 This paper presents a novel methodology for the globalized size reduction of antenna structures. It starts with the detection of the approximate location of the feasible region boundary, followed by the construction of a dimensionality reduced metamodel and global optimization thereof. Finally, the last stage involves the miniaturization-oriented local refinement of geometry parameters. The proposed methodology is verified using four broadband microstrip antennas and benchmarked against multi-start local search-in-nature inspired methods. The results demonstrate superior size reduction rates compared to other methods while maintaining reasonable computational costs. This article was authored by Slavomir Kozil, Anapyotrinko Dabrowska, and Lukash Golonsky.