  In a wireless system with a large number of distributed nodes, the quality of
communication can be greatly improved by pooling the nodes to perform joint
transmission/reception. In this paper, we consider the problem of optimally
selecting a subset of nodes from potentially a large number of candidates to
form a virtual multi-antenna system, while at the same time designing their
joint linear transmission strategies. We focus on two specific application
scenarios: 1) multiple single antenna transmitters cooperatively transmit to a
receiver; 2) a single transmitter transmits to a receiver with the help of a
number of cooperative relays. We formulate the joint node selection and
beamforming problems as cardinality constrained optimization problems with both
discrete variables (used for selecting cooperative nodes) and continuous
variables (used for designing beamformers). For each application scenario, we
first characterize the computational complexity of the joint optimization
problem, and then propose novel semi-definite relaxation (SDR) techniques to
obtain approximate solutions. We show that the new SDR algorithms have a
guaranteed approximation performance in terms of the gap to global optimality,
regardless of channel realizations. The effectiveness of the proposed
algorithms is demonstrated via numerical experiments.
