## 4 ADMM Formulation and Solution  
The optimization problem Equation (16), like Equation (7), is non-convex and NP-hard. Analytical solutions to problems such as this are challenging to obtain and alternatives such as numerical or heuristic techniques are employed instead to obtain approximate solutions. Even using heuristic techniques, such as a genetic algorithm (GA), it may be difficult to formulate the CM constraint. Therefore, we use the ADMM-based iterative technique to approximate a solution to this problem. An auxiliary variable \(\mathbf{\bar{r}}_{r}\) is introduced in Equation (16) and the following equivalent version is obtained:  
\[\underset{\mathbf{\bar{r}}_{r},\mathbf{\bar{s}}_{r}}{\text{minimize}} \mathbf{\bar{r}}_{r}^{T}\mathbf{A}_{r}\left(\widetilde{\Theta} \right)\mathbf{A}_{r}^{T}\left(\widetilde{\Theta}\right)\mathbf{\bar{s}}_{r}\] (18) subject to \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\] \[\mathbf{T}(\mathbf{\bar{r}}_{r})\mathbf{\bar{s}}_{r}-\mathbf{1} =\mathbf{0}\] \[\mathbf{\bar{r}}=\mathbf{\bar{s}}.\]  
It can be observed that for the WS constraint in Equation (16), \(\mathbf{\bar{s}}_{r}\) is expressed as two times \(\mathbf{\bar{s}}_{r}\) and one of them is replaced by \(\mathbf{\bar{r}}_{r}\) in Equation (18). Moreover, the CM constraint, consisting of \(MN\) equations in Equation (16), is expressed in its compact form in (18) as  
\[\mathbf{G}(\mathbf{\bar{r}}_{r},\mathbf{\bar{s}}_{r})=\mathbf{G}(\mathbf{\bar {s}}_{r},\mathbf{\bar{r}}_{r})=\mathbf{0} \tag{19}\]  
where \(\mathbf{G}(\mathbf{\bar{r}}_{r},\mathbf{\bar{s}}_{r})\in\mathbb{R}^{2 MN\times 2 MN}\) is a vector given by  
\[\mathbf{G}(\mathbf{\bar{r}}_{r},\mathbf{\bar{s}}_{r})=\mathbf{T}(\mathbf{\bar {r}}_{r})\mathbf{\bar{s}}_{r}-\mathbf{1}, \tag{20}\]  
where \(\mathbf{1}\) and \(\mathbf{0}\) are \(2 MN\times 1\) vectors, all \(1\)s and \(0\)s, respectively, and  
\[\mathbf{T}(\mathbf{\bar{r}}_{r})=\left[\mathbf{\bar{r}}_{r}^{T}\mathbf{E}_{1}; \mathbf{\bar{r}}_{r}^{T}\mathbf{E}_{2};\cdots;\mathbf{\bar{r}}_{r}^{T}\mathbf{ E}_{MN}\right]\in\mathbb{R}.^{MN\times 2 MN} \tag{21}\]  
The augmented Lagrangian of Equation (18) is given as  
\[\mathcal{L}\left\{\mathbf{\bar{r}}_{r},\mathbf{\bar{s}}_{r}, \mathbf{u},\mathbf{v},\mathbf{w},\right\} = \mathbf{\bar{r}}_{r}^{T}\mathbf{A}_{r}\left(\widetilde{\Theta} \right)\mathbf{A}_{r}^{T}\left(\widetilde{\Theta}\right)\mathbf{\bar{s}}_{r}\] \[+ \frac{\rho_{1}}{2}\left\|\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r} }_{r}+\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{s}}_{r}-2\mathbf{\bar{x}}_{r}+ \mathbf{u}\right\|_{2}^{2}\] \[+ \frac{\rho_{2}}{2}\left\|\mathbf{T}(\mathbf{\bar{r}}_{r})\mathbf{ \bar{s}}_{r}-\mathbf{1}+\mathbf{v}\right\|_{2}^{2}\] \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]0  
where \(\mathbf{u}\in\mathbb{R}^{4N\times 1},\mathbf{v}\in\mathbb{R}^{MN\times 1}\), and \(\mathbf{w}\in\mathbb{R}^{2 MN\times 1}\) are the dual variables and \(\rho_{1},\rho_{2},\rho_{3}>0\) are the adjustable penalty parameters. The \((m+1)\)th iteration of the algorithm, in terms of the different variables, is given as follows:  
\[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]1 \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]2 (23b) \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]3 (23c) \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]4 (23d) \[\mathbf{A}_{r}^{T}(\Theta)\mathbf{\bar{r}}_{r}+\mathbf{A}_{r}^{T }(\Theta)\mathbf{\bar{s}}_{r}=2\mathbf{\bar{x}}_{r}\]5  
As can be seen from Equation (23), the updates Equations (23c)-(23e) are straightforward. The subequations Equations (23a) and (23b) are convex and give closed-form solutions.