Outage Constrained Robust Transmit Optimization for Multiuser MISO Downlinks: Tractable Approximations by Conic Optimization

In this paper, we study a probabilistically robust transmit optimization problem under imperfect channel state information (CSI) at the transmitter and under the multiuser multiple-input single-output (MISO) downlink scenario. The main issue is to keep the probability of each user's achievable rate outage as caused by CSI uncertainties below a given threshold. As is well known, such rate outage constraints present a significant analytical and computational challenge. Indeed, they do not admit simple closed-form expressions and are unlikely to be efficiently computable in general. Assuming Gaussian CSI uncertainties, we first review a traditional robust optimization-based method for approximating the rate outage constraints, and then develop two novel approximation methods using probabilistic techniques. Interestingly, these three methods can be viewed as implementing different tractable analytic upper bounds on the tail probability of a complex Gaussian quadratic form, and they provide convex restrictions, or safe tractable approximations, of the original rate outage constraints. In particular, a feasible solution from any one of these methods will automatically satisfy the rate outage constraints, and all three methods involve convex conic programs that can be solved efficiently using off-the-shelf solvers. We then proceed to study the performance-complexity tradeoffs of these methods through computational complexity and comparative approximation performance analyses. Finally, simulation results are provided to benchmark the three convex restriction methods against the state of the art in the literature. The results show that all three methods offer significantly improved solution quality and much lower complexity.