This paper deals with joint source-relay-destination beamforming (BF) design for an amplify-and-forward (AF) relay network. Considering the channel state information (CSI) from the relay to the destination is imperfect, we first aim to maximize the worst case received SNR under per-antenna power constraints. The associated optimization problem is then solved in two steps. In the first step, by revealing the rank-one property of the optimal relay BF matrix, we establish the semi-closed form solution of the joint optimal BF design that only depends on a vector variable. Based on this result, in the second step, we propose a low-complexity iterative algorithm to obtain the remaining unknown variable. We also study the problem for minimizing the maximum per-antenna power at the relay while ensuring a received signal-to-noise ratio (SNR) target, and show that it reduces to the SNR maximization problem. Thus the same methods can be applied to solve it. The differences between our result and the existing related work are also discussed in details. In particular, we show that in the perfect CSI case, our algorithm has the same performance but with much lower cost of computational complexity than the existing method. Finally, in the simulation part, we investigate the impact of imperfect CSI on the system performance to verify our analysis.