]