## 3 Validation
### Antenna Structures and Numerical Experiment Setup  
The proposed algorithm is validated with the use of four broadband microstrip antenna structures, as presented in Figure 4. Table 3 provides additional information concerning the antennas (see also [89; 90; 91; 92]). All the simulations have been performed using Intel Xeon 2.1 GHz dual-core CPU with 128 GB RAM. The low-fidelity models constitute coarse-discretization versions of the high-fidelity representations. The last two rows of Table 3 provide information about the simulation times for the respective models. As the average time evaluation ratio between the high- and low-fidelity models is over four, considerable time savings while executing the first few stages of the proposed algorithm are obtained. The proposed algorithm is compared to several benchmark methods, which are characterized in Table 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description and Recommendations** \\ \hline \multirow{3}{*}{\(N_{r}\)} & Cardinality of the observable set \\  & Recommended value: from 50 to 100; the higher the design space \\  & dimensionality \(n\), the larger \(N_{r}\) should be used \\ \hline \multirow{3}{*}{\(p\)} & Dimensionality of the surrogate domain \\  & Recommended value: \(p=3\) (in order to ensure good model scalability \\ \cline{1-1}  & as a function of the number of training samples; should take into \\ \cline{1-1}  & account the eigenvalues \(\lambda_{k}\)) \\ \hline \multirow{3}{*}{\(N_{B}\)} & Cardinality of the training data set (for surrogate model construction) \\  & Recommended values to ensure a relative RMS error of a few percent \\ \cline{1-1}  & from 200 to 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Globalized size reduction procedure: control parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & \multicolumn{2}{c}{**Operating Principles**} \\ \hline  & Local gradient-based miniaturization with the trust region algorithm used as a search engine (cf. Section 2.6). The size reduction task is expressed as (2) and (3). The penalty coefficient \(\beta\) is kept fixed throughout the algorithm run. \\ I & Algorithm performance highly depends on the choice of \(\beta\): small values ensure better miniaturization rates but lead to larger constraint violations; a large \(\beta\) leads to a more precise constraint control but inferior size reduction because of the objective function (2) steepness near the feasible region boundary. \\  & The algorithm is executed for different values of \(\beta\) = 103, 104, and 105. \\ \hline II & Local trust-region search with adaptively adjusted penalty coefficients [89]. This method adjusts the value of \(\beta\) during the algorithm run based on the constraint violation detected in the current iteration. This leads to a better overall performance with regard to size reduction and constraint control [89]. \\ \hline III & The particle swarm optimizer (PSO) [93] is a widely used population-based algorithm. The following PSO setup is used: the swarm size is set to 10, the maximal number of iterations is 100, and the typical control parameters are (\(\chi\) = 0.73, c\({}_{1}\) = c\({}_{2}\) = 2.05), cf. [89]. The penalty coefficient of problem (2) is set to 104. \\  & Note that the PSO is set up here with a relatively small computational budget (1000 objective function evaluations) in order to avoid excessive computational costs, which are still in the range of a few days of the CPU time per algorithm run. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Benchmark procedures. Figure 4: Benchmark antenna structures: (**a**) Antenna I, (**b**) Antenna II, (**c**) Antenna III, and (**d**) Antenna IV. Ground planes are marked using the light gray shade.