r2_exact: Exact R2 indicator

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Exact R2 indicator

Description

Computes the exact R2 indicator with respect to a given ideal/utopian reference point assuming minimization of all objectives.

Usage

r2_exact(x, reference, maximise = FALSE)

Arguments

x	matrix()|data.frame() Matrix or data frame of numerical values, where each row gives the coordinates of a point.
reference	numeric() Reference (ideal) 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.

Details

The unary R2 indicator is a quality indicator for a set A \subset \mathbb{R}^m w.r.t. an ideal or utopian reference point \vec{r} \in \mathbb{R}^m. It was originally proposed by \citetHanJas1998 and is defined as the expected Tchebycheff utility under a uniform distribution of weight vectors (w.l.o.g. assuming minimization):

R2(A) := \int_{w \in W} \min_{a \in A} \left\{ \max_{i=1,\dots,m} w_i (a_i - r_i) \right\} \, dw,

where W denotes the uniform distribution across weights:

W = \{w \in \mathbb{R}^m \mid w_i \geq 0, \sum_{i=1}^m w_i = 1\}.

The R2 indicator is to be minimized and has an optimal value of 0 when \vec{r} \in A.

The exact R2 indicator is strongly Pareto-compliant, i.e., compatible with Pareto-optimality:

\forall A, B \subset \mathbb{R}^m: A \prec B \Rightarrow R2(A) < R2(B).

Given an ideal or utopian reference point, which is available in most scenarios, all non-dominated solutions always contribute to the value of the exact R2 indicator. However, it is scale-dependent and care should be taken such that all objectives contribute approximately equally to the indicator, e.g., by normalizing the Pareto front to the unit hypercube.

The current implementation exclusively supports bi-objective solution sets and runs in O(n \log n) following \citetSchKer2025r2v2.

Value

numeric(1) A single numerical value.

References

\insertAllCited

Examples

dat <- matrix(c(5, 5, 4, 6, 2, 7, 7, 4), ncol = 2, byrow = TRUE)
r2_exact(dat, reference = c(0, 0))

# This function assumes minimisation by default. We can easily specify maximisation:
r2_exact(dat, reference = c(10, 10), maximise = TRUE)

# Merge all the sets of a dataset by removing the set number column:
extdata_path <- system.file(package="moocore","extdata")
dat <- read_datasets(file.path(extdata_path, "example1_dat"))[, 1:2]
nrow(dat)

# Dominated points are ignored, so this:
r2_exact(dat, reference = 0)

# gives the same exact R2 value as this:
dat <- filter_dominated(dat)
nrow(dat)
r2_exact(dat, reference = 0)

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