Electromagnetic simulation tools have been playing an increasing role in the design of contemporary antenna structures. The employment of electromagnetic analysis ensures reliability of evaluating antenna characteristics but also incurs considerable computational expenses whenever massive simulations are involved (e.g., parametric optimization, uncertainty quantification). This high cost is the most serious bottleneck of simulation-driven design procedures, and may be troublesome even for local tuning of geometry parameters, let alone global optimization. On the one hand, globalized search is often necessary because the design problem might be multimodal (i.e., the objective function features multiple local optima) or a reasonably good initial design may not be available. On the other hand, the computational efficiency of popular algorithmic approaches, primarily, nature-inspired population-based algorithms, is generally poor. Combining metaheuristics procedures with surrogate modelling techniques and sequential sampling methods alleviates the problem to a certain extent but modelling of nonlinear antenna responses over broad frequency ranges is extremely challenging, and the aforementioned solutions are normally limited to rather simple structures described by a few parameters. This paper proposes a novel approach to global optimization of multi-band antennas. The major component of the presented framework is the knowledge-based inverse surrogate constructed at the level of response features (e.g., frequency and level locations of the antenna resonances). The surrogate facilitates decision-making process of inexpensive identification of the most promising regions of the parameter space, and a rendition of the good-quality initial design for further local tuning. Our methodology is validated using three examples of dual- and triple-band antennas. The average optimization cost is only 150 full-wave antenna analyses while ensuring precise allocation of the antenna resonances at the target frequencies. This performance is demonstrated superior over both local optimizers and population-based metaheuristics.