## 3 Multiaccess Resource Allocation Algorithms and Power Control
### Power-Optimization Algorithms
The absence of a guarantee on the concavity of the overall function prevents the possibility of solving it using a convex optimization toolbox at this point. Hence, additional modification is required. The utilization of a first-order convex approximation enables the transformation of Problem 4 into a convex problem, facilitating its resolution. The gradients of \(s_{i}(W)\) and \(s_{j}(W)\) can be represented in the following manner. \[q_{i}(W)=\sum_{\
u=1}^{V}\log(W_{i}^{\
u}a_{i}^{\
u}+\theta^{2}+\sum_{i^{ \prime}\in B_{\
u^{\prime}},i^{\prime}\
eq i}W_{i}^{\
u}a_{i^{\prime},i}^{ \
u}+\sum_{j\in E_{\
u}}W_{j}^{\
u}a_{jj}^{\
u}), \tag{9}\]1\[q_{i}(W)=\sum_{\
u=1}^{V}\log(W_{i}^{\
u}a_{i}^{\
u}+\theta^{2}+\sum_{i^{ \prime}\in B_{\
u^{\prime}},i^{\prime}\
eq i}W_{i}^{\
u}a_{i^{\prime},i}^{ \
u}+\sum_{j\in E_{\
u}}W_{j}^{\
u}a_{jj}^{\
u}), \tag{9}\]2  
where \(\stackrel{{\rightarrow}}{{e_{i}}}\) is a matrix of \((X+Y)\ *\ V\), represented as \(\stackrel{{\rightarrow}}{{\mathrm{e}_{i}}}(i^{\prime},v)=\frac{ \mathrm{a}_{i,j}^{v}}{\mathrm{Im}2},\mathrm{e}_{i}(\mathrm{j},v)=\frac{ \mathrm{a}_{i,j}^{v}}{\mathrm{Im}2},i^{\prime}\
eq\mathrm{i}\), where \(v\) is the subcarrier assigned to the \(i\)th VUE and \(j\) is the \(j\)th CUE using the the subcarrier \(v\)th; otherwise, \(\stackrel{{\rightarrow}}{{e_{i}}}(i^{\prime}orj,v)=0\). Similarly, \(\stackrel{{\rightarrow}}{{e_{j}}}=\frac{a_{i,D}^{v}}{\mathrm{Im}2}\) is an \((X+Y)\ *\ V\) matrix, defined as \(\stackrel{{\rightarrow}}{{e_{j}}}(i,v)=\frac{a_{i,D}^{v}}{\mathrm{Im }2}\), where \(v\) is the subcarrier assigned to the \(j\)th CUE and \(i\) is the VUE using the \(v\)th subcarrier; otherwise, \(\stackrel{{\rightarrow}}{{e_{j}}}(i,v)=0\). The utilization of the first-order Taylor expansion is subsequently employed to estimate the values of \(s_{i}(W)\) and \(s_{j}(W)\) within their respective tables. This is demonstrated as  
\[q_{i}(W)=\sum_{\
u=1}^{V}\log(W_{i}^{\
u}a_{i}^{\
u}+\theta^{2}+\sum_{i^{ \prime}\in B_{\
u^{\prime}},i^{\prime}\
eq i}W_{i}^{\
u}a_{i^{\prime},i}^{ \
u}+\sum_{j\in E_{\
u}}W_{j}^{\
u}a_{jj}^{\
u}), \tag{9}\]3 \[q_{i}(W)=\sum_{\
u=1}^{V}\log(W_{i}^{\
u}a_{i}^{\
u}+\theta^{2}+\sum_{i^{ \prime}\in B_{\
u^{\prime}},i^{\prime}\
eq i}W_{i}^{\
u}a_{i^{\prime},i}^{ \
u}+\sum_{j\in E_{\
u}}W_{j}^{\
u}a_{jj}^{\
u}), \tag{9}\]4  
After bringing Equations (17) and (18) into Equation (14), _Problem 4_ can be represented as  
\[ \begin{split}\text{\emph{Problem }}5:\underset{\{W_{i}^{v}\}}{max}& \sum_{i=1}^{X}q_{i}(W)-s_{i}\big{(}W^{t}\big{)}+\
abla s_{i}^{T}\big{(}W^{t} \big{)}\big{(}W-W^{t}\big{)},\\ &\text{s.t}\\ & c3:W_{i}^{v}\leq W_{i}^{max},\forall i\in X,\\ & c4:W_{j}^{v}\leq W_{i}^{max},\forall j\in Y,\\ & c5:q_{j}(W)-s_{j}\big{(}w^{t}\big{)}+\
abla s_{j}^{T}\big{(}W^ {t}\big{)}\big{(}W-W^{t}\big{)}\geq\
abla s_{j}^{size},\forall j\in Y,\end{split}  \tag{19}\]  
The aforementioned Problem 5 represents a conventional convex optimization problem that can be effectively resolved using the CVX convex optimization toolbox. The power variable \(W^{0}\) is initialized, and the variable \(W^{t}\) is obtained by solving Problem 5 using a convex optimization toolkit. The variable \(W^{t}\) is then updated iteratively using the algorithm, such that the first-order Taylor expansion provides a continuous approximation of the variables \(\mathsf{o}\) and \(\mathsf{u}\). This allows for the calculation of the system throughput of the variable \(s_{i}(W)\) and \(s_{j}(W)\) VUEs after power optimization. The specific details of the power control algorithm are presented in Algorithm 2. ```
1. Begin
2.\(W^{0}\): Initial power
3.whileInitialization:\(W^{0}\), \(\epsilon=10^{-4}\)
4.If\(t=0\), number of iterations then
5. Calculate\(\Psi^{0}\) by bringing \(W^{0}\) into the objective function in Equations (17) and (18). 6.for\(t=t+1\), \(W^{t}=W^{s}\)do
7. SolveProblem 5 and obtain\(W^{*}\)
8. Endfor
9. ElseBring\(W^{t}\) into Equation (14)
10. Calculate the \(\Psi^{t}\) of the objective function
11. Endif
12. Calculate\(\Delta\Psi=\big{|}\Psi^{t}-\Psi^{t-1}\big{|}\)
13.