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Recently, studies on sub-optimal precoding techniques for multiple-input multiple-output broadcast channels (MIMO-BC), which achieve performance near to that of the dirty paper coding (DPC), have drawn attention to vector perturbation (VP). In practice, each antenna or more generally each antenna group has its own limit on the transmitted power, which makes per-antenna-group (per-AG) power constraints more meaningful than the sum power constraint. In this paper, we assume the per-antenna-group constraints and first investigate antenna-group power-constrained VP inspired by the p-sphere encoder for the case when channel inversion is used by the front-end linear precoder. Furthermore, we consider joint optimization of the front-end precoder and the perturbing vector subject to per-AG power constraints and propose precoding based on a minimum mean square error (MMSE) criterion. The results show that the proposed algorithm outperforms conventional VP and the p-sphere encoding in the case of per-AG power constraints.