This paper investigates a power allocation problem in an optimal and closed-form manner for a relay-aided cognitive network composed of one primary and secondary user/destination pair (e.g., a cellular link coexisting with a device-to-device link in a device-to-device enabled cellular communication setup) and one secondary full-duplex relay. By exploiting the monotonic properties of the objective function and the geometry of the feasible set, we provide the optimal and closed-form power allocation policy assuming the relay performs either Decode-and-Forward (DF) or Compress-and-Forward (CF). We then conjecture that if the secondary direct link is missing, then DF always outperforms CF, irrespective from the system parameters. This conjecture is sustained by numerous numerical simulations.