Partially connected hybrid beamforming (HBF) is a promising approach to alleviate the implementation of large scale millimeter-wave multiple-input multiple-output (MIMO) systems. In this paper, we develop rate maximizing algorithms for the full array- and subarray-based processing strategies of partially connected HBF. We formulate the rate maximization problem as a weighted mean square error minimization problem and use alternating optimization to tackle it. Numerical results show that partially connected HBF provides a good balance between hardware complexity and performance in comparison to optimal fully digital and analog beamforming. Moreover, the simpler subarray-based HBF algorithm achieves comparable performance to that of the full array-based approach in medium and high SNRs. The rate maximizing results serve as upper bounds for lower complexity heuristic methods.