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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 are able to coordinate their transmission towards 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. It is first shown that the system is equivalent to a MIMO interference channel with generalized linear constraints (MIMO-IFC-GC). The problem of minimizing the weighted sum mean square error (WSMSE) of the data estimates is non-convex, and suboptimal solutions with reasonable complexity need to be devised. 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.