On Gaussian MIMO BC-MAC Duality With Multiple Transmit Covariance   Constraints
  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. 
Lan Zhang, Rui Zhang, Ying-Chang Liang, Yan Xin, H. Vincent Poor