For two point sets x and y the function returns the value
of the binary \(\varepsilon\)-indicator [1].
eps(x, y, ...)
| x | [ |
|---|---|
| y | [ |
| ... | [any] |
Single numeric indicator value.
The \(\varepsilon\)-indicator, often denoted as \(I_{\varepsilon}\),
requires the concept of \(\varepsilon\)-dominance.
A vector \(x \in R^m\), for some \(\varepsilon > 0\), \(\varepsilon\)-dominates
another vector \(y \in R^m\), \(x \preceq_{\varepsilon} y\), if and only if
$$
\Leftrightarrow x_i \leq \varepsilon y_i \quad \forall i = 1,\ldots,m.
$$
Equipped with this, Zitzler et al. [1] define the (multiplicative) binary
\(\varepsilon\)-indicator for two point sets \(X = \{x_1, \ldots, x_{|X|}\}\)
and \(Y = \{y_1, \ldots, y_{|Y|}\}\) as follows:
$$
I_{\varepsilon}(X, Y) = \inf_{\varepsilon > 0}\{y \in Y \mid \exists x \in X: x \preceq_{\varepsilon} y\}.
$$
It means that \(I_{\varepsilon}(X, Y)\) is the smallest \(\varepsilon\)
such that there exists a point in \(X\) that dominates a point \(y \in Y\)
in the \(\varepsilon\)-dominance sense. It can be calculated the following way
$$
I_{\varepsilon}(X, Y) = \max_{y \in Y} \min_{x \in X} \max_{1 \le i \le m} \frac{x_i}{y_i}.
$$
Given a reference set \(R\), e.g., the known true Pareto-front or a good
approximation of it, the unary version is simply
$$
I_{\varepsilon}(X) := I_{\varepsilon}(R, X).
$$
Function eps implements the binary \(\varepsilon\)-indicator. It
should be obvious how to calculate the unary indicator.
Keep in mind that this function assumes all objectives to be minimized. In case at least one objective is to be maximized, the data needs to be transformed accordingly in advance.
[1] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, V. G. Da Fonseca, Performance assessment of multiobjective optimizers: An analysis and review, IEEE Transactions on evolutionary computation 7 (2) (2003) 117–132.