## 1 Introduction
Let \(d^{*}=\max\{d_{1},...,d_{N}\}\), then \(d^{*}\leq s_{N}\leq 2d^{*}\)[19]. ### Game Model

This subsection studies the noncooperative game model on an undirected connected graph \(\mathcal{G}\). Player \(i\) in the set \(\mathcal{N}\) learns other players' decisions \(x_{-i}\) through paths in the graph and then controls the local decision \(x_{i}\). Then, the global decision set \(x\) is \(\text{col}(x_{1},...x_{N})\). Denote \(\sum_{i=1}^{N}A_{i}x_{i}\leq\sum_{i=1}^{N}b_{i}\) as global coupling constraint, then we get the cost function:

\[\forall i\in\mathcal{N},\quad \begin{cases}\operatorname*{arg\,min}_{x_{i}\in \mathcal{R}^{n_{i}}}&\mathbb{J}_{i}(x_{i},x_{-i})=f_{i}(x_{i},x_{-i})+g_{i}(x _{i}),\\ \text{s.t.}&\sum_{i=1}^{N}A_{i}x_{i}\leq\sum_{i=1}^{N}b_{i},\end{cases}  \tag{1}\]

where \(i\in\mathcal{N}\), \(A_{i}\in\mathcal{R}^{m\times n_{i}}\), \(b_{i}\in\mathcal{R}^{m}\), \(x_{i}\in\Omega_{i}\). \(\Omega_{i}\subseteq\mathcal{R}^{n_{i}}\) is private data of player i, and \(x_{-i}=\text{col}(x_{i})_{j\in\mathcal{N}_{j}\
eq i}\) denotes the decisions of all players except player \(i\), \(x_{-i}\in\mathcal{R}^{n-n_{i}}\), \(n=\sum_{i=1}^{N}n_{i}\). **Assumption 1**. :
1. _The domain of the function_ \(\text{dom}(g_{i})=\Omega_{i}\subseteq\mathcal{R}^{n_{i}}\) _is nonempty, compact and convex._
2. \(g_{i}\) _is lower semicontinuous on its domain._
3. _The function satisfies the convexity inequality on its domain._

Denote

\[\mathcal{X}:=\prod_{i=1}^{N}\Omega_{i}\cap\{x|x\in R^{n},\sum_{i=1}^{N}A_{i}x _{i}-b_{i}\leq 0\}, \tag{2}\]

which is the set of feasible values of the global decision vector \(x\). The feasible decision set of agent \(i\) is \(\mathcal{X}_{i}(x_{-i}):=\{x_{i}\in\Omega_{i}|(x_{i},x_{-i})\in\mathcal{X}\}\). **Assumption 2**. :
1. _Every locally feasible set_ \(\mathcal{X}_{i}\) _is a non-empty closed convex set._
2. _Globally shared feasible set_ \(\mathcal{X}\) _is nonempty._
3. _For any feasible point, there exists a nonzero vector that is orthogonal to all gradients of the constraint functions, i.e., (MFCQ) holds._

Let \(A=[A_{1},...,A_{N}]\in\mathcal{R}^{m\times N}\) and \(b=\sum_{i=1}^{N}b_{i}\), then the objective function is equal to

\[\forall i\in\mathcal{N},\quad \begin{cases}\operatorname*{arg\,min}_{x_{i}\in R ^{n_{i}}}&\mathbb{J}_{i}(x_{i},x_{-i})=f_{i}(x_{i},x_{-i})+g_{i}(x_{i}),\\ \text{s.t.}&Ax\leq b,\end{cases}  \tag{3}\]

which consists of two parts, i.e., \(f_{i}(x_{i},x_{-i})\) and \(g_{i}(x_{i})\), the former representing the local impact of the global decision set and the latter representing the local cost. A GNE of game (3) is (4)

\[\forall i\in\mathcal{N},\quad \begin{cases}x_{i}^{*}\in\operatorname*{arg\,min }_{x_{i}\in R^{n_{i}}}\mathbb{J}_{i}(x_{i},x_{-i}^{*}),\\ \text{s.t.}&x_{i}\in\mathcal{X}_{i}(x_{-i}^{*}).\end{cases}  \tag{4}\]

**Assumption 3**. :
1. _When given_ \(x_{-i}\)_,_ \(\Omega_{i}\) _is a non-empty compact convex set and_ \(\forall i\in\mathcal{N}\)_,_ \(\mathbb{J}_{i}(x_{i},x_{-i})\) _is_
2. _continuously differentiable at_ \(x_{i}\)_._
3. \(\mathbb{J}_{i}(x_{i},x_{-i})\) _is a convex function and_ \(\mathcal{X}\) _is a nonempty set satisfying Slater-constraint-qualification._

### Nash Equilibrium for Variational Problems(V-GNE) and KKT Condition

If the original problem is a convex optimization problem, then v-GNE and GNE are equivalent. If the original problem is a nonconvex optimization problem, then v-GNE and GNE may be different, but v-GNE can be used as an approximate solution or a lower bound for GNE [8]. Define a local Lagrangian function for agent \(i\) as \(\mathbb{L}_{i}(x_{i},x_{-i},\lambda_{i})=\mathbb{J}_{i}(x_{i},x_{-i})+\lambda_{ i}^{T}(Ax-b)\).