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This paper considers robust transmit beamforming for multiuser multi-input single-output (MISO) downlink transmission, where imperfect channel state information (CSI) is assumed at the base station (BS). The imperfect CSI is captured by a moment-based random error model, in which the BS knows only the mean and covariance of each CSI error, but not the exact distribution. Under this error model, we formulate a distributionally robust beamforming (DRB) problem, in which the total transmit power at the BS is to be minimized, while each user's SINR outage probability, evaluated w.r.t. any distribution with the given mean and covariance, is kept below a given threshold. The DRB problem is a semi-infinite chance-constrained problem. By employing recent results in distributionally robust optimization, we show that the DRB problem admits an explicit conic reformulation, which can be conveniently turned into a convex optimization problem after semidefinite relaxation (SDR). We also consider the case where the mean and covariance are not perfectly known. We show that the resulting DRB problem still admits a conic reformulation and can be approximately solved using SDR. The robustness of the proposed designs are demonstrated by numerical simulations.

Consider a wireless scenario in which a multi-antenna transmitter wants to send a confidential message to a single-antenna information receiver (IR) while transferring wireless energy to a number of multi-antenna energy receivers (ERs). In order to keep the ERs from retrieving the confidential message, an artificial noise (AN)-aided physical-layer secrecy approach is employed at the transmitter. The AN has dual purpose: First, it can interfere with the ERs' information receptions and thus help improve security. Secondly, it provides wireless energy for the ERs to harvest. Assuming imperfect channel state information at the transmitter, we jointly optimize the co-variances of confidential information and AN such that the secrecy rate at the IR is maximized, while each ER receives a prescribed amount of wireless energy. Although this secrecy-rate maximization problem is non-convex, we show that it can be handled by solving a sequence of convex optimization problems. Numerical results are provided to demonstrate the efficacy of the proposed design.

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.