his paper considers the efficient design of precoding matrices for sum throughput maximization under throughput quality of service (QoS) constraints and energy harvesting (EH) constraints for energy-constrained devices in a full-duplex (FD) multicell multi-user multiple-input-multiple-output network. Both time splitting (TS) and power splitting are considered to ensure practical EH and information decoding. These problems are quite complex due to non-concave objectives and nonconvex constraints. Especially, with TS, which is implementation-wise quite simple, the problem is even more challenging because the time splitting variable is not only coupled with the downlink throughput function but also coupled with the self-interference in the uplink throughput function. New path-following algorithms are developed for their solutions, which require only a single convex quadratic program for each iteration and ensure rapid convergence. Moreover, the FD EH maximization problem under throughput QoS constraints with TS is also considered. The performance of the proposed algorithms is compared with that of the modified problems assuming half-duplex systems. Finally, the merit of the proposed algorithms is demonstrated through extensive simulations.