Robust SINR-constrained MISO downlink beamforming: when is semidefinite programming relaxation tight?

Abstract

We consider the multiuser beamforming problem for a multi-input single-output downlink channel that takes into account the errors in the channel state information at the transmitter side (CSIT). By modeling the CSIT errors as elliptically bounded uncertainty regions, this problem can be formulated as minimizing the transmission power subject to the worst-case signal-to-interference-plus-noise ratio constraints. Several methods have been proposed to solve this nonconvex optimization problem, but none can guarantee a global optimal solution. In this article, we consider a semidefinite relaxation (SDR) for this multiuser beamforming problem, and prove that the SDR method actually solves the robust beamforming problem to global optimality as long as the channel uncertainty bound is sufficiently small or when the transmitter is equipped with at most two antennas. Numerical examples show that the proposed SDR approach significantly outperforms the existing methods in terms of the average required power consumption at the transmitter.

Robust Monotonic Optimization Framework for Multicell MISO Systems

“…Lemma 7: Given , with , the expression (33) holds if and only if (34) Proof: The proof is given in [57, Proposition 2].…”

“…Under these conditions, the transmission should obviously satisfy the first property. The use of transmit beamforming is actually optimal under single-user detection (i.e., the second property) if perfect CSI is available [6], [32], [33], while [34] provides conditions on its optimality under channel uncertainty. From an information theoretic perspective, transmit beamforming and simple receivers are suboptimal [35] but these assumptions are of practical importance to achieve low-complexity receivers and power efficiency.…”

Low-complexity robust MISO downlink precoder optimization for the limited feedback case

“…When each Wi has rank one, the vectors Wi are optimal. Numerical evidence, and theoretical results for closely related problems with small uncertainty sets [8], suggest that such a solution almost always exists.…”

“…Under the uncertainty model in (8), the problem in (6) of maximizing the zero-outage region subject to a power constraint is This problem is difficult to solve for two reasons. First the rank con straint is non-convex.…”

“…One way to do that is to model the ac tual channel as lying in a compact set around the channel estimate at the BS, and to design the precoder so that the transmission power is minimized subject to the SINR constraints holding for all channels in the set; e.g., [5][6][7][8]. An alternative to that "worst-case " approach is to model the uncertainties probabilistically, and to seek to mini mize the transmission power subject to the SINR constraints holding with a given probability; i.e.…”