  This paper investigates the energy efficiency (EE) optimization in downlink
multi-cell massive multiple-input multiple-output (MIMO). In our research, the
statistical channel state information (CSI) is exploited to reduce the
signaling overhead. To maximize the minimum EE among the neighbouring cells, we
design the transmit covariance matrices for each base station (BS).
Specifically, optimization schemes for this max-min EE problem are developed,
in the centralized and distributed ways, respectively. To obtain the transmit
covariance matrices, we first find out the closed-form optimal transmit
eigenmatrices for the BS in each cell, and convert the original transmit
covariance matrices designing problem into a power allocation one. Then, to
lower the computational complexity, we utilize an asymptotic approximation
expression for the problem objective. Moreover, for the power allocation
design, we adopt the minorization maximization method to address the
non-convexity of the ergodic rate, and use Dinkelbach's transform to convert
the max-min fractional problem into a series of convex optimization
subproblems. To tackle the transformed subproblems, we propose a centralized
iterative water-filling scheme. For reducing the backhaul burden, we further
develop a distributed algorithm for the power allocation problem, which
requires limited inter-cell information sharing. Finally, the performance of
the proposed algorithms are demonstrated by extensive numerical results.
