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In this paper, we address the problems of power allocation at relays to maximize the transmission rate in a nonorthogonal amplify-and-forward (AF) relay-assisted network, in which multiple relays simultaneously transmit the amplified signals in the same time and frequency. We first formulate a power allocation problem that aims to maximize the minimum of transmission rates in a multicast under a sum power constraint. Then, we prove that the optimal solution can be alternatively obtained by solving its inverse problem, which minimizes the sum power consumption at relays while achieving the target signal-to-noise ratio (SNR) with the same constraint. The inverse problem can be solved by using linear programming, and the inverse problem's complexity can be further reduced by a proposed suboptimal power allocation algorithm. As in the special case of multicast, the optimal solution is derived in unicast, and two suboptimal algorithms are proposed to reduce the feedback overhead and complexity, in which multirelay diversity is also investigated.