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Resource allocation promises significant benefits in wireless networks. In order to fully reap these benefits, it is important to design efficient resource allocation algorithms. Here, we develop relay power allocation (RPA) algorithms for coherent and noncoherent amplify-and-forward (AF) relay networks. The goal is to maximize the output signal-to-noise ratio under individual as well as aggregate relay power constraints. We show that these RPA problems, in the presence of perfect global channel state information (CSI), can be formulated as quasiconvex optimization problems. In such settings, the optimal solutions can be efficiently obtained via a sequence of convex feasibility problems, in the form of second-order cone programs. The benefits of our RPA algorithms, however, depend on the quality of the global CSI, which is rarely perfect in practice. To address this issue, we introduce the robust optimization methodology that accounts for uncertainties in the global CSI. We show that the robust counterparts of our convex feasibility problems with ellipsoidal uncertainty sets are semi-definite programs. Our results reveal that ignoring uncertainties associated with global CSI often leads to poor performance, highlighting the importance of robust algorithm designs in practical wireless networks.