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A cellular multiple-input-multiple-output (MIMO) downlink system is studied, in which each base station (BS) transmits to some of the users so that each user receives its intended signal from a subset of the BSs. This scenario is referred to as network MIMO with partial cooperation since only a subset of the BSs is able to coordinate their transmission toward any user. The focus of this paper is on the optimization of linear beamforming strategies at the BSs and at the users for network MIMO with partial cooperation. Individual power constraints at the BSs are enforced, along with constraints on the number of streams per user. It is first shown that the system is equivalent to a MIMO interference channel with generalized linear constraints (MIMO-IFC-GC). The problems of maximizing the sum rate (SR) and minimizing the weighted sum mean square error (WSMSE) of the data estimates are nonconvex, and suboptimal solutions with reasonable complexity need to be devised. Based on this, suboptimal techniques that aim at maximizing the SR for the MIMO-IFC-GC are reviewed from recent literature and extended to the MIMO-IFC-GC where necessary. Novel designs that aim at minimizing the WSMSE are then proposed. Extensive numerical simulations are provided to compare the performance of the considered schemes for realistic cellular systems.