# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 4 Distributed Algorithm with Partial Information
### Algorithm Development  
This section presents an algorithm for GNE-seeking in game (3) in a fully distributed manner. Each agent \(i\) has a cost function \(J_{i}\) and a feasible set \(\Omega_{i}\) but does not know the full state \(x_{-i}\) of other agents. Agent \(i\) can only exchange information with its neighboring agents over a network \(\mathcal{G}(\mathcal{I},\mathcal{E})\). The edge \((i,j)\) belongs to \(\mathcal{E}\) if agents \(i\) and \(j\) can exchange information mutually. \(\mathbf{x}^{i}:=\mathrm{col}\bigg{(}\Big{(}\mathbf{x}^{i}_{j}\Big{)}_{j\in\mathcal{I}} \bigg{)}\in\mathbb{R}^{n}\) is defined as agent \(i\)'s estimate of agent \(j\)'s decision. Then \(\mathbf{x}^{i}_{i}:=x_{i}\) and \(\mathbf{x}^{j}_{-i}:=\mathrm{col}\bigg{(}\Big{(}\mathbf{x}^{j}_{\ell}\Big{)}_{\ell\in \mathcal{I}\setminus\{i\}}\bigg{)}\) denotes \(j\)'s estimate of all other agents except \(i\). If \(\mathbf{x}^{i}=\mathbf{x}^{j}\), we can replace the cost function of agent \(i\) with \(J_{i}(\mathbf{x}^{i}_{i},\mathbf{x}^{i}_{-i})\). Then, we equivalently transform the game (3) as the following:  
\[ \begin{array}{ll}\min_{x^{i}}&J_{i}\big{(}\mathbf{x}^{i}_{i},\mathbf{x}^{i}_{-i} \big{)}=f_{i}\big{(}\mathbf{x}^{i}_{i},\mathbf{x}^{i}_{-i}\big{)}+g_{i}\big{(}\mathbf{x}^{ i}_{i}\big{)},\\ \text{s.t.}&\mathbf{x}^{i}_{i}\in\mathcal{\bar{X}}_{i},\\ &\mathbf{x}^{i}=\mathbf{x}^{j},\forall j\in\mathcal{I}.\end{array}  \tag{22}\]  
It is worth noting that problems (3) and (22) are equivalent under a certain condition. We will explain this point below. Let \(\hat{\mathcal{X}}_{i}=\left\{x\in\Omega_{i}\mid A_{i}x_{i}^{i}+\sum\limits_{j\
eq i }A_{j}x_{j}^{i}\leq\sum\limits_{i=1}^{m}b_{i}\right\}\) be the set of feasible solutions for agent \(i\) under a consistency constraint. Then, agent \(i\)'s cost function \(J_{i}\) only depends on its own local information \(x^{i}\). We introduce matrices \(\mathcal{R}_{i}\) and \(\mathcal{S}_{i}\) to develop a distributed algorithm that uses partial decision information for game (22). \[\mathcal{R}_{i}:=\left[ \begin{array}{cc}\mathbf{0}_{n_{i}\times n_{<i}}&I_{ n_{i}}&\mathbf{0}_{n_{i}\times n_{>i}}\end{array} \right], \tag{23}\]  
\[\mathcal{S}_{i}:=\left[ \begin{array}{cc}I_{n<i}&\mathbf{0}_{n<i\times n_{i} }&\mathbf{0}_{n<i\times n>i}\\ \mathbf{0}_{n>i\times n<i}&\mathbf{0}_{n>i\times n_{i}}&I_{n>i}\end{array} \right]. \tag{24}\]  
where \(n_{<i}:=\sum_{j=1}^{i-1}n_{j},n_{>i}:=\sum_{j=i+1}^{N}n_{j}\). Then, \(\mathcal{R}_{i}\boldsymbol{x}_{i}=\boldsymbol{x}_{i}^{i}=\boldsymbol{x}_{i}\) and \(\mathcal{S}_{i}\boldsymbol{x}_{i}=\boldsymbol{x}_{-i}^{i}\). Let \(\mathcal{R}:=\text{diag}\big{(}(\mathcal{S}_{i})_{i\in\mathcal{I}}\big{)}\), \(\mathcal{S}:=\text{diag}\big{(}(\mathcal{S}_{i})_{i\in\mathcal{I}}\big{)}\). Hence \(\mathcal{R}\mathcal{R}=x\) and \(\mathcal{R}x=\text{col}\Big{(}\big{(}\boldsymbol{x}_{-i}^{i}\big{)}_{i\in \mathcal{I}}\Big{)}\in\mathbb{R}^{(N-1)n}\). Moreover, \(\boldsymbol{x}=\mathcal{R}^{\top}x+\mathcal{S}^{\top}\mathcal{S}\boldsymbol{x}\). Similar to (13), we design sufficient conditions of (22) as follows:  
\[\left\{ \begin{array}{l}0\in\mathcal{R}^{\mathrm{T}}\hat{\mathbb{R}}( \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}}^{*}\\ 0=\boldsymbol{\mathrm{L}}\boldsymbol{\lambda}^{*}\\ 0\in\boldsymbol{A}\mathcal{R}\hat{\boldsymbol{x}}^{*}-\boldsymbol{b}+ \boldsymbol{\mathrm{L}}\boldsymbol{z}^{*}-\boldsymbol{\mathrm{N}}_{\mathbb{R} \geq 0}^{\mathbb{N}m}(\boldsymbol{\lambda}^{*}),\end{array} \right.