## 4 Numerical Results
### Bi-Objective Antenna Optimization  
The design objectives for the considered antenna optimization problem included the following: the minimization of reflection \(F_{1}\), such that \(\mid\)\(S_{11}\mid\) = \(\mid\)\(R(\mathbf{x})\)\(\mid\) = \(\mid\)\(R(\mathbf{x},f)\)\(\mid\), where \(f\) = \(f_{L}\), \(\ldots\),\(f_{H}\) denotes the frequency points at which the radiator is evaluated (\(f_{L}\) = 3.1 GHz, \(f_{H}\) = 10.6 GHz), as well as the reduction of the structure footprint \(F_{2}\), defined as \(A(\mathbf{x})\) = \(X\)\(\times\)\(Y\) rectangle (see Section 2). The starting points for determination of the extreme Pareto designs were \(\mathbf{x}_{1.1}\)\({}^{(0)}\) = [10 6 16 0.8 1 0 0.35 0.6]\({}^{T}\) and \(\mathbf{x}_{2.1}\)\({}^{(0)}\) = [24 10 27 0.2 1.5 0 0.4 0.8]\({}^{T}\). It is worth noting that, due to being represented using only one variable, the radiators and ground planes of the initial designs were in the forms of circles and rectangles, respectively. Both antenna geometries are shown in Figure 6. The objective function for optimization with respect to the first performance requirement was defined as  
\[F_{1}(\mathbf{x})=\alpha_{1}E_{1}(\mathbf{x})+\min\{0,E_{2}(\mathbf{x})\} \tag{11}\]  
where  
\[E_{1}(\mathbf{x})=\frac{1}{M}\sum\left|\frac{\left|S_{11}\right|_{f_{ \mathrm{S}}}-S_{\mathrm{max}}}{S_{\mathrm{max}}}\right|^{2} \tag{12}\] \[E_{2}(\mathbf{x})=\ \max(\left|S_{11}\right|)+S_{\mathrm{max}} \tag{13}\]  
Figure 6: In-scale visualizations of the spline-parameterized antenna at the initial designs (**a**) \(\mathbf{x}_{1.1}\)\({}^{(0)}\) and (**b**) \(\mathbf{x}_{2.1}\)\({}^{(0)}\). Here, \(\mid S_{11}\mid_{f\
u}\) represents the reflection at \(M\) frequency points \(f_{\
u}\in f_{\
u}\) the value of which was higher than the \(S_{\max}=-15\) dB threshold. Note that the definition of Equation (12) in the least squares sense provided smooth changes of the objective function values in the course of the optimization process [51]. On the other hand, \(E_{1}(\mathbf{x})=0\) when the antenna response was below \(S_{\max}\) for all\(f_{\
u}\) points. In this case, the control over optimization was taken over by the min-max component in Equation (13). The scaling coefficient was set to \(\alpha_{1}=10^{4}\) in order to ensure a smooth transition between the \(E_{1}(\mathbf{x})\) and \(E_{2}(\mathbf{x})\) functions. The second design objective was given as  
\[F_{2}(\mathbf{x})=\alpha_{2}A(\mathbf{x})^{2}+E_{1}(\mathbf{x}) \tag{14}\]  
where \(\alpha_{2}=0.01\) in order to ensure that the contribution of \(E_{1}(\mathbf{x})\) to \(F_{2}(\mathbf{x})\) was noticeable for a relatively small violation of \(S_{\max}=F_{1.\max}=-11\) dB. It should be noted that for the majority of the antenna structures, a typical design requirement is to ensure that the reflection is below the \(-10\) dB threshold. However, in practice, minimization of Equation (14) often leads to violation of the condition [43]. Setting a lower-than-required threshold \(S_{\max}\) allowed us to ensure that \(\mid S_{11}\mid\leq-10\) dB within the frequency band of interest. The vector representing the dimensionality of the antenna (spline-based features of the structure) was \(\mathbf{d}=[d_{1}\dots\ d_{6}]^{T}=[1\ 4\ 8\ 12\ 16\ 20]^{T}\). Hence, the number of nested optimizations per corner solution was \(N=6\). Moreover, \(d_{m}=d_{4}=12\) (see Section 3.3). Figure 7 shows the antenna responses (corner points) obtained at each stage of the nested loop.