## 3 Validation
### Results  
Tables 5-8 present the results achieved using the presented procedure and the benchmark routines. Figures 5-8 illustrate the reflection characteristics of the antennas optimized using our technique for three selected algorithm runs for each structure. The numerical data includes the average footprint area and its standard deviation, the average constraint violation and its standard deviation, and the average optimization cost. The standard deviations can be viewed as a measure of result repeatability. The results are discussed at length in Section 3.3. The experimental validation of the antenna structures themselves can be found in the source papers (e.g., [89; 90; 91; 92]). It is not included here as it is irrelevant to the main topic of the work. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Optimization Algorithm**} & \multicolumn{5}{c}{**Performance Figure**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Antenna Footprint \(A\)**} & \multicolumn{2}{c}{**Std(\(A\)) 2**} & \multicolumn{2}{c}{**Constraint Violation \(D\) (dB) 3**} & \multicolumn{1}{c}{**Std(\(D\)) (dB) 4**} & \multicolumn{1}{c}{**CPU Cost 5**} \\  & \multicolumn{2}{c}{**(mm2) 1**} & & & & & \\ \hline \multirow{4}{*}{Algorithm I} & \(\beta=10^{3}\) & 250.4 & 24.0 & 1.2 & 0.5 & \(124.2\times\mathbf{R}_{f}\) (14.6 h) \\  & \(\beta=10^{4}\) & 318.6 & 60.0 & 0.1 & 0.1 & \(180.3\times\mathbf{R}_{f}\) (21.2 h) \\  & \(\beta=10^{5}\) & 331.6 & 63.4 & 0.1 & 0.1 & \(133.2\times\mathbf{R}_{f}\) (15.7 h) \\ \cline{1-1}  & Algorithm II & 281.6 & 37.1 & 0.2 & 0.2 & \(181.7\times\mathbf{R}_{f}\) (21.4 h) \\ \cline{1-1}  & Algorithm III & 399.4 & 143.6 & 0.6 & 0.4 & \(1000\times\mathbf{R}_{f}\) (118 h) \\ \cline{1-1}  & reduction (this work) & 205.6 & 18.2 & 0.07 & 0.07 & \(472.6\times\mathbf{R}_{f}\) (56 h) \\ \hline \hline \end{tabular}  
* Optimized antenna footprint \(A\) averaged over ten algorithm runs. 2 Std(\(A\))—standard deviation of the optimized antenna footprint averaged over ten algorithm runs. 3 Constraint violation, defined as \(D=\{3.1\,\mathrm{GHz}\leq\leq f\leq 10.6\,\mathrm{GHz}\}\) : \(\max|S_{11}(f)|+10\), averaged over ten algorithm runs. 5 Cost expressed in terms of the equivalent number of high-fidelity EM antenna analyses. Numbers in brackets correspond to the running time in hours.  
\end{table} 
Table 6: Optimization results for Antenna II. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{5}{c}{**Performance Figure**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Antenna Footprint \(A\)**} & \multicolumn{2}{c}{**Std(\(A\)) 2**} & \multicolumn{2}{c}{**Constraint Violation \(D\) (dB) 3**} & \multicolumn{1}{c}{**Std(\(D\)) (dB) 4**} & \multicolumn{1}{c}{**CPU Cost 5**} \\  & \multicolumn{2}{c}{**(mm2) 1**} & & & & & \\ \hline \multirow{4}{*}{Algorithm I} & \(\beta=10^{3}\) & 318.1 & 42.6 & 1.2 & 0.4 & \(43.8\times\mathbf{R}_{f}\) (1.8 h) \\  & \(\beta=10^{4}\) & 317.7 & 42.3 & 0.4 & 0.7 & \(42.2\times\mathbf{R}_{f}\) (1.8 h) \\  & \(\beta=10^{5}\) & 318.8 & 43.3 & 0.1 & 0.2 & \(41.4\times\mathbf{R}_{f}\) (1.7 h) \\ \cline{1-1}  & Algorithm II & 314.1 & 42.3 & 0.3 & 0.2 & \(50.0\times\mathbf{R}_{f}\) (21.1 h) \\ \cline{1-1}  & Algorithm III & 360.9 & 67.5 & 0.5 & 0.9 & \(1000\times\mathbf{R}_{f}\) (42 h) \\ \cline{1-1}  & Globalized search with dimensionality reduction (this work) & 265.4 & 9.6 & 0.1 & 0.1 & \(692.7\times\mathbf{R}_{f}\) (29 h) \\ \hline \hline \end{tabular}  
* Optimized antenna footprint \(A\) averaged over ten algorithm runs. 2 Std(\(A\))—standard deviation of the optimized antenna footprint averaged over ten algorithm runs. 3 Constraint violation, defined as \(D=\{3.1\,\mathrm{GHz}\leq f\leq 10.6\,\mathrm{GHz}\}\) : \(\max|S_{11}(f)|+10\), averaged over ten algorithm runs. 5 Cost expressed in terms of the equivalent number of high-fidelity EM antenna analyses. Numbers in brackets correspond to the running time in hours.  
\end{table} 
Table 5: Optimization results for Antenna I.