Stringent performance specifications along with constraints imposed on physical dimensions make the design of contemporary microwave components a truly onerous task. In recent years, the latter demand has been growing in importance with the innovative application of areas such as the Internet of Things coming into play. The need to employ full-wave electromagnetic (EM) simulations for response evaluation, reliable, yet CPU-heavy, only aggravates the issue. This paper proposes a reduced-cost miniaturization algorithm that employs a trust-region search procedure and multi-resolution EM simulations. In our approach, the resolution of the EM model is adjusted throughout the optimization process based on its convergence status starting from the lowest admissible fidelity. As the algorithm converges, the resolution is increased up to the high-fidelity one, used at the final phase to ensure reliability. Four microwave components have been utilized as verification structures: an impedance matching transformer and three branch-line couplers. Significant savings in terms of the number of EM analyses required to conclude the size reduction process of 41, 42, 38 and 50 percent have been obtained (in comparison to a single-fidelity procedure). The footprint area of the designs optimized using the proposed approach are equal to 32, 205, 410 and 132 mm², in comparison to 52, 275, 525 and 213 mm² of the initial (and already compact) design.