On the Duality of the Max–Min Beamforming Problem With Per-Antenna and Per-Antenna-Array Power Constraints

This paper considers a downlink unicast transmission in a multicell network with multiple users. In a network with frequency reuse factor of one, intercell interference is a limiting factor. The max-min beamforming technique enables a balancing of the signal-to-interference-plus-noise ratio (SINR) among all users in a network under a power budget. Thus, a fair distribution of the achievable rate can be achieved. The max-min beamforming problem (MBP) is nonconvex in general. However, if instantaneous channel-state information (CSI) is available, the MBP has an equivalent quasiconvex form and can optimally be solved with an efficient algorithm based on a convex solver. In addition to this convex-solver-based solution, this paper considers the so-called surrogate dual problem of the MBP with per-antenna and per-antenna-array power constraints. The surrogate dual problem combines multiple power constraints to a single power constraint. Furthermore, the surrogate dual problem can efficiently be solved for long-term CSI in the form of spatial correlation matrices. Strong duality is proved for instantaneous and long-term CSI in the form of higher rank spatial correlation matrices. With the surrogate dual problem, a fast algorithm for the MBP is presented. The convergence issue is discussed. Numerical results verify the convergence and the performance of the proposed algorithm.