Inerters are a novel type of mechanical actuation devices that are able to produce large inertial forces with a relatively small mass. Due to this property, inerters can provide an effective solution to the main drawbacks of tuned mass-dampers and, consequently, they are gaining an increasing relevance in the field of passive structural vibration control. In this paper, a computational design strategy for inerter-based vibration control schemes is presented. The proposed approach combines a computationally effective reduced-frequency H8 cost-function and a constrained global optimization solver to design different configurations of a shared tuned mass-inerter-damper system for the seismic protection of a multi-story two-building structure. To assess the effectiveness of the obtained configurations, the frequency characteristics and the seismic response of the interstory drifts and interbuilding approaches are investigated with positive results.