This paper develops a multi-stage optimization framework for determining periodic inspection intervals for geo-distributed infrastructure systems subject to hidden failures. We assume that the unit's failure can only be rectified at periodic inspection when a perfect repair is carried out to restore the unit to as-good-as-new condition. For large-scale systems such as fire hydrants in a city, water supply and power transformer distribution networks in urban areas, firstly, we partition the units of a system into groups based on their pre-specified features by a clustering algorithm; then compute an optimal periodic inspection interval for each group based on the expected cost per cycle; and finally, we find the inspection schedule for the whole system that not only minimizes the total inspection and maintenance cost but also satisfies the workforce and budget resource constraints. An integer nonlinear programming formulation and a heuristics are introduced to determine the inspection schedule. The proposed approach is applied to scheduling periodic inspection of large amounts of fire hydrants in a major U.S. city.