## 3 Validation
### Result Analysis  
The results presented in Tables 5-8 permit us to summarize the performance of the introduced globalized size reduction algorithm and how it compares to the benchmark methods. The obtained results confirm that the considered miniaturization tasks are indeed multimodal, which is implied by the high standard deviation values of the antenna size A for Algorithms I and II, which are both local methods. In particular, depending on the  
Figure 8: Reflection responses of Antenna IV for three representative designs yielded using the introduced miniaturization framework: (**a**) design 1 (footprint area 418 mm\({}^{2}\)), (**b**) design 2 (footprint area 399 mm\({}^{2}\)), and (**c**) design 3 (footprint area 427 mm\({}^{2}\)). Target operating band marked using the horizontal line. Figure 6: Reflection responses of Antenna II for representative designs yielded using the introduced miniaturization framework: (**a**) design 1 (footprint area 189 mm\({}^{2}\)), (**b**) design 2 (footprint area 182 mm\({}^{2}\)), and (**c**) design 3 (footprint area 228 mm\({}^{2}\)). Target operating band marked using the horizontal line. Figure 7: Reflection responses of Antenna III for three representative designs yielded using the introduced miniaturization framework: (**a**) design 1 (footprint area 218 mm\({}^{2}\)), (**b**) design 2 (footprint area 206 mm\({}^{2}\)), and (**c**) design 3 (footprint area 231 mm\({}^{2}\)). Target operating band marked using the horizontal line. initial design, the obtained footprint area may vary significantly and by as much as thirty percent with respect to the average size. A comparison of the proposed procedure with the benchmark algorithms allows us to draw the following conclusions. As expected, and already demonstrated in the literature (e.g., [89]), the outcome of Algorithm I depends on the penalty coefficient setup. When using low values of \(\beta\), excessive constraint violations occur, whereas using coefficients that are too high leads to inferior miniaturization rates. Algorithm II resolves these issues to a certain extent by ensuring both the constraint control and competitive (w.r.t. Algorithm I) antenna sizes. The computational cost of Algorithm II is only slightly higher than that of Algorithm I. As far as the nature-inspired optimization (Algorithm III) is concerned, it performs poorly for all considered antenna structures, both with regard to the achieved antenna footprint areas and constraint violations. The reasons are twofold: (i) the intrinsic challenges of the size reduction task and (ii) the low computational budget assigned for the algorithm. As explained before, we use this setup to avoid excessive computational costs. The most important takeaway here is that 1000 EM analyses are--to a large extent--insufficient to yield satisfactory optimization outcomes using population-based techniques. Overall, the size reduction procedure proposed in this work is demonstrated to outperform all benchmark algorithms in terms of size reduction. On the one hand, it yields designs featuring considerably smaller average footprint area (e.g., the improvement over Algorithm II is almost thirty percent) except for Antenna III, where the obtained average size is comparable to Algorithm II. Yet, the standard deviation of the antenna size is significantly lower for most of the considered structures (by almost 80% for Antenna I, over 50% for Antenna II, and over 95% for Antenna IV; all with respect to Algorithm II) and comparable for Antenna III. The latter corroborates the global search capability of the presented approach. In terms of computational cost, the proposed method is clearly more expensive than local routines, although the difference is not significant for Antennas II and III. Still, the overall expenses equal to a few hundred EM simulations is very much practical given the huge advantages in terms of the optimization process reliability ensured with our technique.