hv_approx: Approximate the hypervolume indicator.

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Approximate the hypervolume indicator.

Description

Approximate the value of the hypervolume metric with respect to a given reference point assuming minimization of all objectives. Methods "Rphi-FWE+" and "DZ2019-HW" are deterministic and ignore the parameter seed, while method="DZ2019-MC" relies on Monte-Carlo sampling \citepDenZha2019approxhv. All methods tend to get more accurate with higher values of nsamples, but the increase in accuracy is not monotonic, as shown in the vignette \HTMLVignettehv_approxApproximating the hypervolume.

Usage

hv_approx(
  x,
  reference,
  maximise = FALSE,
  nsamples = 262144L,
  seed = NULL,
  method = c("Rphi-FWE+", "DZ2019-HW", "DZ2019-MC")
)

Arguments

x	matrix()|data.frame() Matrix or data frame of numerical values, where each row gives the coordinates of a point.
reference	numeric() Reference point as a vector of numerical values.
maximise	logical() Whether the objectives must be maximised instead of minimised. Either a single logical value that applies to all objectives or a vector of logical values, with one value per objective.
nsamples	integer(1) Number of samples for Monte-Carlo sampling. Higher values typically produce more accurate approximations of the true hypervolume, but require more time.
seed	integer(1) Random seed.
method	character(1) Method to generate the sampling weights. See ‘Details’.

Details

All available methods approximate the hypervolume as a (m-1)-dimensional integral over the surface of hypersphere \citepDenZha2019approxhv:

\widehat{HV}_r(A) = \frac{2\pi^\frac{m}{2}}{\Gamma(\frac{m}{2})}\frac{1}{m 2^m}\frac{1}{n}\sum_{i=1}^n \max_{y \in A} s(w^{(i)}, y)^m

where m is the number of objectives, w^{(i)} are weights uniformly distributed on S_{+}, i.e., the positive orthant of the (m-1)-D unit hypersphere, n is the number of weights sampled, \Gamma() is the gamma function gamma(), i.e., the analytical continuation of the factorial function, and s(w, y) = \min_{k=1}^m (r_k - y_k)/w_k.

In the default method="Rphi-FWE+" \citepLop2026hvapprox, the weights w^{(i)}, i=1\ldots n are defined using the deterministic low-discrepancy sequence R_\phi \citepRob2018unreasonable mapped to the positive orthant of the hypersphere using a modified version of Fang and Wang efficient mapping \citepFanWan1994numtheory.

In method="DZ2019-HW" \citepDenZha2019approxhv, the weights w^{(i)}, i=1\ldots n are defined using a deterministic low-discrepancy sequence. The weight values depend on their number (nsamples), thus increasing the number of weights may not necessarily increase accuracy because the set of weights would be different.

In method="DZ2019-MC" \citepDenZha2019approxhv, the weights w^{(i)}, i=1\ldots n are sampled from the unit normal vector such that each weight w = \frac{|x|}{\|x\|_2} where each component of x is independently sampled from the standard normal distribution \citepMuller1959sphere.

The original source code in C++/MATLAB for both "DZ2019-HW" and "DZ2019-MC" methods can be found at https://github.com/Ksrma/Hypervolume-Approximation-using-polar-coordinate.

\citet

Lop2026hvapprox empirically shows that "Rphi-FWE+" typically produces an approximation error as low as the other methods, with a computational cost similar to "DZ2019-MC" and significantly faster than "DZ2019-HW".

Value

A single numerical value.

Author(s)

Manuel López-Ibáñez

References

\insertAllCited

Examples

x <- matrix(c(5, 5, 4, 6, 2, 7, 7, 4), ncol=2, byrow=TRUE)
hypervolume(x, ref=10)
hv_approx(x, ref=10, method="Rphi-FWE+")
hv_approx(x, ref=10, method="DZ2019-HW")
hv_approx(x, ref=10, seed=42, method="DZ2019-MC")

moocore documentation built on July 12, 2026, 5:06 p.m.