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We study a downlink non-orthogonal multiple access system with cooperative full-duplex relaying, where the near user in terms of the base station (BS) is enabled to act as a full-duplex relay for the far user. In particular, we first derive the outage probability and ergodic sum rate with closed-form expressions when the power allocations at the BS and relay (or the near user) are fixed. Then, we analytically obtain the optimal power allocations with closed-form expressions at the BS and relay to minimize the outage probability. Furthermore, by taking the fairness between the near user and far user into account, we characterize the optimal power allocations with closed-form expressions at the BS and relay to maximize the minimum achievable rate of users. Simulation results validate the correctness of the theoretical analysis and demonstrate the advantages of the proposed algorithms over the state of the art.