  In this paper, we investigate the problem of optimal base station (BS) ON-OFF
switching and user association in a heterogeneous network (HetNet) with massive
MIMO, with the objective to maximize the system energy efficiency (EE). The
joint BS ON-OFF switching and user association problem is formulated as an
integer programming problem. We first develop a centralized scheme, in which we
relax the integer constraints and employ a series of Lagrangian dual methods
that transform the original problem into a standard linear programming (LP)
problem. Due to the special structure of the LP, we prove that the optimal
solution to the relaxed LP is also feasible and optimal to the original
problem. We then propose a distributed scheme by formulating a repeated bidding
game for users and BS's, and prove that the game converges to a Nash
Equilibrium (NE). Simulation studies demonstrate that the proposed schemes can
achieve considerable gains in EE over several benchmark schemes in all the
scenarios considered.
