Weak Duality and Iterative Beamforming Algorithms for Ad Hoc Networks

A unicasting Ad hoc network is considered, where each node i is equipped with a receive/transmit beamformer pair (wi,gi). Each node’s SNR Γi satis£es Γi ≥ γ0. It is £rst shown that the minimum sum power beamformers for the network satisfy a weak duality condition, in which the pairs (g i ,w ∗ i ) achieve the same sum power as the primal network. However, the optimum receive beamformer wi is not in general equal to the optimum g i , in contrast to the case of cellular and time-division duplexing (TDD) networks. A suboptimal iterative beamforming algorithm is then proposed in which wi = g i is enforced. The algorithm is shown to be an instance of the Power Algorithm in which gi is the maximizing eigenvector of an SNR-related objective matrix. The beamforming algorithm is shown to have a Game Theory interpretation, in which the payoff is SNR, and the tax is related to interference caused to other nodes. The algorithm is also proven to have a quasi-Nash equilibrium(a). A sequential beamforming algorithm for rank-1 channels is also presented, which is proven to converge to an equilibrium and minimize a total interference function.