hv_contributions: Hypervolume contribution of a set of points

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Hypervolume contribution of a set of points

Description

Computes the hypervolume contribution of each point of a set of points with respect to a given reference point. Duplicated and dominated points have zero contribution. By default, dominated points are ignored, that is, they do not affect the contribution of other points. See the Notes below for more details. For details about the hypervolume, see hypervolume().

Usage

hv_contributions(x, reference, maximise = FALSE, ignore_dominated = TRUE)

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.
ignore_dominated	logical(1) Whether dominated points are ignored when computing the contribution of nondominated points. The value of this parameter has an effect on the return values only if the input contains dominated points. Setting this to FALSE slows down the computation significantly. See the Notes below for a detailed explanation.

Details

The hypervolume contribution of point \vec{p} \in X is defined as \text{hvc}(\vec{p}) = \text{hyp}(X) - \text{hyp}(X \setminus \{\vec{p}\}). This definition implies that duplicated points have zero contribution even if not dominated, because removing one of the duplicates does not change the hypervolume of the remaining set. Moreover, dominated points also have zero contribution. However, a point that is dominated by a single (dominating) nondominated point reduces the contribution of the latter, because removing the dominating point makes the dominated one become nondominated.

Handling this special case is non-trivial and makes the computation more expensive, thus the default (ignore_dominated=TRUE) ignores all dominated points in the input, that is, their contribution is set to zero and their presence does not affect the contribution of any other point. Setting ignore_dominated=FALSE will consider dominated points according to the mathematical definition given above, but the computation will be slower.

When the input only consists of mutually nondominated points, the value of ignore_dominated does not change the result, but the default value is significantly faster.

The current implementation uses a O(n\log n) dimension-sweep algorithm for 2D. With ignore_dominated=TRUE, the 3D case uses the HVC3D algorithm \citepGueFon2017hv4d, which has O(n\log n) complexity. Otherwise, the implementation uses the naive algorithm that requires calculating the hypervolume |X|+1 times.

Value

numeric()
A numerical vector

Author(s)

Manuel López-Ibáñez

References

\insertAllCited

See Also

hypervolume()

Examples

x <- matrix(c(5,1, 1,5, 4,2, 4,4, 5,1), ncol=2, byrow=TRUE)
hv_contributions(x, reference=c(6,6))
# hvc[(5,1)] = 0 = duplicated
# hvc[(1,5)] = 3 = (4 - 1) * (6 - 5)
# hvc[(4,2)] = 3 = (5 - 4) * (5 - 2)
# hvc[(4,4)] = 0 = dominated
# hvc[(5,1)] = 0 = duplicated
hv_contributions(x, reference=c(6,6), ignore_dominated = FALSE)
# hvc[(5,1)] = 0 = duplicated
# hvc[(1,5)] = 3 = (4 - 1) * (6 - 5)
# hvc[(4,2)] = 2 = (5 - 4) * (4 - 2)
# hvc[(4,4)] = 0 = dominated
# hvc[(5,1)] = 0 = duplicated
data(SPEA2minstoptimeRichmond)
# The second objective must be maximized
# We calculate the hypervolume contribution of each point of the union of all sets.
hv_contributions(SPEA2minstoptimeRichmond[, 1:2], reference = c(250, 0),
            maximise = c(FALSE, TRUE))

# Duplicated points show zero contribution above, even if not
# dominated. However, filter_dominated removes all duplicates except
# one. Hence, there are more points below with nonzero contribution.
hv_contributions(filter_dominated(SPEA2minstoptimeRichmond[, 1:2], maximise = c(FALSE, TRUE)),
                 reference = c(250, 0), maximise = c(FALSE, TRUE))

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