  This paper studies a multi-UAV wireless network, in which multiple UAV users
share the same spectrum to send individual messages to their respectively
associated ground base stations (GBSs). The UAV users aim to optimize their
locations to maximize the weighted sum rate. While most existing work considers
simplified line-of-sight (LoS) or statistic air-to-ground (A2G) channel models,
we exploit the location-specific channel knowledge map (CKM) to enhance the
placement performance in practice. However, as the CKMs normally contain
discrete site- and location-specific channel data without analytic model
functions, the corresponding weighted sum rate function becomes
non-differentiable in general. In this case, conventional optimization
techniques relying on function derivatives are inapplicable to solve the
resultant placement optimization problem. To address this issue, we propose a
novel iterative algorithm based on the derivative-free optimization. In each
iteration, we first construct a quadratic function to approximate the
non-differentiable weighted sum rate under a set of interpolation conditions,
and then update the UAVs' placement locations by maximizing the approximate
quadratic function subject to a trust region constraint. Numerical results show
the convergence of the proposed algorithm. It is also shown that the proposed
algorithm achieves a weighted sum rate close to the optimal design based on
exhaustive search with much lower implementation complexity, and it
significantly outperforms the conventional optimization method based on
simplified LoS channel models and the heuristic design with each UAV hovering
above its associated GBS.
