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```
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This paper has more information as well.
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# Supplementary information
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## Preliminaries
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1. Define a multi-output function as $f: \mathbb{R}^D \rightarrow \mathbb{R}^M$,
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2. Assume we run $N$ independent optimization runs and obtain the Pareto sets for each run $\forall i \in \\{1,\dots,M\\}, \mathcal{F}_i \subseteq \mathbb{R}^M$,
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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
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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 $F \preceq \boldsymbol{y}$
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## Attainment surface
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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.
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It is simple to obtain the attainment surface if we have only one experiment;
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however, it is hard to show the aggregated results from multiple experiments.
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To address this issue, we use the $k$% attainment surface.
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Credit: Figure 4. in [Indicator-Based Evolutionary Algorithm with Hypervolume Approximation by Achievement Scalarizing Functions](https://dl.acm.org/doi/pdf/10.1145/1830483.1830578?casa_token=wAx-0-6HgLYAAAAA:LTZmyz4H20nnS9aaTJhQA84UejRISpWK_iCkl33LIT2ER6higBIahESB3x9-yZEq8jVkR9BzSjzMPQ).
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## $k$% attainment surface
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First, we define the following empirical attainment function:
The best, median, and worst attainment surfaces could be fetched by $\\{1/N,1/2,1\\}\times 100$% attainment surface, respectively.
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Please check the following references for more details:
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[2][An Approach to Visualizing the 3D Empirical Attainment Function](https://dl.acm.org/doi/pdf/10.1145/2464576.2482716?casa_token=b9vWo8MI3i8AAAAA:4UaDmmM1YgQFVo-vEQdNKvk9-12RTT8sO7n16CQIvneP_J33w_eGo2wYhfphwufqY5OcYPYj_Gc3mA)
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[3][On the Computation of the Empirical Attainment Function](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.705.1929&rep=rep1&type=pdf)
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This paper has more information as well and we also provide a markdown file [here](supp.md).
1. Define a multi-output function as $f: \mathbb{R}^D \rightarrow \mathbb{R}^M$,
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2. Assume we run $N$ independent optimization runs and obtain the Pareto sets for each run $\forall i \in \\{1,\dots,M\\}, \mathcal{F}_i \subseteq \mathbb{R}^M$,
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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
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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 $F \preceq \boldsymbol{y}$
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## Attainment surface
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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.
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It is simple to obtain the attainment surface if we have only one experiment;
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however, it is hard to show the aggregated results from multiple experiments.
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To address this issue, we use the $k$% attainment surface.
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+

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Credit: Figure 4. in [Indicator-Based Evolutionary Algorithm with Hypervolume Approximation by Achievement Scalarizing Functions](https://dl.acm.org/doi/pdf/10.1145/1830483.1830578?casa_token=wAx-0-6HgLYAAAAA:LTZmyz4H20nnS9aaTJhQA84UejRISpWK_iCkl33LIT2ER6higBIahESB3x9-yZEq8jVkR9BzSjzMPQ).
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## $k$% attainment surface
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First, we define the following empirical attainment function:
The best, median, and worst attainment surfaces could be fetched by $\\{1/N,1/2,1\\}\times 100$% attainment surface, respectively.
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Please check the following references for more details:
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[2][An Approach to Visualizing the 3D Empirical Attainment Function](https://dl.acm.org/doi/pdf/10.1145/2464576.2482716?casa_token=b9vWo8MI3i8AAAAA:4UaDmmM1YgQFVo-vEQdNKvk9-12RTT8sO7n16CQIvneP_J33w_eGo2wYhfphwufqY5OcYPYj_Gc3mA)
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[3][On the Computation of the Empirical Attainment Function](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.705.1929&rep=rep1&type=pdf)
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