  This paper gives the long sought network version of water-filling named as
polite water-filling. Unlike in single-user MIMO channels, where no one uses
general purpose optimization algorithms in place of the simple and optimal
water-filling for transmitter optimization, the traditional water-filling is
generally far from optimal in networks as simple as MIMO multiaccess channels
(MAC) and broadcast channels (BC), where steepest ascent algorithms have been
used except for the sum-rate optimization. This is changed by the polite
water-filling that is optimal for all boundary points of the capacity regions
of MAC and BC and for all boundary points of a set of achievable regions of a
more general class of MIMO B-MAC interference networks, which is a combination
of multiple interfering broadcast channels, from the transmitter point of view,
and multiaccess channels, from the receiver point of view, including MAC, BC,
interference channels, X networks, and most practical wireless networks as
special case. It is polite because it strikes an optimal balance between
reducing interference to others and maximizing a link's own rate. Employing it,
the related optimizations can be vastly simplified by taking advantage of the
structure of the problems. Deeply connected to the polite water-filling, the
rate duality is extended to the forward and reverse links of the B-MAC
networks. As a demonstration, weighted sum-rate maximization algorithms based
on polite water-filling and duality with superior performance and low
complexity are designed for B-MAC networks and are analyzed for Interference
Tree (iTree) Networks, a sub-class of the B-MAC networks that possesses
promising properties for further information theoretic study.
