## 3 Validation
### Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Performance Figure**} \\ \cline{3-6}
**Optimization Algorithm** & \multicolumn{2}{c}{**Antenna**} & \multicolumn{2}{c}{**Constraint**} & \multirow{2}{*}{**Std(\(A\))**2**} & \multicolumn{2}{c}{**Violation \(D\)**} & \multirow{2}{*}{**Std(\(D\)) (dB)**4**} & \multicolumn{2}{c}{**CPU Cost 5**} \\  & **Footprint \(A\)** & **Std(\(A\))**2** & & & \\  & **(mm\({}^{2}\))**1** & & & **(dB)**3** & & \\ \hline \multirow{3}{*}{Algorithm I} & \(\beta=10^{3}\) & 212.8 & 14.3 & 1.0 & 0.4 & 164.9 \(\times\)\(R_{f}\) (12.1 h) \\  & \(\beta=10^{4}\) & 255.0 & 25.1 & 0.2 & 0.1 & 138.1 \(\times\)\(R_{f}\) (10.2 h) \\ \cline{1-1}  & \(\beta=10^{5}\) & 280.1 & 47.4 & 0.1 & 0.1 & 154.0 \(\times\)\(R_{f}\) (11.3 h) \\ \cline{1-1}  & Algorithm II & 215.6 & 3.6 & 0.3 & 0.1 & 189.9 \(\times\)\(R_{f}\) (14.0 h) \\ \cline{1-1}  & Algorithm III & 425.7 & 145.8 & 0.2 & 0.2 & 1000 \(\times\)\(R_{f}\) (74 h) \\ \cline{1-1}  & Globalized search with & & & & & \\  dimensionality reduction (this work) & 226.2 & 12.4 & 0.1 & 0.1 & 689.7 \(\times\)\(R_{f}\) (51 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 GHz \(\leq\)\(f\)\(\leq\) 10.6 GHz : max \(1\)\(S_{11}\)(\(f\))] + 10, averaged over ten algorithm runs; \(1\)\(S_{11}\)(\(f\)) ]—antenna reflection. 4 Std(\(D\))—standard deviation of the constraint violation \(D\), 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 7: Optimization results for Antenna III. Figure 5: Reflection responses of Antenna I for three representative designs yielded using the introduced miniaturization framework: (**a**) design 1 (footprint area 260 mm\({}^{2}\)), (**b**) design 2 (footprint area 285 mm\({}^{2}\)), and (**c**) design 3 (footprint area 270 mm\({}^{2}\)). Target operating band marked using the horizontal line. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Performance Figure**} \\ \cline{2-7}
**Optimization Algorithm** & \multicolumn{2}{c}{**Antenna**} & \multicolumn{2}{c}{**Constraint**} & \multirow{2}{*}{**Std(\(A\))**2**} & \multirow{2}{*}{**Violation \(D\)**} & \multirow{2}{*}{**Std(\(D\)) (dB)**4**} & \multicolumn{2}{c}{**CPU Cost 5**} \\  & **Footprint \(A\)** & **Std(\(A\))**2** & & & \\  & **(mm\({}^{2}\))**1** & & & **(dB)**3** & & \\ \hline \multirow{3}{*}{Algorithm I} & \(\beta=10^{3}\) & 727.9 & 236.0 & 1.7 & 1.5 & 180.3 \(\times\)\(R_{f}\) (5.3 h) \\  & \(\beta=10^{4}\) & 829.5 & 206.4 & 1.0 & 1.9 & 211.2 \(\times\)\(R_{f}\) (6.2 h) \\ \cline{1-1}  & \(\beta=10^{5}\) & 842.8 & 130.2 & 0.4 & 0.9 & 248.0 \(\times\)\(R_{f}\) (7.2 h) \\ \cline{1-1}  & Algorithm II & 753.9 & 243.0 & 0.9 & 0.8 & 230.3 \(\times\)\(R_{f}\) (6.7 h) \\ \cline{1-1}  & Algorithm III & 457.8 & 59.1 & 0.7 & 0.4 & 1000 \(\times\)\(R_{f}\) (29 h) \\ \cline{1-1}  & Globalized search with & & & & & \\  dimensionality reduction (this work) & 414.8 & 10.4 & 0.3 & 0.1 & 922.5 \(\times\)\(R_{f}\) (26.9 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 GHz \(\leq\)\(f\)\(\leq\) 10.6 GHz : max \(1\)\(S_{11}\)(\(f\))] + 10, averaged over ten algorithm runs; \(1\)\(S_{11}\)(\(f\)) ]—antenna reflection. 4 Std(\(D\))—standard deviation of the constraint violation \(D\), 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 8: Optimization results for Antenna IV.