Equivalent Quasi-Convex Form of the Multicast Max–Min Beamforming Problem

Multicast downlink transmission in a multicell network with multiple users is investigated. Max-min beamforming (MB) enables a fair distribution of the signal-to-interference-plus-noise ratio (SINR) among all users in a network for given power constraints at the base stations (BSs) of the network. The multicast MB problem (MBP) is proven to be NP-hard and nonconvex in general. However, the MBP has an equivalent quasi-convex (QC) form and can be optimally solved with an efficient algorithm for special instances, depending on the structure of the available channel state information (CSI). This paper derives the equivalent QC form of the MBP for the practically relevant scenario of long-term CSI in the form of Hermitian positive semi-definite Toeplitz (HPST) matrices and per-antenna array power constraints. The optimization problem is then given by a convex feasibility check problem with finite autocorrelation sequences (FASs) as optimization variables. Using FASs, the MBP can be expressed as a QC fractional program (FP). Based on the theory of QC programming, this paper proposes a fast root-finding algorithm with superlinear convergence.