# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 3 Iterative Algorithms with Global Information
### Algorithm Development
\tag{14}\]  
We regard the iterative algorithm as a special case of proximal-point algorithms (PPA) [16] for finding a zero of \(\mathcal{A}\). The general form of PPA can be written as  
\[v^{k+1}\in\text{J}_{\mathcal{A}}\Big{(}v^{k}\Big{)}, \tag{15}\]  
where \(\text{J}_{\mathcal{A}}\Big{(}v^{k}\Big{)}\
eq\varnothing\). We apply the interaction rule (9) to the operator \(\Phi^{-1}\mathcal{A}\), where \(\Phi\) is defined as the following:  
\[\Phi=\left[ \begin{array}{ccc}\mathbf{0}&\mathbf{0}&-\mathbf{A}^{\mathrm{T}}\\ \mathbf{0}&\mathbf{0}&-\mathbf{L}\\ -\mathbf{A}&-\mathbf{L}&\mathbf{0}\end{array} \right]+\left[ \begin{array}{ccc} \alpha^{-1}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\tau^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\gamma^{-1}\end{array} \right]. \tag{16}\]  
We choose the step sizes \(\alpha=\text{diag}(\alpha_{i}I_{n_{i}})\), \(\tau=\text{diag}(\tau_{i}I_{m})\) and \(\gamma=\text{diag}(\gamma_{i}I_{m})\) such that \(\Phi>0\).The next lemma provides sufficient conditions for \(\Phi>0\) based on [PERSON]'s circle theorem. Lemma 2: _For any agent \(i\in\mathcal{I}\) and any \(\delta>0\), the preconditioning matrix \(\Phi\) in (16) is positive definite if_  
\[ \begin{array}{ll}0<\alpha_{i}\leq\left(\max_{j\in\{1,\ldots,n_{i}\}}\sum_{k= 1}^{m}\Big{|}\Big{[}A_{i}^{\tau}\Big{]}_{jk}\Big{|}+\delta\right)^{-1},\\ 0<\tau_{i}\leq\left(2d_{i}+\delta\right)^{-1},\\ 0<\gamma_{i}\leq\left(\max_{j\in\{1,\ldots,m\}}\sum_{k=1}^{n_{i}}\Big{|}\big{[} A_{i}\big{]}_{jk}\Big{|}+2d_{i}+\delta\right)^{-1}.\end{array}  \tag{17}\]  
Assuming that \(\Phi>0\) for all chosen step sizes, we can state the following results. Lemma 3: _The iterative algorithm is equivalent to the following:_  
\[(\forall k\in\mathbb{N})\quad v^{k+1}\in\text{J}_{\Phi^{-1}\mathcal{A}}\Big{(} v^{k}\Big{)}, \tag{18}\]_where \(\mathcal{A}\) is given by (14), \(\Phi\) is given by (16)._  
**Proof of Lemma 3. ** We use the definition of the inverse operator and obtain that  
\[ \begin{split} v^{k+1}&\in\Big{(}\mathrm{Id}+\Phi^{- 1}\mathcal{A}\Big{)}^{-1}\Big{(}v^{k}\Big{)}\\ \Leftrightarrow&\mathbf{0}\in\Phi^{-1}\mathcal{A} \Big{(}v^{k+1}\Big{)}-v^{k}+v^{k+1}\\ \Leftrightarrow&\mathbf{0}\in\Phi\Big{(}v^{k+1}-v^{ k}\Big{)}+\mathcal{A}\Big{(}v^{k+1}\Big{)}.\end{split}  \tag{19}\]  
Substituting \(\mathcal{A}\) and \(\Phi\) into (19) and simplifying them, we obtain that  
\[ \begin{split}& x_{k+1}+\alpha\mathbb{F}(\mathbf{x}_{k+1})+\alpha G( \mathbf{x}_{k+1})=-\alpha\mathbf{A}^{\mathrm{T}}\mathbf{\lambda}_{k}+\mathbf{x}_{k},\\ & z_{k+1}=-\tau\mathbf{\mathrm{L}}\mathbf{\lambda}_{k}+\mathbf{z}_{k},\\ &\lambda_{k+1}+\mathrm{N}_{\mathbb{R}_{\geq 0}^{Nm}}(\mathbf{\lambda}_{k +1})=-\gamma\mathbf{A}\mathbf{x}_{k}-\gamma\mathbf{\mathrm{L}}\mathbf{z}_{k}+\mathbf{\lambda}_{k}+ 2\gamma\mathbf{A}\mathbf{x}_{k+1}+2\gamma\mathbf{\mathrm{L}}\mathbf{z}_{k+1}-\gamma\mathbf{\mathrm{ b}}.\end{split}  \tag{20}\]  
Using the definition of the proximal operator [16], we can write (20) as  
\[ \begin{split}&\mathbf{x}_{k+1}=\arg\min_{x}\big{(}\frac{1}{2}\|\mathbf{x}- \mathbf{x}_{k}\|^{2}+\Big{\langle}\mathbf{x},\alpha\mathbf{A}^{\mathrm{T}}\mathbf{\lambda}_{k} \Big{\rangle}+\alpha f(\mathbf{x})+\alpha g(\mathbf{x})\big{)},\\ &\mathbf{z}_{k+1}=-\tau\mathbf{\mathrm{L}}\mathbf{\lambda}_{k}+\mathbf{z}_{k},\\ &\mathbf{\lambda}_{k+1}=\mathrm{proj}_{\mathbb{R}_{\geq 0}^{Nm}}(\mathbf{ \lambda}_{k}-\gamma\mathbf{A}\mathbf{x}_{k}-\gamma\mathbf{\mathrm{L}}\mathbf{z}_{k}+2\gamma \mathbf{A}\mathbf{x}_{k+1}+2\gamma\mathbf{\mathrm{L}}\mathbf{z}_{k+1}-\gamma\mathbf{\mathrm{b}}). \end{split}  \tag{21}\]  
\(\Box\)