# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 4 Distributed Algorithm with Partial Information
### Algorithm Development
\tag{25}\]  
where \(\hat{\boldsymbol{L}}=(L\otimes I_{n})\in\mathbb{R}^{Nn\times Nn}\), \(\mathcal{R}=\text{diag}\{\mathcal{R}_{i}\}\), \(\mathbb{F}(\boldsymbol{x})=\text{col}\Big{(}\
abla_{x_{i}^{i}}f_{i}\big{(}x_{ i^{\prime}}^{i}\boldsymbol{x}_{-i}^{i}\big{)}\Big{)}_{i\in I}\in\mathbb{R}^{n}\), \(\mathbb{G}(\boldsymbol{x})=\text{col}\big{(}\
abla_{\mathcal{S}_{i}}\big{(}x_{ i}^{i}\big{)}\big{)}_{i\in I}\in\mathbb{R}^{n}\). Moreover, \(\mathrm{c}\) is a parameter about the dual variable associated with the constraint \(0=-\hat{L}x^{*}\). Define the following operator:  
\[\overline{\mathbb{A}}:\left[ \begin{array}{c}\hat{\boldsymbol{x}}\\ \boldsymbol{z}\\ \boldsymbol{\lambda}\end{array} \right]\mapsto\left[ \begin{array}{c}\mathcal{ R}^{\mathrm{T}}\hat{\mathbb{F}}(\hat{\boldsymbol{x}})+\mathcal{R}^{\mathrm{T}} \mathbb{G}(\mathcal{R}\hat{\boldsymbol{x}})+\mathcal{R}^{\mathrm{T}} \boldsymbol{A}^{\mathrm{T}}\boldsymbol{\lambda}+c\hat{\boldsymbol{L}}\hat{ \boldsymbol{x}}\\ \boldsymbol{L}\boldsymbol{\lambda}\\ -\boldsymbol{A}\mathcal{R}\hat{\boldsymbol{x}}+\boldsymbol{b}-\boldsymbol{ \mathrm{L}}\boldsymbol{z}+\boldsymbol{\mathrm{N}}_{\mathbb{R}\geq 0}(\boldsymbol{\lambda})\end{array} \right], \tag{26}\]  
\[\Phi=\left[ \begin{array}{ccc}c\hat{\boldsymbol{W}}&\mathbf{0}&-\mathcal{R} ^{\mathrm{T}}\boldsymbol{A}^{\mathrm{T}}\\ \mathbf{0}&\mathbf{0}&-\boldsymbol{\mathrm{L}}\\ -\boldsymbol{A}\mathcal{R}&-\boldsymbol{\mathrm{L}}&\mathbf{0}\end{array} \right]+ \left[ \begin{array}{ccc}\mathbf{\alpha}^{-1}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{\tau}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{\gamma}^{-1}\end{array} \right]. \tag{27}\]  
The variables \(\boldsymbol{x},\boldsymbol{z}\) and \(\boldsymbol{\lambda}\) satisfy condition (25) if \(\mathbf{0}\in\overline{\mathbb{A}}v\) where \(v=\left[\hat{\boldsymbol{x}}^{\mathrm{T}},\boldsymbol{z}^{\mathrm{T}}, \boldsymbol{\lambda}^{\mathrm{T}}\right]^{\mathrm{T}}\). Lemma 4: _Algorithm 1 is equivalent to the following:_  
\[(\forall k\in\mathbb{N})\quad v^{k+1}\in\boldsymbol{\mathrm{J}}_{\Phi^{-1} \overline{\mathbb{A}}}\Big{(}v^{k}\Big{)}. \tag{28}\]  
_Let \(\overline{\mathbb{A}}\) be defined as in (26) and \(\Phi\) as in (27). Then, Algorithm 1 generates a sequence \(\Big{(}x^{k},z^{k},\boldsymbol{\lambda}^{k}\Big{)}\) for \(k\in\mathbb{N}\) and any initial condition \(v^{0}=\text{col}\Big{(}\boldsymbol{x}^{0},\boldsymbol{z}^{0},\boldsymbol{ \lambda}^{0}\Big{)}\)._Proof of Lemma 4.: By applying the definition of inverse operations, we obtain that  
\[ \begin{split}&\mathbf{v}^{k+1}\in\left(\mathrm{Id}+\Phi^{-1}\overline{ \mathbb{A}}\right)^{-1}\left(\mathbf{v}^{k}\right)\\ \Leftrightarrow&\mathbf{0}\in\Phi^{-1}\overline{\mathbb{A }}\left(\mathbf{v}^{k+1}\right)-\mathbf{v}^{k}+\mathbf{v}^{k+1}\\ \Leftrightarrow&\mathbf{0}\in\Phi\left(\mathbf{v}^{k+1}-\mathbf{ v}^{k}\right)+\overline{\mathbb{A}}\left(\mathbf{v}^{k+1}\right)\\ \Leftrightarrow&\begin{cases}\mathbf{0}\in\alpha^{-1}( \hat{\mathbf{x}}_{k+1}-\hat{\mathbf{x}}_{k})+c\mathbf{W}\hat{\mathbf{x}}_{k+1}-c\mathbf{W}\hat{\mathbf{ x}}_{k}+c\mathbf{L}\hat{\mathbf{x}}_{k+1}-\mathcal{R}^{\top}\mathbf{A}^{\top}\mathbf{A}_{k+1}+ \mathcal{R}^{\top}\mathbf{A}^{\top}\mathbf{A}_{k}\\ &\quad\quad\quad\quad\quad+\mathcal{R}^{\top}\hat{\mathbb{F}}(\hat{\mathbf{x}}_{k +1})+\mathcal{R}^{\top}G(\mathcal{R}\hat{\mathbf{x}}_{k+1})-\mathcal{R}^{\top} \mathbf{A}^{\top}\mathbf{\lambda}_{k+1}\\ \mathbf{0}\in\gamma^{-1}\left(\mathbf{z}^{k+1}-\mathbf{z}^{k}\right)+\mathbf{\Lambda}_{k+1}+ \mathbf{\Lambda}_{k}-\mathbf{\Lambda}_{k+1}\\ \mathbf{0}\in\gamma^{-1}(\mathbf{\lambda}_{k+1}-\mathbf{\lambda}_{k})+\mathrm{N}_{ \mathbb{R}^{mN}_{\geq 0}}\left(\mathbf{\lambda}^{k+1}\right)+\mathbf{b}-A\mathcal{R}(2 \hat{\mathbf{x}}_{k+1}-\hat{\mathbf{x}}_{k})+\mathbf{\mathrm{L}}(2\hat{\mathbf{z}}_{k+1}-\mathbf{ z}_{k}).\end{cases}\end{split}