# The Optimization of UAV-Assisted Downlink Transmission Based on RSMA
## 3 Mathematic Modeling
### Model Solving Optimization  
Due to the significant complexity of solving a problem directly (1), this paper decomposes it into three sub-problems: UAV position parameter optimization, user group parameter optimization and RSMA parameter optimization, and designs a ternary cross iterative optimization algorithm to complete the original problem solving. By grouping the UAV position \(pos_{u}\) as the UAV parameter, the bandwidth allocation \(B_{\text{down}}\) as well as the inter-group power allocation in \(P_{\text{down}\_\text{aloc}}\) act as the user group resource parameter. The intra-group power allocation in \(P_{\text{down}\_\text{aloc}}\) is grouped into RSMA parameters, and the RSMA parameters of the downlink optimization system can still be solved independently for each user group. * UAV parameter's optimization problem Optimizing UAV parameters such as position, altitude, and power is crucial for enhancing communication performance, especially when UAVs are used as flying base stations in areas with limited infrastructure. This improves coverage, reduces interference, and ensures efficient energy use, making it vital for applications like disaster recovery and remote connectivity. The optimization problem regarding the UAV position parameter can be abstracted into the following form: \[\left\{ \begin{array}{ll}\arg\!\min_{pos_{u}}V_{\text{target}}\\ \text{subject to }uav_{lc}\\ v_{rc}\end{array} \right.\] (2) where \[V_{\text{target}}\] is the opposite number of the sum of downlink transmission rate. \[uav_{lc}\] is the location constraint of the UAV. \[v_{rc}\] is the user's downlink transmission rate constraint. In this case, the user group parameters as well as the RSMA parameters are treated as constants. In Equation (2), the first constraint determines the deployable region of the UAV, i.e., the search space of the solution, and the second constraint is the rate constraint, which corresponds to the rate demand constraint of Equation (1), which are inequality constraints. The Augmented Lagrange Method (ALM) is used in this paper to transform an inequality constrained optimization problem into an unconstrained optimization problem for further solution. We transform the rate requirement constraints of Equation (1) into the forms of Equation (3), then the left-hand side of the inequality is denoted as \(g_{i,j}(X)\). The objective function of (2) is denoted as \(f(X)\), where \(X=(x,y,z)\) is the coordinates of the UAV. According to reference [10], the augmented Lagrange function of (2) is calculated as (4), where \(nc\) is the number of user groups, \(nu(i)\) is the number of users in group \(i\), \(\lambda^{i,j}\) is the Lagrange multiplier, and \(\rho(\rho>0)\) is the penalty parameter. The Lagrange multiplier is updated by \(\lambda^{i,j}_{t}=\min\Bigl{(}\max\Bigl{(}0,\lambda^{i,j}_{t-1}-\rho g_{i,j}(X_{ t-1})\Bigr{)},\lambda^{ij}_{\max}\Bigr{)}\), where \(\lambda^{i,j}_{t}\) represents the value of \(\lambda^{ij}\) when the number of iterations reaches \(t\), \(\lambda^{i,j}_{\max}\) represents the upper limit of \(\lambda^{i,j}\), and \(X_{t-1}\) represents the value of solution obtained when the number of iterations reaches \(t-1\). In this paper, we use dynamic penalty function; the penalty parameter \(\rho\) is updated by \(\rho_{t}=\rho t_{\min}+(\rho t_{\max}-\rho t_{\min})\frac{t\sqrt{t-1}}{t\sqrt{t -1}}\), where \(t\) is the number of iterations, \([\rho t_{\min},\rho t_{\max}]\) is the range of values of \(\rho\), and \(t_{m}\) is the maximum number of iterations. As the number of iterations increases, the penalty parameter \(\rho\) also increases.