We consider multi-pair two-way amplify and forward (AF) relaying, where
multiple full-duplex source-node pairs exchange information via a shared
full-duplex massive multiple-input multiple-output (MIMO) relay. Most of the
previous massive MIMO relay works maximize spectral efficiency (SE). We
maximize global energy efficiency (GEE) with quality-of-service (QoS)
constraints that are expressed as the rate required by source nodes. The
problem is non-convex and is solved by approximating it as a pseudo-concave
problem, which is then solved using the Dinkelbach method. We also consider the
max-min EE problem which maximizes the EE of the worst energy efficient user.
For solving the EE optimization, we derive approximate closed-form lower bounds
for the ergodic achievable rate for maximal-ratio and zero-forcing processing
at the relay by using minimum mean squared error channel estimation. We
numerically show the accuracy of the derived lower bounds and the improved GEE
with optimal power allocation, with and without QoS constraints.