Underlay cognitive networks are being promoted as an implementable solution for the congestion in the wireless spectrum. With some nodes employed as relays, the throughput of such networks is enhanced through cooperative gains. In this paper, we address the problem of maximizing the throughput of an underlay cognitive network through optimal power allocation of non-orthogonal amplify-and-forward relays. In order to maintain stable communication at the primary network, the interference due to the activities of all the cognitive nodes is kept below a limit. The optimal power allocation problem is formulated and then transformed to a quadratically constrained quadratic problem (QCQP). The maximum throughput of the cognitive network is derived as an eigen-solution of a channel-dependent matrix and the corresponding signal-to-noise ratio (SNR) is shown to be the dominant eigenvalue of this matrix. The proposed power allocation achieves significant throughput improvement over equal power allocation scheme and opportunistic relay selection scheme.