# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 4 Distributed Algorithm with Partial Information
### Algorithm Development
 \tag{29}\]  
By \(\hat{\mathbf{\mathrm{L}}}=\hat{\mathbf{\mathrm{D}}}-\hat{\mathbf{\mathrm{W}}}\), \(\mathcal{R}\alpha\mathcal{R}^{\top}=\check{\mathbf{\mathrm{x}}},\mathcal{S}\alpha \mathcal{R}^{\top}=\mathbf{0},\mathcal{R}\alpha\hat{\mathbf{\mathrm{L}}}=\check{\pi} \mathcal{R}\hat{\mathbf{\mathrm{L}}}\), and \(\mathcal{S}\alpha\hat{\mathbf{\mathrm{L}}}=\check{\alpha}\hat{\mathcal{S}}\), we have  
\[ \begin{split}&\begin{cases}0\in\mathcal{S}\left(\left(I+c\hat{ \alpha}\hat{\mathbf{\mathrm{D}}}\right)\mathbf{x}_{k+1}-\mathbf{x}_{k}-c\hat{\alpha}\hat{ \mathbf{W}}\mathbf{x}_{k}\right)\\ 0\in\mathcal{R}\left(\left(I+c\check{\alpha}\hat{\mathbf{\mathrm{D}}}\right)\mathbf{x }_{k+1}-\mathbf{x}_{k}-c\check{\alpha}\hat{\mathbf{W}}\mathbf{x}_{k}\right)+\check{\pi} \hat{\mathbb{F}}(\hat{\mathbf{x}}_{k+1})+\check{\pi}\mathbb{G}(\mathcal{R}\mathbf{x}_{ \mathbf{k+1}})+\check{\pi}\mathbb{G}(\mathcal{R}\mathbf{x}_{\mathbf{k+1}})+\check{\pi} \mathbf{A}^{\top}\mathbf{\lambda}_{k}\\ \Leftrightarrow&\begin{cases}\mathbf{x}_{-i,k+1}^{i}=\frac{1}{1+c \alpha_{i}d_{i}}\left(\mathbf{x}_{-i,k}^{i}+\alpha_{i}\sum_{j=1}^{N}w_{ij}\mathbf{x}_{ -i,k}^{i}\right)\\ 0_{n_{i}}\in\partial_{x_{i}^{k+1}}\left(\frac{1}{2}\|\mathbf{x}-x_{i,k}\|^{2}+ \alpha_{i}x^{\top}A_{i}^{\top}\lambda_{i,k}+\frac{\xi}{2}\alpha_{i}d_{i}\| \mathbf{x}-\frac{1}{d_{i}}\sum_{j=1}^{N}w_{ij}\mathbf{x}_{i,k+1}^{i}\|^{2}\\ \quad\quad\quad\quad\quad\quad+\alpha_{i}f_{i}\Big{(}\mathbf{x}_{-i,k+1}^{i}\Big{)} +\alpha_{i}g_{i}(x)\Big{)}.\end{cases}\end{split}  \tag{30}\]  
Hence, by the property that the subdifferential of a convex function contains only zeros at its minima [16], we can reformulate (30) as follows. \[ \begin{split}&\forall i\in\mathcal{I}:\left\{\begin{array}{l}\mathbf{x}_{-i,k+1}^{i}= \frac{1}{1+\tau_{i}d_{i}}\left(\mathbf{x}_{-i,k}^{i}+\tau_{i}\sum_{j=1}^{N}w_{ij} \mathbf{x}_{-i,k}^{j}\right)\\ &\mathbf{x}_{i,k+1}=\underset{x}{\mathrm{argmin}}\left(\frac{1}{2}\|\mathbf{x}-x_{i, k}\|^{2}+\alpha_{i}x^{\top}A_{i}^{\top}\lambda_{i,k}+\alpha_{i}f_{i}\Big{(}\mathbf{x},\mathbf{x}_{-i,k+1}^{i}\Big{)}\\ \quad\quad\quad\quad\quad\quad+\alpha_{i}g_{i}(x)+\frac{\xi}{2}\alpha_{i}d_{i }\Big{\|}\mathbf{x}-\frac{1}{d_{i}}\sum_{j=1}^{N}w_{ij}\mathbf{x}_{i,k+1}^{i}\Big{\|}^{2 }\right),\\ &\mathbf{z}_{k+1}=-\tau\mathbf{\mathrm{L}}\mathbf{\lambda}_{k}+\mathbf{z}_{k},\\ &\lambda_{k+1}=\text{proj}_{\mathbb{R}^{mN}_{\geq 0}}\big{(}\ \mathbf{\lambda}_{k}+\gamma A\mathcal{R}(2\hat{\mathbf{x}}_{k+1}-\hat{\mathbf{x}}_{k})+2 \gamma\mathbf{\mathrm{L}}\mathbf{z}_{k+1}-\gamma\mathbf{\mathrm{L}}\mathbf{z}_{k}-\gamma\mathbf{ \mathrm{b}}\ \ \big{)}.\end{array}\right.\end{split}  \tag{31}\]  
Write (31) as a distributed algorithm as Algorithm 1. ```
Initialize: For all \(i\in\mathcal{I}\), set \(x_{i,0}\in\Omega_{i}\
u_{i-,i,0}^{i}\in\mathbb{R}^{n-n_{i}}\), \(\lambda_{i,0}\in\mathbb{R}^{m}_{+}\), \(z_{i,0}\in\mathbb{R}^{m}\) for\(k=1,2,3,\dots\)do \(\mathbf{x}_{-i,k+1}^{i}=\frac{1}{1+\tau_{i}d_{i}}\left(\mathbf{x}_{-i,k}^{i}+\tau_{i}\sum_{j= 1}^{N}w_{ij}\mathbf{x}_{-i,k}^{i}\right)\) \(x_{i,k+1}=\underset{x}{\mathrm{argmin}}\left(\frac{1}{2}\|\mathbf{x}-x_{i,k}\|^{2}+ \alpha_{i}x^{\top}A_{i}^{\top}\Lambda_{i,k}+\frac{\xi}{2}\alpha_{i}d_{i}\Big{\|} \mathbf{x}-\frac{1}{d_{i}}\sum_{j=1}^{N}w_{ij}\mathbf{x}_{i,k+1}^{i}\right)^{2}\) \(+\alpha_{i}g_{i}(x)\Big{)}\) \(z_{i,k+1}=z_{i,k}-\tau_{i}\sum_{j=1}^{N}w_{ij}(\lambda_{i,k}-\lambda_{j,k})\) \(\lambda_{i,k+1}=\text{prox}_{\mathbb{R}^{mN}_{\geq 0}}\Big{(}\lambda_{i,k}+2\gamma_{i}A_{i}x_{i,k+1}- \gamma_{i}A_{i}x_{i,k}+2\gamma_{i}\sum_{j=1}^{N}w_{ij}(z_{i,k+1}-z_{i,k+1})- \gamma_{i}b_{i}\Big{)}\) endfor Return: The sequence \(\left(x_{i,k}\right)_{k=1}^{\infty}\) will eventually approximate the optimal solution. ```  
**Algorithm 1** Distributed Algorithm with Partial Information