  Owing to the structure of the Gaussian multiple-input multiple-output (MIMO)
broadcast channel (BC), associated optimization problems such as capacity
region computation and beamforming optimization are typically non-convex, and
cannot be solved directly. One feasible approach to these problems is to
transform them into their dual multiple access channel (MAC) problems, which
are easier to deal with due to their convexity properties. The conventional
BC-MAC duality is established via BC-MAC signal transformation, and has been
successfully applied to solve beamforming optimization,
signal-to-interference-plus-noise ratio (SINR) balancing, and capacity region
computation. However, this conventional duality approach is applicable only to
the case, in which the base station (BS) of the BC is subject to a single sum
power constraint. An alternative approach is minimax duality, established by Yu
in the framework of Lagrange duality, which can be applied to solve the
per-antenna power constraint problem. This paper extends the conventional
BC-MAC duality to the general linear constraint case, and thereby establishes a
general BC-MAC duality. This new duality is applied to solve the capacity
computation and beamforming optimization for the MIMO and multiple-input
single-output (MISO) BC, respectively, with multiple linear constraints.
Moreover, the relationship between this new general BC-MAC duality and minimax
duality is also presented. It is shown that the general BC-MAC duality offers
more flexibility in solving BC optimization problems relative to minimax
duality. Numerical results are provided to illustrate the effectiveness of the
proposed algorithms.
