## 2 Motivation and General Problem Statement
### General Problem Statement  
_Design points, design parameters, or designs_ are represented by an \(m\)-dimensional vector \(\mathbf{x}=(x_{1},\ldots,x_{m})\). The _design space_\([\mathbf{x}^{l},\mathbf{x}^{u}]=\{\mathbf{x}=(x_{1},\ldots,x_{m}):x_{i}^{l}\leq x_{i}\leq x _{i}^{u},i=1,\ldots,m\}\) is assumed to be continuous. Here, \(\mathbf{x}^{l}=(x_{1}^{l},\ldots,x_{m}^{l})\) represents the lower bound of the design parameters, and \(\mathbf{x}^{u}=(x_{1}^{u},\ldots,x_{m}^{u})\) is the upper bound of the design parameters. It is assumed that \(f_{i}(\mathbf{x})\) is a performance function with the threshold value \(f_{i}^{l}\), \(i=1,\ldots,p\). The analytical form of \(f_{i}(\mathbf{x})\) is generally unknown, so its value may need to be computed numerically. A design \(\mathbf{x}\) that meets the performance requirements \(f_{i}(\mathbf{x})\geq f_{i}^{l}\) for \(i=1,\ldots,p\) is termed a good design. Conversely, a design \(\mathbf{x}\in[\mathbf{x}^{l},\mathbf{x}^{u}]\) that violates any of these constraints is termed a bad design. The _complete solution space_ includes all good design points within the design space. Its shape depends on the performance functions and threshold values. Generally, the boundary of one parameter range depends on the values of other parameters. To determine interval boundaries for each parameter independently of other parameters, box-shaped solution spaces are considered. The box-shaped solution space \([\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}}]\) is defined by \(\mathbf{x}^{\text{low}}=(x_{1}^{\text{low}},\ldots,x_{m}^{\text{low}})\) and \(\mathbf{x}^{\text{up}}=(x_{1}^{\text{up}},\ldots,x_{m}^{\text{up}})\) as follows:  
\[[\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}}]=\{\mathbf{x}=(x_{1},\ldots,x_{m}):x_{i}^{ \text{low}}\leq x_{i}\leq x_{i}^{\text{up}},i=1,\ldots,m\}. \tag{1}\]  
A box-shaped solution space that includes only good design points is termed a _feasible box-shaped solution space_. Since the performance requirement is expressed as \(f_{i}(\mathbf{x})\geq f_{i}^{l}\), higher performance is preferable. As a result, the performance of the box-shaped solution space is defined by the global minimum of the performance function \(f_{i}(\mathbf{x})\) within the solution space. Additionally, a larger solution space implies a higher tolerance to parameter variations. Thus, the robustness of the box-shaped solution space is measured by its volume. In this paper, both the performance and robustness of the box-shaped solution space are considered simultaneously. Specifically, our goal is to identify a feasible box-shaped solution space  
Figure 3: The results of the approximated Pareto-optimal front. with optimal performance and high robustness. Typically, optimizing performance and increasing robustness are conflicting objectives, which means that a trade-off exists between performance and robustness. Therefore, we propose the following multi-objective optimization problem:  
\[ \begin{split}\max_{\mathbf{x}^{\text{low}},\text{ar}^{\text{up}}}& v(\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}}),\\ \max_{\mathbf{x}^{\text{low}},\text{ar}^{\text{up}}}& f_{i}^{*}(\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}}), \quad i=1,\ldots,p,\\ \text{subject to}&\begin{cases}f_{i}^{*}(\mathbf{x}^{ \text{low}},\mathbf{x}^{\text{up}})\geq f_{i}^{!},\quad i=1,\ldots,p,\\ \mathbf{x}^{!}\leq\mathbf{x}^{\text{low}}\leq\mathbf{x}^{\text{up}}\leq\mathbf{x}^{\text{up}},\end{cases}\end{split}  \tag{2}\]  
where \(v(\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}})=\prod_{i=1}^{m}(x_{i}^{\text{up}}-x_{ i}^{\text{low}})\) represents the volume of the box-shaped solution space, and \(f_{i}^{*}(\mathbf{x}^{\text{low}},\mathbf{x}^{\text{up}})=\min_{\mathbf{x}\in[\text{ar}^{ \text{low}},\text{ar}^{\text{up}}]}f_{i}(\mathbf{x})\) is the performance of the box-shaped solution space.