Function hypervolume_wfg 

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

The hypervolume indicator for any number of objectives, computed exactly by the WFG algorithm.

The same quantity, reference-point contract and point-skipping rules as hypervolume: a point that is malformed, non-finite, or that does not strictly beat reference in every objective encloses no volume and is skipped; duplicates and dominated points add nothing. The difference is the dimension: up to MAX_EXACT_HYPERVOLUME_DIM objectives this delegates to hypervolume and returns exactly its result, and above that it runs WFG instead of refusing.

Algorithm and cost

WFG (While, Bradstreet, Barone 2012; Optuna optuna/_hypervolume/wfg.py) sorts the non-dominated points by the first objective and sums exclusive contributions: the hypervolume of p₁ … pₙ is the box of pₙ plus, for every earlier pᵢ, the box of pᵢ minus the hypervolume of the later points each limited to pᵢ (componentwise max). Limiting makes most of those later points dominated, so they are filtered out before the recursion — the pruning that makes WFG fast in practice.

Exact hypervolume is #P-hard in the number of objectives, and WFG’s worst case is exponential in it. Its recursive slicing also makes runtime grow with the number of points; pruning often reduces the rows passed to later recursive calls, but does not make point count irrelevant. It is practical for the population sizes a sampler works with (tens to a few hundred points in 4–6 objectives); it is not a tool for scoring ten thousand points in ten objectives. That is why this is a separate entry point rather than a change to hypervolume.

Near the exponent limits, volume arithmetic uses scaled exponents with f64 precision. A final result that overflows f64 is reported as +∞; intermediate overflow alone does not force an infinite result.

Errors

Error::InvalidSpace if directions is empty, if reference does not have one entry per direction, or if reference carries a non-finite value.

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

// Four 4-D points, each one unit better than (2, 2, 2, 2) in one objective,
// against the reference (3, 3, 3, 3): the union of their boxes is the shared
// unit cube plus one unit cube per point.
let front = vec![
    vec![1.0, 2.0, 2.0, 2.0],
    vec![2.0, 1.0, 2.0, 2.0],
    vec![2.0, 2.0, 1.0, 2.0],
    vec![2.0, 2.0, 2.0, 1.0],
];
let hv = hypervolume_wfg(&front, &[3.0; 4], &[Minimize; 4])?;
assert_eq!(hv, 5.0);