In this paper, source and relay precoders are derived which optimize upper and lower bounds on the Gaussian MIMO relay channel capacity. First, the prior art on the cut-set upperbound on capacity is extended by showing that the optimization of the source and relay codebooks can be formulated as a convex problem without having to introduce a scalar parameter that captures their cross-correlation. Both the Full-Duplex and Time Division Duplex (TDD) relay channels are addressed, assuming perfect

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... owledge of all channels, and two procedures are proposed which solve the problem efficiently by relying on analytical expressions of gradients, subgradients and projection operators: the first one solves the dual problem while the second one applies the barrier method. Similar techniques are then used to maximize the achievable rate of Decode-and-Forward (DF) TDD MIMO relaying strategies with either partial or full decoding at the relay. Sub-optimum precoders are also proposed which have a closed-form expression that can be obtained from the KKT conditions, thus reducing the computational complexity at the expense of a lower rate. Simulations in a cellular downlink scenario show that the partial DF strategy can achieve a rate very close to capacity for realistic values of the Source to Relay signalto-noise ratio. Finally, the availability of Channel State Information (CSI) in a real system is discussed.