Optimization under multiple linear constraints is important for practical systems with individual power constraints, per-antenna power constraints, and/or interference constraints as in cognitive radios. While for single-user multiple-input multiple-output (MIMO) channel transmitter optimization, no one uses general purpose convex programming because water-filling is optimal and much simpler, it is not true for MIMO multiaccess channels (MAC), broadcast channels (BC), and the nonconvex optimization of interference networks because the traditional water-filling is far from optimal for networks. We recently found the right form of water-filling, polite water-filling, for capacity or achievable regions of the general MIMO interference networks, named B-MAC networks, which include BC, MAC, interference channels, X networks, and most practical wireless networks as special cases. In this paper, we extend the polite water-filling results from a single linear constraint to multiple linear constraints and use weighted sum-rate maximization as an example to show how to design high efficiency and low complexity algorithms, which find optimal solution for convex cases and locally optimal solution for nonconvex cases. Several times faster convergence speed and orders of magnitude higher accuracy than the state-of-the-art are demonstrated by numerical examples.
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