In this paper we study the maximum throughput achievable with optimal scheduling in multi-hop networks with highly directive antenna arrays capable of Space Division Multiplexing (SDM) at the transmitter and Space Division Multiple Access (SDMA) at the receiver. This network model is relevant for future millimeter wave (mmWave) systems, which are expected to implement self-backhauled cellular networks with very high data rates, relying on carrier frequencies between 10-300 GHz, channels with a very large bandwidth, and a large number of antenna elements, even in mobile devices. We adapt mmWave channel propagation, antenna array and link rate models to the classic throughput-optimality and Network Utility Maximization (NUM) scheduling framework for multi-hop networks. Directional antenna gains, transmission towards multiple destinations at once, and simultaneous reception of signals from multiple sources, are all new characteristics not featured in the existing NUM literature. We verify that the classic NUM convergence lemmas are still valid under this new set of constraints, and discuss a series of algorithms to achieve or approximate the performance of the optimal Maximum Back Pressure (MBP) solution. Finally, our analysis serves as an estimation of the potential improvement in cellular network throughput capacity due to the integration of SDM/SDMA techniques and multi-hop.