Optimal multi-antenna relay beamforming with per-antenna power control

We consider amplify-and-forward multi-antenna relaying between a single pair of source and destination under per-antenna power constraints. Our objective is to obtain the optimal relay processing matrix to minimize the maximum individual antenna power for a given received SNR target. The problem is not convex, but it can be shown to satisfy strong Lagrange duality. We reveal a prominent structure of this problem, by establishing its duality with direct point-to-point SIMO beamforming with an uncertain noise. This enables us to derive a semi-closed form expression for the optimal relay processing matrix that depends on a set of dual variables, thus converting the original optimization of a N×N matrix with (N+1) constraints, to a dual problem with (N+1) variables and three constraints. We further show that the dual problem has a semi-definite programming form, so that the proposed solution has polynomial worst-case complexity.