Multi-hop communication is a spectral-efficient approach for connecting multiple pairs when direct links are absent or have insufficient strength. In this paper, we highlight the benefits of improper Gaussian signaling for a multi-pair full-duplex MIMO relay network in terms of power efficiency. By employing improper Gaussian transmission, the power minimization problem under rate constraints is intrinsically a non-convex problem due to non-convex rate expressions in the constraint set. We utilize Fenchel's inequality to linearize the non-convex part of the constraint, which results in feasible solutions of the problem in polynomial time. This approximation results in an upper-bound for the original problem which gets tighter in the number of iterations performed. The solution is compared with analytical beamforming solutions, namely zero-forcing (ZF) and maximum-ratio transmission/combining (MRT/MRC). Due to the feasibility of single rank real-valued transmission in improper Gaussian signaling, the optimal solution switches to single rank transmission depending on the constraints defined by the requests of the users, which can not be captured by proper Gaussian signaling.