# Covert Communications in Active-RIS-Aided NOMA Systems: Element Grouping or Not?
## 3 Problem Formulation and Proposed Solutions
### Problem Formulation
\[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]1 \[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]2 \[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]3  
To tackle the rank-one constraint in problem (P5), we add a penalty term \(\rho_{3}\mathcal{R}\big{(}\mathbf{W},\mathbf{W}^{(k)}\big{)}=\rho_{3}\big{(} \|\mathbf{W}\|_{*}+\widehat{\mathbf{W}}^{(k)}\big{)}\) into the objective function, where \(\rho_{3}\) is the penalty factor, \(k\) is the iteration number with respect to the penalty term, and \(\widehat{\mathbf{W}}^{(k)}=-\|\mathbf{W}^{(k)}\|_{2}-\text{Tr}(\widehat{ \mathbf{v}}_{\text{max}}^{(k)}(\widehat{\mathbf{v}}_{\text{max}}^{(k)})^{H}( \mathbf{W}-\mathbf{W}^{(k)}))\) with \(\mathbf{W}^{(k)}\) denoting the solution obtained in the \(k\)th iteration and \(\widehat{\mathbf{v}}_{\text{max}}^{(k)}\) denoting the eigenvector corresponding to the largest eigenvalue of \(\mathbf{W}^{(k)}\). Let \(f_{3}(\mathbf{W})=\text{Tr}(\mathbf{\Omega}_{ab}\mathbf{W})\) and \(f_{4}(\mathbf{W})=P_{a}\text{Tr}(\mathbf{\Psi}_{ab}\mathbf{W})+\sigma_{r}^{2} \text{Tr}(\mathbf{F}\mathbf{W})+\sigma_{b}^{2}\). Then, with the introduced penalty term and applying the Dinkelbach transform, problem (P5) is reformulated as:  
\[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]4 (32a) s.t. \[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]5  
In problem (P5.1), the coefficient \(v_{2}\) is updated by  
\[P_{i}\leq p_{i}^{\max},\forall i\in\{a,c\},\]6  
where the Dinkelbach transform coefficient \(v_{2}\) is updated by (33) in the \(k\)th iteration. Now, a convex problem solver can be applied to obtain the optimal solution of problem (P5.1). Based on the optimized solutions for problems (P4.2) and (P5.1), the proposed AO algorithm to obtain the solution for problem (P1) is summarized in Algorithm 1. The complexity of solving problems (P4.2) and (P5.1) are \(\mathcal{O}(I_{a}I_{d}I_{p}(N+1)^{3.5})\) and \(\mathcal{O}(I_{a}I_{d}I_{p}(M)^{3.5})\), where \(I_{a}\), \(I_{d}\), and \(I_{p}\) are the number of iterations of the AO procedure, iteration numbers with respect to the penalty term, and Dinkelbach coefficients, respectively. ```
1:Initialize \(t\gets 0\), \(P_{a}^{(0)}\), \(P_{c}^{(0)}\), \(\varphi^{(0)}\), \(\mathbf{A}_{m}^{(0)}\), and \(\mathbf{W}^{(0)}\)
2:repeat\(t\gets t+1\)
3: Update \(P_{a}^{(t)}\) according to (21)
4: Initialize \(\rho_{1}>0\), \(\rho_{2}>0\), \(v_{1}^{(0)}>0\), \(k\gets 0\)
5:repeat\(k\gets k+1\)
6: Update \(\mathbf{A}_{m}^{(k)}\) by solving (P4.2)
7: Update \(\theta_{m}^{(k+1)}=\sqrt{(\gamma_{n}^{j})^{(k)}/(\gamma_{n}^{(t)})^{(k)}}\)
8: Update \(v_{1}^{(k+1)}\) according to (30)
9: Update \(\rho_{1}\gets c_{1}\rho_{1}\) and \(\rho_{2}\gets c_{2}\rho_{2}\)
10:until\(\rho_{1}\sum\limits_{n=1}^{N}\mathcal{A}(\gamma_{n}^{\prime},\gamma_{n}^{\prime}) \leq\xi_{2}\) and
11:\(\rho_{2}\sum\limits_{n=\{c,j\}}\mathcal{R}(\mathbf{A}_{m},\mathbf{A}_{m}^{(k)} )\leq\xi_{3}\)
12:\(\mathbf{A}_{m}^{(t)}\leftarrow\mathbf{A}_{m}^{(k)}\)
13: Initialize \(\rho_{3}>0\), \(v_{2}^{(0)}>0\), \(k\gets 0\)
14:repeat\(k\gets k+1\)
15: Update \(\mathbf{W}^{(k)}\) by solving (P5.1)
16: Update \(v_{2}^{(k+1)}\) according to (33)
17: Update \(\rho_{3}\gets c_{3}\rho_{3}\)
18:until\(\rho_{3}\mathcal{R}(\mathbf{W},\mathbf{W}^{(k)})\leq\xi_{4}\)
19:\(\mathbf{W}^{(t)}_{i}\leftarrow\mathbf{W}^{(k)}\)
20:until\(\gamma_{n}^{(t)}-\gamma_{n}^{(t-1)}\leq\xi_{1}\)
```  
**Algorithm 1** AO algorithm for optimizing \(\{P_{a},P_{c},\mathbf{A}_{m},\mathbf{W}\}\)