SDR approximation bounds for the robust multicast beamforming problem with interference temperature constraints

In this work, we consider the robust beamforming design for secondary downlink multicasting channels, where primary users are present with norm-bounded channel errors. In particular, the max-min-fair formulation is considered and the resulting design problem is a quadratically constrained quadratic program (QCQP) with a set of semi-infinite constraints, which is NP-hard in general. As a remedy, we apply the semidefinite relaxation (SDR) technique and S-lemma to approximate the problem into a tractable form. The key contribution of this paper is to study the approximation quality. Our analytical results show that, the SDR solution achieves an objective value that is at least Ω(1/MN log J) times the optimal objective value, where M is the number of secondary users, J is the number of primary users, and N is the number of antennas at the secondary base station. This is a fundamentally new result for SDR applied to robust QCQPs. Practically, it provides a performance guarantee for the robust beamforming design. All these results are verified by our numerical simulations.