Robust beamforming in two-way relay networks: Quartically perturbed chance constrained formulation and tractable approximation

In this paper, we consider an outage-based robust beamforming problem in two-way relay networks under the imperfect channel state information (CSI) scenario. Specficially, our goal is to minimize the relay transmit power while keeping the probability of each user´s signal-to-interference-plus-noise ratio (SINR) outage as caused by the imperfect CSI below a given threshold. Assuming that the CSI errors follow a complex Gaussian distribution, the probabilistic SINR constraints involve quartic polynomials of complex Gaussian random variables, which, to the best of our knowledge, have not been treated from a computational perspective before. Using moment inequalities for Gaussian polynomials and the semidefinite relaxation technique, we propose a new tractable approximation approach for tackling such constraints. Simulation results show that the proposed method outperforms the existing robust approaches when the CSI errors are large.