We consider two-way amplify and forward relaying, where multiple full-duplex user pairs exchange information via a shared full-duplex massive multiple-input multiple-output (MIMO) relay. Most of the previous massive MIMO relaying works maximize the spectral efficiency (SE). By contrast, we maximize the non-convex energy efficiency (EE) metric by approximating it as a pseudo-concave problem, which is then solved using the classic Dinkelbach approach. We also maximize the EE of the least energy-efficient user relying on the max-min approach. For solving these optimization problems, we derive closedform lower bounds for the ergodic achievable rate both for maximal-ratio combining and zero-forcing processing at the relay, by using minimum mean squared error channel estimation. We numerically characterize the accuracy of the lower bounds derived. We also compare the SE and EE of the proposed design to those of the existing full-duplex systems and quantify the significant improvement achieved by the proposed algorithm. We also compare the EE of the proposed full-duplex system to that of its half-duplex counterparts, and characterize the self-loop and inter-user interference regimes, for which the proposed full-duplex system succeeds in outperforming the half-duplex ones.

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