  In this paper, energy efficient power control for small cells underlaying a
macro cellular network is investigated. We formulate the power control problem
in self-organizing small cell networks as a non-cooperative game, and propose a
distributed energy efficient power control scheme, which allows the small base
stations (SBSs) to take individual decisions for attaining the Nash equilibrium
(NE) with minimum information exchange. Specially, in the non-cooperative power
control game, a non-convex optimization problem is formulated for each SBS to
maximize their energy efficiency (EE). By exploiting the properties of
parameter-free fractional programming and the concept of perspective function,
the non-convex optimization problem for each SBS is transformed into an
equivalent constrained convex optimization problem. Then, the constrained
convex optimization problem is converted into an unconstrained convex
optimization problem by exploiting the mixed penalty function method. The
inequality constraints are eliminated by introducing the logarithmic barrier
functions and the equality constraint is eliminated by introducing the
quadratic penalty function. We also theoretically show the existence and the
uniqueness of the NE in the non-cooperative power control game. Simulation
results show remarkable improvements in terms of EE by using the proposed
scheme.
