# Distributed GNE-Seeking under Partial Information Based on Preconditioned Proximal-Point Algorithms
## 2 Game Formulation  
We study a non-cooperative generalized game with a crowd of agents. Each agent \(i\in\mathcal{I}\left(\mathcal{I}=1,\ldots,N\right)\) has a local decision set \(\Omega_{i}\subseteq\mathbb{R}^{n_{i}}\) and chooses its own decision \(x_{i}\in\Omega_{i}\), where \(\mathbb{R}\) is the set of real numbers and \(n_{i}\) is the dimension of agent i's decision-making. The global decision space is \(\Omega=\prod_{i=1}^{N}\Omega_{i}\subset\mathbb{R}^{n}\) where \(n=\sum\limits_{i=1}^{N}n_{i}\) and the stacked vector of all agents' decisions is \(\mathbf{x}=col\big{(}\big{(}x_{i}\big{)}_{i\in\mathcal{I}}\big{)}\). Let \(\mathbf{x}_{-i}=col\big{(}\big{(}x_{i}\big{)}_{j\in\mathcal{I}_{j}\
eq i}\big{)}\) denote the decision profile of all other agents except for agent \(i\), then \(\mathbf{x}=(x_{i},\mathbf{x}_{-i})\). The feasible set for any agent depends on the coupling constraints shared with other agents and their own decisions. Each agent aims to minimize its objective function over this feasible set as follows:  
\[\min_{x_{i}}J_{i}(x_{i},\mathbf{x}_{-i}):=f_{i}(x_{i},\mathbf{x}_{-i})+g_{i}(x_{i}). \tag{1}\]The objective function (1) can describe how multiple participants can optimize their interests in a non-cooperative situation while being affected by sharing constraints. This model can be applied to a variety of practical scenarios, such as smart grids, competitive markets, and electric vehicle charging. The cost functions \(f_{i}\) and \(g_{i}\) of each agent \(i\in\mathcal{I}\) are convex. The common cost function \(f_{i}\) is continuously differentiable and the local idiosyncratic cost function \(g_{i}\) is lower semicontinuous. \(\Omega_{i}\) is bounded and closed for any \(i\in\mathcal{I}\). We denote \(A:=[A_{1},\cdots,A_{N}]\) and \(b:=\sum_{i=1}^{N}b_{i}\), where \(A_{i}\in\mathbb{R}^{m\times n_{i}}\) and \(b_{i}\in\mathbb{R}^{m}\) are local parameters. By the affine function above, the set of overall feasible decisions via affine coupling constraints and cost functions is denoted as follows:  
\[\mathcal{X}:=\Omega\bigcap\Big{\{}\mathbf{x}\in\mathbb{R}^{nN}\mid A\mathbf{x}-b\leq \mathbf{0}_{m}\Big{\}}\subseteq\mathbb{R}^{nN}, \tag{2}\]  
where \(\mathcal{X}\) is a closed and convex set of decisions that is nonempty. It is used to simulate some phenomena in nature, such as group behavior, competition, and cooperation. Such constraints can reflect the interactions and dependencies among participants, making the game more complex and interesting. [PERSON]'s constraint qualification holds for the collective set \(\mathcal{X}\). Therefore, each agent \(i\in\mathcal{I}\) in the generalized game tries to solve this interdependent optimization problem as follows:  
\[\forall i\in\mathcal{I}:\left\{ \begin{array}{ll}\min\limits_{x_{i}\in\Omega_ {i}}&J_{i}(x_{i},\mathbf{x}_{-i})\\ \text{s.t.}&A_{i}x_{i}\leq b-\sum\limits_{j\
eq i}^{N}A_{j}x_{j}.\end{array} \right. \tag{3}\]  
To obtain the primal-dual characterization of each agent \(i\in\mathcal{I}\), we define a Lagrangian function with dual variable multiplier \(\lambda_{i}\in\mathbb{R}^{m}_{+}\) as follows:  
\[\mathcal{L}_{i}(x_{i},\mathbf{x}_{-i};\lambda_{i}):=f_{i}(x_{i},\mathbf{x}_{-i})+g_{i }(x_{i})+\lambda_{i}^{T}\Bigg{(}\sum\limits_{i=1}^{N}\left(A_{i}x_{i}-b_{i} \right)\Bigg{)}. \tag{4}\]  
We call a decision \(x^{*}\in\mathcal{X}\) that satisfies (3) a GNE of the game. This implies that for any agent \(i\in\mathcal{I}\),  
\[J_{i}\big{(}\mathbf{x}_{i}^{*},\mathbf{x}_{-i}^{*}\big{)}\leq\inf\big{\{}J_{i}\big{(}y,\mathbf{x}_{-i}^{*}\big{)}\mid y\in\mathcal{X}_{i}\big{(}\mathbf{x}_{-i}^{*}\big{)} \big{\}}, \tag{5}\]  
where \(\mathcal{X}_{i}(\mathbf{x}_{-i}):=\Bigg{\{}y_{i}\in\Omega_{i}\mid A_{i}y_{i}\leq b -\sum\limits_{j\
eq i}^{N}A_{j}x_{j}\Bigg{\}}\). If \(x_{i}^{*}\in\Omega_{i}\) satisfies the KKT conditions below with \(\lambda_{i}^{*}\in\mathbb{R}^{m}_{+}\), then it is an optimal solution to (3). \[\forall i\in\mathcal{I}: