## 3 Proposed Joint Resource Optimization for NOMA-Enabled Vehicular Networks
### Solution Variants for Reducing Computational Complexity
#### 3.3.2 Reformulation with Utility Function and Load Balancing  
One effective way of reducing the complexity of an optimization problem is to reduce the number of binary/integer variables, and further partitioning of the problem so that all or part of the problem can be solved in a distributed manner. To address the computational challenges, we reformulate the first-phase problem through a three-step hierarchical approach in Table 2, aimed at utility maximization with load balancing, per-RSU channel assignment, and independent power control. The reformulated problem still uses binary variables. However, due to the reduced number of binary variables to handle at a time, the computer solvers can (e.g., CPLEX, Gurobi, Mosek, Matlab to name a few) compute the optimal solution quickly. To this end, we introduce a new decision variable \(\mathbf{Y}=[y_{bu}]\) where \(\mathbf{Y}\in\{0,1\}^{N_{RSU}\times N_{VU}}\) to represent the RSU-VU association. In this formulation, \(y_{bu}=1\) indicates that RSU \(b\) is associated with VU \(u\). Here, \(N_{RSU}\) represents the total number of RSUs, and \(N_{VU}\) is the total number of vehicle users. The reformulated problem is given below (referred to as P. (13)). \[\max_{\mathbf{Y}} \frac{\alpha}{m_{1}}\sum_{b}\sum_{u}log_{2}(1+\frac{f(\{g_{bcu}| \forall c\})\cdot y_{bu}\cdot p_{ch}^{max}}{\sigma})-\frac{1-\alpha}{m_{2}}\times l\] (13a) s.t. \[\forall b,u:\quad y_{bu}\leq a_{bu} \tag{13b}\] \[\forall b:\quad\sum_{u}y_{bu}\leq\kappa_{b}\] (13c) \[\forall u:\quad\sum_{b}y_{bu}=1\] (13d) \[\forall b:\quad\sum_{u}y_{bu}\leq l \tag{13e}\]  
In the objective function, the parameter \(\alpha\in[0,1]\) is a weight assigned to the utility function, which is non-decreasing function of \(y_{bu}\). Additionally, \(m_{1}\) and \(m_{2}\) are scaling factors that normalize each term in the objective function to the range [0,1]. The first component of the objective function focuses on maximizing throughput under a fixed transmit power. Achieving high throughput with \(y_{bu}=1\) implies that the user's QoS requirements can be met with lower power, aligning with the primary goal of reducing total power consumption. Since the channel gain varies across different channels, the achievable throughput depends on the channel-to-user mapping. To reduce computational complexity, we use a function to represent the effective channel gain. In this study, the function \(f\) computes either the average gain across all channels or the gain from a randomly selected channel. The second component of the objective function ensures load balancing among RSUs, as enforced by constraint Equation (13e). The load balancing level is represented by \(l\), which limits the number of users assigned to each RSU. Intuitively, reducing the number of users associated with an RSU decreases intra-channel interference by minimizing the need for NOMA channels. Although the variable \(y_{bu}\) remains binary, the reduced number of decision variables in this formulation decreases the computational complexity compared to the original first-phase problem. After determining the RSU-VU association, the subsequent optimization step (S2) is executed on a per-RSU basis. Despite the use of binary variables, the reduced dimensionality and advanced algorithms make this step computationally efficient. To determine the optimal VU-channel mapping, the following optimization problem (referred to as P. (14)) is solved. Using the output \(\mathbf{Y}^{*}\) from P. (13), each RSU processes this problem independently. \[\max_{\mathbf{X},\mathbf{P}} \sum_{c}\sum_{u}\Big{(}p_{bcu}^{h}+p_{bcu}^{l}\Big{)}\] (14a) s.t. \[\forall u:\quad\sum_{c}(x_{bcu}^{h}+x_{bcu}^{l})=y_{bu}\] (14b) \[\text{constraints}\] (1), (5), (6), (7), (8)