## 2 Globalized Miniaturization Using Variable-Fidelity EM Models and PCA
### Globalized Size Reduction: The Concept  
Compact antennas often exhibit parameter redundancy, being a result of introducing multitudinous topological modifications [21] and additional components into the basic radiators (e.g., monopoles [24]). As a result, there might be a number of combinations of designable parameters \(\mathbf{x}\) that correspond to a relatively small antenna size and acceptable electrical performance. From the standpoint of numerical optimization, this is equivalent to the multimodality of size reduction task (2), i.e., the existence of multiple local optima. As explained in Section 1, the direct application of global search methods, in particular, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraint** & **Analytical Description \#** & **Penalty Function** \\ \hline In-band reflection & & \\ coefficient \(1S_{11}(\mathbf{x}f)\,|\) & \(1S_{11}(\mathbf{x}f)\,|\leq-10\) dB for \(f\in F\) & \(c(\mathbf{x})=\left[\frac{\max\{S(\mathbf{x})+10,0\}}{10}\right]^{2}\), \\ not exceeding \(-10\) dB & & & where \(S(\mathbf{x})=\max\{f\in F:\ |\ S_{11}(\mathbf{x}f)\,|\}\) \\ In-band axial ratio \(AR(\mathbf{x}f)\) & \(AR(\mathbf{x}f)\leq 3\) dB for \(f\in F\) & \(c(\mathbf{x})=\left[\frac{\max\{AR(\mathbf{x})-3\}}{3}\right]^{2}\), \\ not exceeding \(3\) dB & & & where \(AR(\mathbf{x})=\max\{f\in F:\ |\ AR(\mathbf{x}f)\}\) \\ In-band variability of & \(\Delta G(\mathbf{x}f)\leq 2\) dB for \(f\in F\), & \(c(\mathbf{x})=\left[\frac{\max\{G(\mathbf{x})-2\}}{2}\right]^{2}\), \\ realized gain \(G(\mathbf{x}f)\) & where \(\Delta G(\mathbf{x}f)=\max\{f\in F:\ |\ \delta(\mathbf{x}f)\}\) & where \(G(\mathbf{x})=\max\{f\in F:\ \delta(\mathbf{x}f)\}\) \\ not exceeding \(2\) dB & \(G(\mathbf{x}f)|-\min\{f\in F:G(\mathbf{x}f)\}\) & \(\Delta G(\mathbf{x}f)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected performance-related constraints in antenna optimization and related penalty functions. population-based algorithms, is impractical due to the poor computational efficiency of such techniques. Similarly, the utilization of generic surrogate-assisted global optimizers (efficient global optimization, EGO [83; 84]) is of limited use as a consequence of design space dimensionality and, to an even greater extent, because of broad parameter ranges. In this work, we propose an alternative approach that allows for at least partial mitigation of the mentioned difficulties, which includes globalized search and making the optimization outcome more immune to the supplied initial design. The foundation of the proposed methodology is an exploration of the feasible region \(X_{y}\) boundary. The feasible region is a subset of the parameter space that comprises all vectors \(x\) satisfying the design constraints, i.e., such that \(g_{k}(x)\leq 0\), \(k=1\),..., \(n_{g}\), and \(h_{k}(x)=0\), \(k=1\),..., \(n_{h}\). Its boundary, \(X_{b}\), is the set of vectors such that \(g_{k}(x)=0\) and \(h_{k}(x)=0\) for at least one of the constraints. As the reduction of antenna size deteriorates electrical and field performance figures, the minimum-size design has to be allocated on the boundary, i.e., at least one of the constraints is active. Consequently, it is beneficial to identify the spatial allocation of the boundary, \(X_{b}\), and focus the search in the vicinity of this set. Figure 1 shows a visualization of the main concepts of the proposed optimization approach. The search process consists of several steps, which are presented in Figure 2. A detailed discussion of each stage will be provided in subsequent sections. To enhance computational efficiency, the first three stages are carried out at the level of a low-fidelity (lower resolution) EM antenna model \(R_{c}\). This is justified by the fact that at these stages, the evaluation accuracy is of minor importance. The reliability of the optimization process is secured by executing the final stages with the use of the high-fidelity EM model \(R_{f}\).