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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.

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Abstract. Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in N -dimensional real or complex Euclidean space, such that M concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by O(Q 2 (M − Q + 1) + M 2 ) in the real case and by O(M (M − Q + 1)) in the complex case. For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by O(ǫ/ ln(M )) for both the real and the complex cases. Moreover, this ratio is tight up to a constant factor.Key words. nonconvex quadratic constrained quadratic programming, semidefinite programming relaxation, approximation bound, NP-hard AMS subject classifications. 90C22, 90C20, 90C591. Introduction. Motivated by applications in wireless communications, we study in this paper two classes of mixed binary nonconvex quadratically constrained quadratic programming (MBQCQP) problems, where the objective functions are quadratic in the continuous variables and the constraints contain continuous and binary variables. Although these two classes of optimization problems are nonconvex, they are amenable to semidefinite programming (SDP) relaxation. The focus of our study is on the approximation bounds of the SDP relaxation for both problems.The minimization model. Consider the following MBQCQP problem: