We consider the problem of maximizing the energy efficiency (EE) for a MIMO interference channel (IC), with the power constraint on each link. To obtain totally distributed solutions, this problem is formulated as a noncooperative game. We show that this game always admits a Nash equilibra (NE). Importantly, the sufficient condition that one can check to guarantee the uniqueness of the NE is derived. To reach the NE of this game, we provide a totally distributed EE algorithm, in which each player employs the fractional programming to update his own solution. These updates can be performed in a completely distributed and asynchronous fashion. Sufficient conditions that guarantee the convergence of the algorithm have been given as well. Simulation results show that the proposed algorithm converges fast and significantly outperforms the existing algorithms in terms of the sum-EE or the sum-rate.
In contrast to the most of the above cited papers which focus on the (weighted) sum SE problem, in this work we consider the EE maximization problem. For SE optimization problems, it is known that all the transmitters use full power during transmission in order to maximize its own SE. Based on this fact, the best response strategy at the NE can be written in a closed-form water-filling solution, which can be interpreted as a projector on the convex and closed set. This interpretation enables the authors to derive the uniqueness of the game's NE @cite_49 . However, the study of EE maximization problem cannot be obtained by employing the methodologies since the transmitters in fact use a portion of the power, instead of full, to achieve energy efficient transmission.
@cite_49
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This paper considers the noncooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general multiple-input multiple-output (MIMO) case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game.