epsilon: Epsilon metric

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Epsilon metric

Description

Computes the epsilon metric, either additive or multiplicative.

Usage

epsilon_additive(x, reference, maximise = FALSE)

epsilon_mult(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	matrix|data.frame Reference set as a matrix or data.frame 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 epsilon metric of a set A \subset \mathbb{R}^m with respect to a reference set R \subset \mathbb{R}^m is defined as

epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 \leq i \leq m} epsilon(a_i, r_i)

where a and r are objective vectors of length m.

In the case of minimization of objective i, epsilon(a_i,r_i) is computed as a_i/r_i for the multiplicative variant (respectively, a_i - r_i for the additive variant), whereas in the case of maximization of objective i, epsilon(a_i,r_i) = r_i/a_i for the multiplicative variant (respectively, r_i - a_i for the additive variant). This allows computing a single value for problems where some objectives are to be maximized while others are to be minimized. Moreover, a lower value corresponds to a better approximation set, independently of the type of problem (minimization, maximization or mixed). However, the meaning of the value is different for each objective type. For example, imagine that objective 1 is to be minimized and objective 2 is to be maximized, and the multiplicative epsilon computed here for epsilon(A,R) = 3. This means that A needs to be multiplied by 1/3 for all a_1 values and by 3 for all a_2 values in order to weakly dominate R.

The multiplicative variant can be computed as \exp(epsilon_{+}(\log(A), \log(R))), which makes clear that the computation of the multiplicative version for zero or negative values doesn't make sense. See the examples below.

The current implementation uses the naive algorithm that requires O(m \cdot |A| \cdot |R|), where m is the number of objectives (dimension of vectors).

Value

numeric(1) A single numerical value.

Author(s)

Manuel López-Ibáñez

References

\insertRef

ZitThiLauFon2003:tecmoocore

Examples

# Fig 6 from Zitzler et al. (2003).
A1 <- matrix(c(9,2,8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A2 <- matrix(c(8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A3 <- matrix(c(10,4,9,5,8,6,7,7,6,8), ncol=2, byrow=TRUE)
if (requireNamespace("graphics", quietly = TRUE)) {
   plot(A1, xlab=expression(f[1]), ylab=expression(f[2]),
        panel.first=grid(nx=NULL), pch=4, cex=1.5, xlim = c(0,10), ylim=c(0,8))
   points(A2, pch=0, cex=1.5)
   points(A3, pch=1, cex=1.5)
   legend("bottomleft", legend=c("A1", "A2", "A3"), pch=c(4,0,1),
          pt.bg="gray", bg="white", bty = "n", pt.cex=1.5, cex=1.2)
}
epsilon_mult(A1, A3) # A1 epsilon-dominates A3 => e = 9/10 < 1
epsilon_mult(A1, A2) # A1 weakly dominates A2 => e = 1
epsilon_mult(A2, A1) # A2 is epsilon-dominated by A1 => e = 2 > 1
# Equivalence between additive and multiplicative
exp(epsilon_additive(log(A2), log(A1)))

# A more realistic example
extdata_path <- system.file(package="moocore","extdata")
path.A1 <- file.path(extdata_path, "ALG_1_dat.xz")
path.A2 <- file.path(extdata_path, "ALG_2_dat.xz")
A1 <- read_datasets(path.A1)[,1:2]
A2 <- read_datasets(path.A2)[,1:2]
ref <- filter_dominated(rbind(A1, A2))
epsilon_additive(A1, ref)
epsilon_additive(A2, ref)
# Multiplicative version of epsilon metric
ref <- filter_dominated(rbind(A1, A2))
epsilon_mult(A1, ref)
epsilon_mult(A2, ref)

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