# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 4 Distributed Algorithm with Partial Information
### Convergence Analysis  
In this section, we will prove that Algorithm 1 applying the game (22) converges to the variational GNE through a rigorous mathematical analysis. We show that any limit point of Algorithm 1 satisfies \(\mathbf{0}\in\Phi^{-1}\overline{\mathbb{A}}\) and has certain properties. Moreover, we demonstrate that every such zero is in the consensus subspace and solves VI\((\mathbb{F},\mathcal{X})\). **Assumption 5**. : \(\mathbb{F}\) _is a Lipschitz continuous mapping, which implies that there exists a positive constant \(\theta\) such that for any \(\mathbf{x}\) and \(\mathbf{y}\), we have \(|\mathbb{F}(\mathbf{x})-\mathbb{F}(\mathbf{y})|\leq\theta|\mathbf{x}-\mathbf{y}|\)._  
Define the operator:  
\[\mathbf{F}_{\rm c}(\mathbf{x}):=\mathcal{R}^{\rm T}\widehat{\mathbb{F}}(\mathbf{x})+ \mathcal{R}^{\rm T}\mathbb{G}(\mathcal{R}\mathbf{x})+c\mathbf{\rm L}\mathbf{x}. \tag{32}\]  
From (26), we know  
\[\overline{\mathbb{A}}\left[ \begin{array}{c}\mathbf{x}\\ \mathbf{z}\\ \mathbf{\lambda}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}_{\rm c}(\hat{ \mathbf{x}})\\ \mathbf{0}\\ \mathbf{b}\end{array} \right]+\left[ \begin{array}{c}\mathcal{R}^{\rm T}\mathbf{A}^{ \rm T}\mathbf{\lambda}\\ \mathbf{L}\mathbf{\lambda}\\ -\mathcal{A}\mathcal{R}\hat{\mathbf{x}}-\mathbf{L}\mathbf{z}\end{array} \right]+\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \mathrm{N}_{\overline{R}_{\geq 0}^{N\overline{\mathbb{A}}}}(\mathbf{\lambda})\end{array}  \right]. \tag{33}\]  
**Lemma 5**. : _Define \(c_{\min}=\frac{(\theta+\theta_{0})^{2}}{4\mu_{\rm S2}(L)}+\frac{\theta}{s_{2} (L)}\)._  
\[\Psi=\left[ \begin{array}{cc}\frac{\mu}{N}&-\frac{\theta+\theta_{0}}{2\sqrt{ N}}\\ -\frac{\theta+\theta_{0}}{2\sqrt{N}}&cs_{2}(L)-\theta\end{array} \right],\mu_{ \rm F_{c}}=s_{\min}(\Psi). \tag{34}\]  
_When \(c>c_{\min}\), we have \(\mu_{\rm F_{c}}=s_{\min}(\Psi)>0\) and \(\overline{\mathbb{A}}\) is restricted monotone if Assumptions 1-5 hold._  
Proof of Lemma 5.: Let \(v^{*}=\operatorname{col}(\mathbf{x}^{*},\mathbf{z}^{*},\mathbf{\lambda}^{*})\in\operatorname {zer}(\overline{\mathbb{A}})\) be any zero of \(\overline{\mathbb{A}}\), which exists by Lemma 4. From (33), we can decompose \(\overline{\mathbb{A}}\) as a sum of three operators, two of which are monotone. Moreover, for any \(c>c_{\min}\), we have \(\Psi\geqslant 0\) for all \(\mathbf{x}\in\mathbb{R}^{Nn}\) by Lemma 1. Then,  
\[\langle\mathbf{x}-\mathbf{x}^{*}\mid\mathbf{F}_{\rm c}(\mathbf{x})-\mathbf{F}_{\rm c}(\mathbf{x}^{*}) \rangle\geq\mu_{\rm F_{c}}\|\mathbf{x}-\mathbf{x}^{*}\|^{2}. \tag{35}\]  
Hence, for any \((v,u)\in\operatorname{gra}(\overline{\mathbb{A}})\), with \(v=\operatorname{col}(\mathbf{x},\mathbf{z},\mathbf{\lambda})\), we obtain \(\langle v-v^{*}\mid u-\mathbf{0}\rangle\geq\mu_{\rm F_{c}}\|\mathbf{x}-\mathbf{x}^{*}\|^{2} \geq 0\). **Lemma 6**. : _Define \(\mathcal{B}:\mathbb{R}^{q}\to\mathbb{R}^{q}\) as a restricted monotone in \(\mathcal{H}_{\Phi}\) and \(\mathrm{J}_{\mathcal{B}}\) is firmly quasinonexpansive in \(\mathcal{H}_{\Phi}\), where \(\mathcal{H}_{\Phi}\) is the Hilbert space induced by the inner product \(\langle\cdot,\cdot\rangle_{\Phi}\). Then for any \((v,u)\in\operatorname{gra}(\mathrm{J}_{\mathcal{B}})\), \(v^{*}\in\operatorname{zer}(\mathcal{B})=\operatorname{fix}(\mathrm{J}_{ \mathcal{B}})\), it holds that_  
\[\langle v-u\mid v-v^{*}\rangle_{\Phi}-\|u-v\|_{\Phi}^{2}=\langle v-u\mid u-v^ {*}\rangle_{\Phi}\geq 0. \tag{36}\]  
Proof of Lemma 6.: We use the definition of resolvent: \(v^{*}\in\mathrm{J}_{\mathcal{B}}(v^{*})\Leftrightarrow v^{*}+\mathcal{B}v^{*} \
i v^{*}\Leftrightarrow\mathbf{0}\in\mathcal{B}(v^{*})\). Moreover, for any \((v,u)\in\operatorname{gra}(\mathrm{J}_{\mathcal{B}})\), we have \(v-u\in\mathcal{B}(u)\). Therefore, (36) follows from the restricted monotonicity of \(\mathcal{B}\) and some simple algebra. Finally, setting \(v=v^{*}\) in (36), we obtain that \(J_{\mathcal{B}}\) is single-valued. **Lemma 7**. : _Let \(\operatorname{zer}(\mathcal{B})\
eq\varnothing\) by Lemma 6.