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. The default method="DZ2019-HW" is deterministic and ignores the parameter seed, while method="DZ2019-MC" relies on Monte-Carlo sampling \citepDenZha2019approxhv. Both methods tend to get more accurate with higher values of nsamples, but the increase in accuracy is not monotonic.

Usage

hv_approx(
  x,
  reference,
  maximise = FALSE,
  nsamples = 100000L,
  seed = NULL,
  method = c("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.
seed	integer(1) Random seed.
method	character(1) Method to generate the sampling weights. See ‘Details’.

Details

This function implements the method proposed by \citetDenZha2019approxhv to approximate the hypervolume:

\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, n is the number of weights w^{(i)} 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="DZ2019-HW", 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", 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. The original source code in C++/MATLAB for both methods can be found at https://github.com/Ksrma/Hypervolume-Approximation-using-polar-coordinate.

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, seed=42, method="DZ2019-MC")
hv_approx(x, ref=10, method="DZ2019-HW")

moocore documentation built on Jan. 11, 2026, 9:06 a.m.