## 3 Theoretical Foundations
### Augmented Lagrange Multiplier Method and Inexact Augmented Lagrange Multiplier Method  
The Augmented Lagrange Multiplier Method (ALM) [36; 37] is primarily used to solve problems with equality constraints. The general form is as follows:  
\[\begin{split}&\min_{\mathbf{x},\mathbf{z}}f(\mathbf{x})+g(\mathbf{z})\\ &\text{s.t}\ \ \mathbf{Ax}+\mathbf{Bz}=\mathbf{C}\end{split} \tag{8}\]  
In the above, \(\mathbf{x}\) and \(\mathbf{z}\) are the free variables; \(f(\mathbf{x})+g(\mathbf{z})\) represents the objective term; \(\mathbf{Ax}+\mathbf{Bz}=\mathbf{C}\) denotes the equality constraint conditions. The optimization problem aims to adjust the magnitudes of \(\mathbf{x}\) and \(\mathbf{z}\) while satisfying the constraint conditions in order to minimize the objective term's function value.  
By introducing the dual variable \(\lambda\), the aforementioned optimization problem with equality constraints is transformed into an unconstrained optimization problem. The augmented Lagrange function is constructed as follows:  
\[L(\mathbf{x},\mathbf{z},\lambda)=f(\mathbf{x})+g(\mathbf{z})+\lambda^{T}(\mathbf{Ax}+\mathbf{Bz}-\mathbf{C })+\frac{\rho}{2}\|\mathbf{Ax}+\mathbf{Bz}-\mathbf{C}\|^{2} \tag{9}\]  
where is the penalty coefficient. The general approach used to solve the above equation is by employing the dual ascent method. The ALM algorithm primarily aims to find the optimal solutions for two problems:  
\[\left(\mathbf{x}^{k+1},\mathbf{z}^{k+1}\right)=\operatorname*{argmin}_{\mathbf{x},\mathbf{z}} L\left(\mathbf{x},\mathbf{z},\lambda^{k}\right) \tag{10}\]  
\[\lambda^{k+1}=\lambda^{k}+\rho\left(\mathbf{Ax}^{k+1}+\mathbf{Bz}^{k+1}-\mathbf{C}\right) \tag{11}\]  
The method is mainly divided into two steps. In the first step, \(\lambda\) is held constant while solving for \(\min_{\mathbf{x},\mathbf{z}}L\left(\mathbf{x},\mathbf{z},\lambda^{k}\right)\). In the second step, the gradient ascent method is utilized, and the solutions obtained for \(\mathbf{x}^{k+1}\) and \(\mathbf{z}^{k+1}\) are substituted into Equation (21). The parameter \(\rho\) is used as the step size to update \(\lambda\).  
Due to the fact that the convergence of the ALM algorithm cannot meet the real-time requirements of practical problems, the IALM was proposed based on ALM. IALM is a distributed optimization algorithm [38] that exhibits a faster convergence rate than ALM. The solving process of IALM follows the same steps as ALM. Firstly, the augmented Lagrange function is constructed:  
\[L(\mathbf{x},\mathbf{z},\lambda)=f(\mathbf{x})+g(\mathbf{z})+\lambda^{T}(\mathbf{Ax}+\mathbf{Bz}-\mathbf{ C})+\frac{\rho}{2}\|\mathbf{Ax}+\mathbf{Bz}-\mathbf{C}\|^{2} \tag{12}\]  
Then, the method employs an alternating computation approach to update \(\mathbf{x},\mathbf{z}\) and \(\lambda\).  
\[\mathbf{x}^{k+1}=\operatorname*{argmin}_{\mathbf{x}}L\left(\mathbf{x},\mathbf{z}^{k},\lambda^{ k}\right) \tag{13}\]  
\[\mathbf{z}^{k+1}=\operatorname*{argmin}_{\mathbf{z}}L\left(\mathbf{x}^{k+1},\mathbf{z},\lambda^ {k}\right) \tag{14}\]  
\[\lambda^{k+1}=\lambda^{k}+\rho\left(\mathbf{Ax}^{k+1}+\mathbf{Bz}^{k+1}-\mathbf{C}\right) \tag{15}\]  
Compared to the ALM algorithm, which updates \((\mathbf{x},\mathbf{z})\) as a whole, IALM treats \(\mathbf{x}\) and \(\mathbf{z}\) as independent entities and alternately updates them. This approach effectively reduces computational complexity. Through this alternating update process of the two matrices, IALM employs element-wise thresholding operations to maintain non-negativity, gradually approaching an approximate solution to the original matrix. Therefore, for large-scale problems, IALM offers advantages such as fast convergence and good numerical stability.