  Consider a multi-input multi-output (MIMO) downlink multi-user channel. A
well-studied problem in such system is the design of linear beamformers for
power minimization with the quality of service (QoS) constraints. The most
representative algorithms for solving this class of problems are the so-called
MMSE-SOCP algorithm [11-12] and the UDD algorithm [9]. The former is based on
alternating optimization of the transmit and receive beamformers, while the
latter is based on the well-known uplink-dowlink duality theory. Despite their
wide applicability, the convergence (to KKT solutions) of both algorithms is
still open in the literature. In this paper, we rigorously establish the
convergence of these algorithms for QoS-constrained power minimization (QCPM)
problem with both single stream and multiple streams per user cases. Key to our
analysis is the development and analysis of a new MMSE-DUAL algorithm, which
connects the MMSE-SOCP and the UDD algorithm. Our numerical experiments show
that 1) all these algorithms can almost always reach points with the same
objective value irrespective of initialization, 2) the MMSE-SOCP/MMSE-DUAL
algorithm works well while the UDD algorithm may fail with an infeasible
initialization.
