# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 4 Distributed Algorithm with Partial Information
### Convergence Analysis
For \(k\in\mathbb{N}\),we assume that \(\beta^{k}\in[0,2]\) and \(e^{k}\in\mathbb{R}^{q}\) such that \(\beta^{k}\Big{\|}e^{k}\Big{\|}_{P}\in\ell^{1}\). If \(v^{0}\in\mathbb{R}^{q}\),_  
\[(\forall k\in\mathbb{N})\quad v^{k+1}=v^{k}+\beta^{k}\Big{(}u^{k}-v^{k}+e^{k} \Big{)},u^{k}\in\mathrm{J}_{\mathcal{B}}\Big{(}v^{k}\Big{)}. \tag{37}\]Then we obtain Algorithm 1 by applying (13) to the operator \(\Phi^{-1}\overline{\mathbb{A}}\), where  
\[\Phi=\left[ \begin{array}{ccc}c\mathbf{\hat{W}}&\mathbf{0}&-\mathcal{R}^{ \mathsf{T}}\mathbf{A}^{\mathsf{T}}\\ \mathbf{0}&\mathbf{0}&-\mathbf{L}\\ -\mathcal{A}\mathcal{R}&-\mathbf{L}&\mathbf{0}\end{array} \right]+\left[ \begin{array}{ccc}\alpha^{-1}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\tau^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\gamma^{-1}\end{array} \right] \tag{38}\]  
is called a preconditioning matrix. It ensures that the agents will be able to compute the resulting iteration in a fully distributed manner. We choose the step sizes \(\alpha=\operatorname{diag}(\alpha_{i}I_{n_{i}})\), \(\tau=\operatorname{diag}(\tau_{i}I_{m})\) and \(\gamma=\operatorname{diag}(\gamma_{i}I_{m})\) such that \(\Phi>0\). This implies that \(\operatorname{zer}\!\left(\Phi^{-1}\overline{\mathbb{A}}\right)=\operatorname {zer}\!\left(\overline{\mathbb{A}}\right)\). The next lemma provides sufficient conditions for \(\Phi>0\) based on [PERSON]'s circle theorem. For any agent \(i\in\mathcal{I}\) and any \(\delta>0\), the preconditioning matrix \(\Phi\) in (38) is positive definite if  
\[ \begin{split} 0&<\alpha_{i}\leq\left(\max_{j\in\{1, \ldots,n_{i}\}}\sum_{k=1}^{m}\left|\left[A_{i}^{\top}\right]_{jk}\right|+\delta \right)^{-1},\\ 0&<\tau_{i}\leq(2d_{i}+\delta)^{-1},\\ 0&<\gamma_{i}\leq\left(\max_{j\in\{1,\ldots,m\}}\sum_{k=1}^{n_{i}} \left|\left[A_{i}\right]_{jk}\right|+2d_{i}+\delta\right)^{-1}.\end{split}  \tag{39}\]  
Using Lemma 4 and Assumption 4, we can show that \(\operatorname{J}_{\Phi^{-1}\overline{\mathbb{A}}}\) is single-valued by applying Equation (28) to \(\Phi^{-1}\overline{\mathbb{A}}\). Let \(c>c_{\min}\), \(c_{\min}\) as in Lemma 3. Then \(\Phi^{-1}\overline{\mathbb{A}}\) is restricted monotone in \(\mathcal{H}_{\Phi}\). It means that finding a zero point of \(\overline{\mathbb{A}}\) is equivalent to finding a variational GNE for problem (3). Moreover, since \(\Phi>0\), we have \(\operatorname{zer}\!\left(\overline{\mathbb{A}}\right)=\operatorname{zer}\! \left(\Phi^{-1}\overline{\mathbb{A}}\right)\) by Lemma 9. Therefore, finding a zero point of \(\Phi^{-1}\overline{\mathbb{A}}\) is also equivalent to finding a variational GNE for problem (3). This establishes the equivalence between problems (3) and (22) under the condition that \(c>c_{\min}\), \(c_{\min}\) as in Lemma 3. Assume that \(c>c_{\min}\), \(c_{\min}\) are as defined in Lemma 8 and that the step sizes \(\alpha,\tau,\gamma\) satisfy Lemma 5. Then, Algorithm 1 generates a sequence \(\left(x^{k},z^{k},\lambda^{k}\right)_{k\in\mathbb{N}}\) that converges to an equilibrium \((x^{*},z^{*},\lambda^{*})\). Proof of Theorem 1.: The set of inequalities (29) is equivalent to Equation (37) when \(\beta^{k}=1\), \(e^{k}=0\) for all \(k\in\mathbb{N}\). By applying Lemma 9, we deduce that \(\Phi^{-1}\overline{\mathbb{A}}\) is restricted monotone on \(\mathcal{H}_{\Phi}\). Then, we define  
\[u^{k}=\operatorname{J}_{\Phi^{-1}\overline{\mathbb{A}}}\!\left(v^{k}\right). \tag{40}\]  
By Lemma 9, \(\Phi^{-1}\overline{\mathbb{A}}\) is restricted monotone in \(\mathcal{H}_{\Phi}\), which implies \(u^{k}=v^{k+1}\). Let \(h^{k}:=v^{k}+\beta^{k}\!\left(u^{k}-v^{k}\right)\), so that \(v^{k+1}=h^{k}+\beta^{k}e^{k}\).