supp.md

Supplementary information

Preliminaries

1. Define a multi-output function as $f: \mathbb{R}^D \rightarrow \mathbb{R}^M$,
2. Assume we run $N$ independent optimization runs and obtain the Pareto sets for each run $\forall i \in \{1,\dots,M\}, \mathcal{F}_i \subseteq \mathbb{R}^M$,
3. Define the objective vector $\boldsymbol{f} \in \mathbb{R}^M$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\forall m \in \{1,\dots,M\}, f_m \leq y_m$ and notate it as $\boldsymbol{f} \preceq \boldsymbol{y}$, and
4. Define a set of objective vectors $F$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\exists \boldsymbol{f} \in F, \boldsymbol{f} \leq \boldsymbol{y}$ and notate it as $F \preceq \boldsymbol{y}$

Attainment surface

As seen in the figure below, the attainment surface is the surface in the objective space that we can obtain by splitting the objective space like a step function by the Pareto front solutions yielded during the optimization.

It is simple to obtain the attainment surface if we have only one experiment; however, it is hard to show the aggregated results from multiple experiments. To address this issue, we use the $k$% attainment surface.

Credit: Figure 4. in Indicator-Based Evolutionary Algorithm with Hypervolume Approximation by Achievement Scalarizing Functions.

$k$% attainment surface

First, we define the following empirical attainment function:

$$ \alpha(\boldsymbol{y}) := \alpha ( \boldsymbol{y} | \mathcal{F}_{1} , \dots , \mathcal{F}_{N}) = \frac{1}{N} \sum_{{n=1}}^{N} \mathbb{I} [ \mathcal{F}_{n} \preceq \boldsymbol{y} ] . $$

The $k$% attainment surface is the attainment surface such that it is achieved by $k$% of independent runs and more formally, it is defined as:

$S = \biggl\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{k}{100}\biggr\}.$

Note that as we only have $N$ independent runs, we define a control parameter level ( $1 \leq L \leq N$ ) and obtain the following set:

$S_L = \biggl\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{L}{N}\biggr\}.$

The best, median, and worst attainment surfaces could be fetched by $\{1/N,1/2,1\}\times 100$% attainment surface, respectively.

Please check the following references for more details:

[2] An Approach to Visualizing the 3D Empirical Attainment Function

[3] On the Computation of the Empirical Attainment Function