  Cooperative beamforming in relay networks is considered, in which a source
transmits to its destination with the help of a set of cooperating nodes. The
source first transmits locally. The cooperating nodes that receive the source
signal retransmit a weighted version of it in an amplify-and-forward (AF)
fashion. Assuming knowledge of the second-order statistics of the channel state
information, beamforming weights are determined so that the signal-to-noise
ratio (SNR) at the destination is maximized subject to two different power
constraints, i.e., a total (source and relay) power constraint, and individual
relay power constraints. For the former constraint, the original problem is
transformed into a problem of one variable, which can be solved via Newton's
method. For the latter constraint, the original problem is transformed into a
homogeneous quadratically constrained quadratic programming (QCQP) problem. In
this case, it is shown that when the number of relays does not exceed three the
global solution can always be constructed via semidefinite programming (SDP)
relaxation and the matrix rank-one decomposition technique. For the cases in
which the SDP relaxation does not generate a rank one solution, two methods are
proposed to solve the problem: the first one is based on the coordinate descent
method, and the second one transforms the QCQP problem into an infinity norm
maximization problem in which a smooth finite norm approximation can lead to
the solution using the augmented Lagrangian method.
