In this paper, power control in the uplink for two-tier small-cell networks is investigated. We formulate the power control problem as a Stackelberg game, where the macrocell user equipment (MUE) acts as the leader and the small-cell user equipment (SUE) acts as the follower. To reduce the cross-tier and cotier interferences and the power consumption of both the MUE and SUE, we propose optimizing not only the transmit rate but also the transmit power. The corresponding optimization problems are

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... solved through a two-layer iteration. In the inner iteration, the SUE items (SUEs) compete with each other, and their optimal transmit powers are obtained through iterative computations. In the outer iteration, the optimal transmit power of the MUE is obtained in a closed form based on the transmit powers of the SUEs through proper mathematical manipulations. We prove the convergence of the proposed power control scheme, and we also theoretically show the existence and uniqueness of the Stackelberg equilibrium (SE) in the formulated Stackelberg game. The simulation results show that the proposed power control scheme provides considerable improvements, particularly for the MUE.