In this paper, we consider an unmanned aerial vehicle (UAV) enabled relaying system where multiple UAVs are deployed as aerial relays to support simultaneous communications from a set of source nodes to their destination nodes on the ground. An optimization problem is formulated under practical channel models to maximize the minimum achievable expected rate among all pairs of ground nodes by jointly designing UAVs three-dimensional (3D) placement as well as the bandwidth-and-power allocation. This problem, however, is non-convex and thus difficult to solve. As such, we propose a new method, called iterative Gibbs-sampling and block-coordinate-descent (IGS-BCD), to efficiently obtain a high-quality suboptimal solution by synergizing the advantages of both the deterministic (BCD) and stochastic (GS) optimization methods. Specifically, our proposed method alternates between two optimization phases until convergence is reached, namely, one phase that uses the BCD method to find locally-optimal UAVs 3D placement and the other phase that leverages the GS method to generate new UAVs 3D placement for exploration. Moreover, we present an efficient method for properly initializing UAVs placement that leads to faster convergence of the proposed IGS-BCD algorithm. Numerical results show that the proposed IGS-BCD and initialization methods outperform the conventional BCD or GS method alone in terms of convergence-and-performance trade-off, as well as other benchmark schemes.
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