  In this paper we consider a probabilistic signal-to-interference and-noise
ratio (SINR) constrained problem for transmit beamforming design in the
presence of imperfect channel state information (CSI), under a multiuser
multiple-input single-output (MISO) downlink scenario. In particular, we deal
with outage-based quality-of-service constraints, where the probability of each
user's SINR not satisfying a service requirement must not fall below a given
outage probability specification. The study of solution approaches to the
probabilistic SINR constrained problem is important because CSI errors are
often present in practical systems and they may cause substantial SINR outages
if not handled properly. However, a major technical challenge is how to process
the probabilistic SINR constraints. To tackle this, we propose a novel
relaxation- restriction (RAR) approach, which consists of two key
ingredients-semidefinite relaxation (SDR), and analytic tools for
conservatively approximating probabilistic constraints. The underlying goal is
to establish approximate probabilistic SINR constrained formulations in the
form of convex conic optimization problems, so that they can be readily
implemented by available solvers. Using either an intuitive worst-case argument
or specialized probabilistic results, we develop various conservative
approximation schemes for processing probabilistic constraints with quadratic
uncertainties. Consequently, we obtain several RAR alternatives for handling
the probabilistic SINR constrained problem. Our techniques apply to both
complex Gaussian CSI errors and i.i.d. bounded CSI errors with unknown
distribution. Moreover, results obtained from our extensive simulations show
that the proposed RAR methods significantly improve upon existing ones, both in
terms of solution quality and computational complexity.
