# The Optimization of UAV-Assisted Downlink Transmission Based on RSMA
## 3 Mathematic Modeling
### Model Solving Optimization
\[v_{\text{down}}(i,j)-ve_{\text{down}}(i,j)\geq 0,1\leq i\leq nc_{\text{down}},1 \leq j\leq nu_{\text{down}}(i) \tag{3}\]  
\[\mathcal{L}_{\rho}(X,\lambda)=f(X)+\frac{1}{2\rho}\sum_{i=1}^{nc}\sum_{j=1}^{ nu(i)}\biggl{\{}\Bigl{[}\max\Bigl{(}0,\lambda^{i,j}-\rho g_{i,j}(X)\Bigr{)} \Bigr{]}^{2}-\Bigl{(}\lambda^{i,j}\Bigr{)}^{2}\biggr{\}} \tag{4}\]  
Eventually, the inequality constraint problem (2) is transformed into the unconstrained problem (4), which can be further solved using an unconstrained optimization algorithm. Considering the complexity of the problem and the shortcomings of traditional methods for solving this type of problems [11], this paper will use a heuristic algorithm to solve the problem and finally designs a dynamic memory fish swarm algorithm, which is described in Section 3. * User group parameters' optimization problem Group parameters optimization involves managing resources such as power and bandwidth among multiple users in multicast or broadcast scenarios. It ensures efficient resource utilization, prevents service degradation, and balances power allocation, providing reliable service for all users in dense networks. Firstly, the problem is abstractly modeled by considering the optimization of the user group parameters of the downlink system, whose optimization parameters include the bandwidth allocation ratio (5) and the power allocation ratio (6) between the groups, and thus the problem can be described as solving for the optimal downlink bandwidth allocation ratio as well as the power allocation ratio \(P_{\text{down}}^{\text{SP}}\) so that the downlink rate and the optimal and satisfy the user's rate requirements. \[\left\{ \begin{array}{ll}\arg&\min_{g_{\text{down}}^{\text{SP}}}V_{\text{ target}}\\ s.t.&B_{\text{down}}^{\text{SP}}=\bigl{(}bf_{\text{down}}^{1},bf_{\text{down}}^{2}, \ldots,bf_{\text{down}}^{n_{\text{down}}}\bigr{)}\\ &\sum_{i=1}^{n_{\text{down}}}bf_{\text{down}}^{i}=1\\ &bf_{\text{down}}^{i}\geq 0,1\leq i\leq nc_{\text{down}}\end{array} \right. \tag{5}\]  
where \(bf_{\text{down}}^{i}\) represents the uplink bandwidth allocated to user group \(i\), \(bf_{\text{down}}^{i}\) represents the proportion of power shared by the \(i\)-th user group. \[\left\{ \begin{array}{ll}\arg&\min_{g_{\text{down}}^{\text{SP}}}V_{\text{ target}}\\ s.t.&P_{\text{down}}^{\text{SP}}=\bigl{(}bf_{\text{down}}^{1},bf_{\text{down}}^{2}, \ldots,bf_{\text{down}}^{n_{\text{down}}}\bigr{)}\\ &\sum_{i=1}^{n_{\text{down}}}bf_{\text{down}}^{i}=1\\ &pf_{\text{down}}^{i}\geq 0,1\leq i\leq nc_{\text{down}}\end{array} \right. \tag{6}\]  
* RSMA parameters' optimization problem RSMA optimizes the splitting of user messages, power allocation, and bandwidth distribution to manage interference and enhance network efficiency. It enables better handling of user heterogeneity, improves spectral efficiency, and plays a key role in next-generation networks like 5G and 6G, particularly in UAV-assisted communications. Consider the optimization of RSMA parameters for the downlink system, which consists of the common message splitting ratio \(R_{\text{down}}^{\text{SP}}\) and the power allocation ratio within the group \(PU_{\text{down}}^{\text{SP}}\). The problem can be described as solving for the optimal \(R_{\text{down}}^{\text{SP}}\) (as in Equation (7)) and \(PU_{\text{down}}^{\text{SP}}\) (as in Equation (8)) such that the transmission rate and is maximized while satisfying the user rate requirements. \[ \begin{cases}\arg&\min_{R_{\text{down}}^{\text{SP}}}V_{\text{target}}\\ s.t.&R_{\text{down}}^{\text{SP}}=\left({prf_{\text{down}}^{1}rf_{\text{down} }^{2}\dots,rf_{\text{down}}^{mt_{\text{down}}}}\right)\\ &\sum_{i=1}^{mt_{\text{down}}}rf_{\text{down}}^{i}=1\\ &rf_{\text{down}}^{i}\geq 0,1\leq i\leq mu_{\text{down}}\end{cases}  \tag{7}\]  
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