Function hypervolume 

pub fn hypervolume(
    points: &[Vec<f64>],
    reference: &[f64],
    directions: &[Direction],
) -> Result<f64>

The hypervolume indicator: the volume of objective space that points dominate, bounded by reference.

The standard scalar quality measure for a Pareto front — it rewards a front for being both close to the true front (convergence) and spread along it (diversity), which no single-objective summary of a multi-objective study can do. Larger is always better, whatever the directions are.

Arithmetic near the exponent limits uses scaled volumes with f64 precision, so intermediate overflow or underflow need not destroy a representable result. The final result is +∞ if it exceeds f64’s range, or zero if it is too small to represent.

The reference point

The hypervolume is only defined relative to a point that every solution is better than in every objective; it is the far corner of the box being measured. A solution that ties or loses to the reference in any objective contributes zero and is silently skipped — it is outside the measured box, not an error.

nadir_point computes the usual choice (the componentwise worst observed value). Note what that implies: the worst solution in each objective sits on the reference and therefore contributes nothing, and a reference recomputed as the study grows makes the numbers from different moments incomparable. A hypervolume history must be computed against one fixed reference, chosen once — for instance the nadir of the final population, or a domain-knowledge bound such as “0 return, 1 second”. That choice is the caller’s, which is why this function does not make it.

Complexity, and the dimension limit

Objectives	Algorithm	Cost
1	the single best value	O(N)
2	staircase sweep after a sort	O(N log N)
3	dimension sweep: slice along objective 3, 2-D sweep per slab	O(N² log N)
≥ 4	refused (MAX_EXACT_HYPERVOLUME_DIM)	—

Errors

Error::InvalidSpace if directions is empty, if it holds more than MAX_EXACT_HYPERVOLUME_DIM objectives, if reference does not have one entry per direction, or if reference carries a non-finite value (there is no finite volume to report against an infinite corner). A point that is malformed or non-finite is skipped rather than fatal, so one diverged trial cannot make a whole report unanswerable.

use atune_core::pareto::hypervolume;
use atune_core::study::Direction::Minimize;

// Three points of a minimisation front against the reference (4, 4).
let front = vec![vec![1.0, 3.0], vec![2.0, 2.0], vec![3.0, 1.0]];
let hv = hypervolume(&front, &[4.0, 4.0], &[Minimize, Minimize])?;
assert_eq!(hv, 6.0);