## 1 Introduction
\begin{cases}\operatorname*{arg\,min}_{x\in\mathcal{R}^{n}}&\left<\hat{ \mathbb{F}}(\mathbf{\hat{x}}^{*}),\mathbf{R}\mathbf{\hat{x}}\right>+g(\mathbf{R}\mathbf{\hat{x}}), \\ \text{s.t.}&\mathbf{AR}\mathbf{\hat{x}}\leq\mathbf{b},\\ &\mathbf{L}\mathbf{\lambda}=\mathbf{0},\\ &\mathbf{L}\mathbf{\hat{x}}=\mathbf{0}.\end{cases}  \tag{15}\]

The Lagrangian function of the optimization problem (15) is defined as \(\mathbb{\hat{L}}=\left<\hat{\mathbb{F}}(\mathbf{\hat{x}}^{*}),\mathbf{R}\mathbf{\hat{x}} \right>+g(\mathbf{R}\mathbf{\hat{x}})+\mathbf{\lambda}^{T}(\mathbf{AR}\mathbf{\hat{x}}-\mathbf{b})+ \mathbf{z}^{T}\mathbf{L}\mathbf{\lambda}+\frac{\varepsilon}{2}\mathbf{\hat{x}}^{T}\mathbf{L}\mathbf{\hat {x}}\). Let \(\hat{\mathbb{G}}(\mathbf{R}\mathbf{\hat{x}})=\text{col}(\partial_{\mathbf{\hat{x}}\mathbf{\hat {x}}}(\mathbf{R}\mathbf{\hat{x}}))_{i\in\mathcal{N}}\), the KKT condition is

\[ \begin{cases}\mathbf{0}\in\mathbf{R}^{T}\hat{\mathbb{F}}(\mathbf{\hat{x}})+\mathbf{R}^{T}\hat {\mathbb{G}}(\mathbf{R}\mathbf{\hat{x}})+\mathbf{R}^{T}\mathbf{A}^{T}\mathbf{\lambda}+c\mathbf{\hat{L} }\mathbf{\hat{x}},\\ \mathbf{0}=\mathbf{L}\mathbf{\lambda},\\ \mathbf{0}\in\mathbf{AR}\mathbf{\hat{x}}-\mathbf{b}+\mathbf{L}\mathbf{z}-N_{\mathbb{R}_{\mathbb{R}}^{ Nm}}(\mathbf{\lambda}).\end{cases}  \tag{16}\]

Next, according to the KKT condition (16), the operator \(\mathbb{T}\) can be defined as:

\[\mathbb{T}:\left[ \begin{array}{c}\mathbf{\hat{x}}\\ \mathbf{z}\\ \mathbf{\lambda}\end{array} \right]\longrightarrow\left[ \begin{array}{c}\mathbf{R}^{ T}\hat{\mathbb{F}}(\mathbf{\hat{x}})+\mathbf{R}^{T}\hat{\mathbb{G}}(\mathbf{R}\mathbf{\hat{x}})+\mathbf{R}^{T} \mathbf{A}^{T}\mathbf{\lambda}+c\mathbf{\hat{L}}\mathbf{\hat{x}}\\ \mathbf{L}\mathbf{\lambda}\end{array} \right]. \tag{17}\]

Let \(\mathbf{\omega}=\text{col}(\mathbf{\hat{x}},\mathbf{z},\mathbf{\lambda})\); the KKT condition can be written as \(0\in\mathbb{T}(\mathbf{\omega})\), then the solution of optimization problem (15) is converted into the operator zero-seeking problem. If \(\text{col}(\mathbf{\hat{x}}^{*},\mathbf{z}^{*},\mathbf{\lambda}^{*})\) is a zero of the operator \(\mathbb{T}\), then \(\mathbf{R}\mathbf{\hat{x}}^{*}\) is a v-GNE of the original game problem (3). The operator \(\mathbb{T}\) is partitioned into \(\overline{\mathbb{A}}\), \(\overline{\mathbb{B}}\) as follows :

\[\overline{\mathbb{A}}:\mathbf{\omega}\longrightarrow\left[ \begin{array}{c}\mathbf{R} ^{T}\hat{\mathbb{F}}(\mathbf{\hat{x}})\\ \mathbf{0}\\ \mathbf{b}\end{array} \right]+\frac{1}{2}\left[ \begin{array}{ccc}c\mathbf{\hat{L}}& \mathbf{0}&\mathbf{R}^{T}\mathbf{A}^{T}\\ \mathbf{0}&\mathbf{0}&\mathbf{L}\\ -\mathbf{AR}&-\mathbf{L}&\mathbf{0}\end{array} \right]\mathbf{\omega}, \tag{18}\]

\[\overline{\mathbb{B}}:\mathbf{\omega}\longrightarrow\left[ \begin{array}{c}\mathbf{R}^ {T}\hat{\mathbb{G}}(\mathbf{R}\mathbf{\hat{x}})\\ \mathbf{0}\\ N_{\mathbb{R}_{\mathbb{R}}^{Nm}}(\mathbf{\lambda})\end{array} \right]+\frac{1}{2} \left[ \begin{array}{ccc}c\mathbf{\hat{L}}&\mathbf{0}&\mathbf{R}^{T}\mathbf{A}^{T}\\ \mathbf{0}&\mathbf{0}&\mathbf{L}\\ -\mathbf{AR}&-\mathbf{L}&\mathbf{0}\end{array} \right]\mathbf{\omega}. \tag{19}\]

In view of the partition operators \(\overline{\mathbb{A}}\), \(\overline{\mathbb{B}}\). To obtain the distributed algorithm, a matrix \(\overline{\Phi}\) is defined, and this matrix is required to be positive definite. One choice is

\[\overline{\Phi}=\frac{1}{2}\left[ \begin{array}{ccc}-c\mathbf{\hat{L}}&\mathbf{0}&- \mathbf{R}^{T}\mathbf{A}^{T}\\ \mathbf{0}&\mathbf{0}&-\mathbf{L}\\ -\mathbf{AR}&-\mathbf{L}&\mathbf{0}\end{array} \right]+\left[ \begin{array}{ccc}\mathbf{\bar{ \alpha}}^{-1}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\overline{\tau}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\overline{\tau}^{-1}\end{array} \right], \tag{20}\]

where \(\overline{\alpha}=diag(\overline{\pi}_{i}I)\), \(\overline{\tau}=diag(\overline{\tau}_{i}I)\), \(\overline{\tau}=diag(\overline{\tau}_{i}I)\), with \(\overline{\pi}_{i}\), \(\overline{\tau}_{i}\) and \(\overline{\tau}_{i}\) for \(i\in\mathcal{N}\).