## 2 Globalized Miniaturization Using Variable-Fidelity EM Models and PCA
### Globalized Size Reduction: Approximating the Boundary of the Feasible Region  
The design space of the antenna under study consists of an interval \(X=[\mathbf{lu}]\). We denote \(I\) and \(\mathbf{u}\) as the lower and upper parameter bounds, respectively, i.e., the following holds \(l_{k}\)\(\leq\)\(x_{k}\)\(\leq\)\(u_{k}\), \(k=1,\,\ldots,\,n\). In the first stage of the optimization procedure, we generate \(\mathbb{N}_{1}\) random vectors \(\mathbf{x}_{r}\)\({}^{(j)}\) meeting the following requirements: (i) \(\mathbf{x}_{r}\)\({}^{(j)}\)\(\in[\mathbf{lu}]\); (ii) \(A_{2}\)\(\leq\)\(A(\mathbf{x}_{r}\)\({}^{(j)})\)\(\leq\)\(A_{1}\); and (iii) the problem-dependent constraints are met at \(\mathbf{x}_{r}\)\({}^{(l)}\). Here, \(A_{1}\) and \(A_{2}\) represent the non-mandatory maximum and minimum antenna sizes, respectively, that might be known from the preceding work carried out on the same structure. It is also possible to introduce additional constraints that restrict the part of the design where sampling is to take place. The formulation of these constraints may involve the researcher's know-how or other data garnered on the antenna at hand.  
In the next step, we calculate low-fidelity antenna outputs, \(\mathbf{R}_{e}(\mathbf{x}_{r}\)\({}^{(0)}\), \(j=1,\,\ldots,\,N_{1}\), and designate a subset comprising \(N_{2}\) entries \(\{\mathbf{x}_{r}^{(j)}\}_{j=1,\,\ldots,\,N_{2}}\subset\{\mathbf{x}_{r}\)\({}^{(0)}_{j=1,\,\ldots,\,N_{1}}\) of vectors, which will serve as initial designs for miniaturization. This selection is carried out using an automated decision-making procedure, which takes into account respective constraint violation levels (with designs featuring violations of smaller magnitude being assessed  
Figure 2: Basic stages of the proposed globalized antenna size reduction algorithm. Notation: \(X\)—conventional design space; \(\mathbf{x}_{r}\)\({}^{(j)}\)—random observables; \(R_{e}(\mathbf{x}_{r}\)\({}^{(0)})\)—EM-simulated antenna response at the low-fidelity level; \(N_{1}\), \(N_{2}\), \(N_{3}\)—the numbers of all observables, the observables selected for pre-optimization, and the pre-optimized designs with the lowest constraint violation, respectively; \(X_{p}\)—surrogate model domain spanned by \(p\) selected eigenvectors.  
as more beneficial). In the numerical experiments of Section 3, we set \(N_{2}=20\), which is a good compromise between computational cost and the amount of information about the geometry of the boundary set \(X_{b}\) that may be attained with the selected designs. All points \(\mathbf{x}_{i}^{(i)}\) are optimized for size reduction. In other words, for \(j=1,\,\ldots,\,N_{2}\), starting from \(\mathbf{x}^{(0)}=\mathbf{x}_{i}^{(j)}\), we find  
\[\mathbf{x}_{c}^{(i)}=\arg\min\{\mathbf{x}:U(\mathbf{x})\} \tag{3}\]  
where \(U\) denotes the penalty-based merit function (2) and \(\mathbf{x}_{c}^{(i)}\) are found with the use of the low-fidelity model \(\mathbf{R}_{c}\). Furthermore, task (3) is solved with relaxed convergence criteria to lower the computational expenditures associated with the optimization process: at this stage, the resolution of identifying the optimum design is not of major concern. Problem (3) is solved using a trust-region (TR), gradient-based algorithm [86]. The details of the TR algorithm will be given in Section 2.6.  
In the last step of this stage, an \(N_{3}\)-element of \(\{\mathbf{x}_{c}^{(i)}\}_{j=1,\,\ldots,\,N_{2}}\) has to be selected, such that it encompasses the vectors of satisfactory quality with regard to constraint violation. This eliminates vectors at which optimization process (3) failed for any reason. For example, a typical selection criterion for impedance matching constraints would be to eliminate designs for which constraint violation at \(\mathbf{x}_{c}^{(i)}\) exceeds 2 dB. In the next section, we will refer to the selected subset simply as \(\{\mathbf{x}_{c}^{(i)}\}_{j=1,\,\ldots,\,N_{3}}\).