  In this paper, the joint power and subcarrier allocation problem is solved in
the context of maximizing the energy-efficiency (EE) of a multi-user,
multi-relay orthogonal frequency division multiple access (OFDMA) cellular
network, where the objective function is formulated as the ratio of the
spectral-efficiency (SE) over the total power dissipation. It is proven that
the fractional programming problem considered is quasi-concave so that
Dinkelbach's method may be employed for finding the optimal solution at a low
complexity. This method solves the above-mentioned master problem by solving a
series of parameterized concave secondary problems. These secondary problems
are solved using a dual decomposition approach, where each secondary problem is
further decomposed into a number of similar subproblems. The impact of various
system parameters on the attainable EE and SE of the system employing both EE
maximization (EEM) and SE maximization (SEM) algorithms is characterized. In
particular, it is observed that increasing the number of relays for a range of
cell sizes, although marginally increases the attainable SE, reduces the EE
significantly. It is noted that the highest SE and EE are achieved, when the
relays are placed closer to the BS to take advantage of the resultant
line-of-sight link. Furthermore, increasing both the number of available
subcarriers and the number of active user equipment (UE) increases both the EE
and the total SE of the system as a benefit of the increased frequency and
multi-user diversity, respectively. Finally, it is demonstrated that as
expected, increasing the available power tends to improve the SE, when using
the SEM algorithm. By contrast, given a sufficiently high available power, the
EEM algorithm attains the maximum achievable EE and a suboptimal SE.
