  In this paper, we optimize the energy efficiency (bits/s/Hz/J) of
device-to-multi-device (D2MD) wireless communications. While the
device-to-device scenario has been extensively studied to improve the spectral
efficiency in cellular networks, the use of multicast communications opens the
possibility of reusing the spectrum resources also inside the groups. The
optimization problem is formulated as a mixed integer non-linear joint
optimization for the power control and allocation of resource blocks (RBs) to
each group. Our model explicitly considers resource sharing by letting
co-channel transmission over a RB (up to a maximum of r transmitters) and/or
transmission through s different channels in each group. We use an iterative
decomposition approach, using first matching theory to find a stable even if
sub-optimal channel allocation, to then optimize the transmission power vectors
in each group via fractional programming. Additionally, within this framework,
both the network energy efficiency and the max-min individual energy efficiency
are investigated. We characterize numerically the energy-efficient capacity
region, and our results show that the normalized energy efficiency is nearly
optimal (above 90 percent of the network capacity) for a wide range of
minimum-rate constraints. This performance is better than that of other
matching-based techniques previously proposed.
