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To satisfy the capacity requirements of future mobile systems, under-utilized millimeter wave frequencies can be efficiently exploited by employing massive MIMO technology with highly directive beamforming. Hybrid analog-digital beamforming has been recognised as a promising approach for large-scale MIMO implementations with a reduced number of costly and power-hungry RF chains. In comparison to fully connected architecture, hybrid beamforming (HBF) with partially connected RF architecture is particularly appealing for the practical implementation due to less complex RF power division and combining networks. In this paper, we first formulate single- and multi-user rate maximization problems as weighted minimum mean square error (WMMSE) and derive solutions for hybrid beamformers using alternating optimization. The algorithms are designed for the full-array- and sub-array-based processing strategies of partially connected HBF architecture. In addition to the rate maximizing WMMSE solutions, we propose lower complexity sub-array-based zero-forcing algorithms. The performance of the proposed algorithms is evaluated in two different channel models, i.e., a simple geometric model and a realistic statistical millimeter wave model known as NYUSIM. The performance results of the WMMSE HBF algorithms are meant to reveal the potential of partially connected HBF and serve as upper bounds for lower complexity methods. Numerical results imply that properly designed partially connected HBF has the potential to provide an good compromise between hardware complexity and system performance in comparison to fully digital beamforming.