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

Multicast downlink transmission in a multi-cell network with multiple users is investigated. Max–min beamforming (MB) enables a fair distribution of the signal-to-interferenceplus-noise-ratio among all users in a network for given power constraints at the base stations of the network. The multicast MB problem (MBP) is proved to be NP-hard and non-convex in general. However, the MBP has an equivalent quasi-convex 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 quasiconvex form of the MBP for the practically relevant scenario of long-term CSI in the form of Hermitian positive semi-definite Toeplitz matrices and per-antenna array power constraints. The optimization problem is then given by a convex feasibility check problem with finite auto-correlation sequences (FASs) as optimization variables. Using FASs the MBP can be expressed as a quasi-convex fractional program. Based on the theory of quasiconvex programming, this paper proposes a fast root-finding algorithm with super-linear convergence.