## 1 Introduction
According to Theorem 26.11 in [23], when \(\overline{\mathbb{A}}\) and \(\overline{\mathbb{B}}\) are maximally monotone operators, the sequence \(\{\mathbf{\xi}_{k}\}\) and \(\{\lambda_{k}\}\) obtained by (21) converges to \(x^{*}\) and \(\lambda^{*}\). Based on Assumptions 1-5, Algorithm 1 can be shown to be convergent using existing research, i.e., Theorem 1. The operator \(\widetilde{\mathbb{A}}\) is restricted monotone on \(\mathbb{Q}\) if Assumption 5 does not hold and Assumption 6 holds. Next, referring to the key concept of restricted monotonicity in the literature [13], it is possible to analyze the convergence of the algorithm based on Assumption 6. Lemma 4: _Define \(\sigma=s_{2}(L)\) and \(\widetilde{\mathbb{A}}_{p}:col(\mathbf{\hat{x}},\mathbf{z},\mathbf{\lambda})\longrightarrow col (\mathbf{R}^{T}\widehat{\mathbb{F}}(\mathbf{\hat{x}})+\frac{1}{2}c\mathbf{L}\mathbf{\hat{x}}, 0,\mathbf{b})\), assuming that Assumptions 2-4 hold and that the monotonicity of the operator \(\widetilde{\mathbb{A}}_{p}\) on the inner product space \(\mathbb{Q}\) is restricted when \(c\) satisfies \(c\geq\frac{2}{\sigma}(\frac{(\Theta_{2}+\Theta_{2})^{2}+4\theta_{1}\Theta_{2}} {4\theta_{1}})\)._

**Proof. **\(\forall\omega\) and \(\omega^{*}\), \(\langle\omega-\omega^{*},\widetilde{\Lambda}_{p}(\omega)-\widetilde{\Lambda}_{p}( \omega^{*})\rangle=\langle\hat{\mathbf{x}}-\hat{\mathbf{x}}^{*},\mathbf{R}^{T}(\widehat{ \mathbb{F}}(\hat{\mathbf{x}})-\widehat{\mathbb{F}}(\hat{\mathbf{x}}^{*}))\rangle+ \langle\hat{\mathbf{x}}-\hat{\mathbf{x}}^{*},\)\(\frac{1}{2}c\hat{\mathbf{L}}(\hat{\mathbf{x}}-\hat{\mathbf{x}}^{*})\rangle\). Denote \(\hat{\mathbf{x}}^{x}=(1_{N\times N}\bigotimes L)\hat{\mathbf{x}}\) and \(\hat{\mathbf{x}}^{y}=(I_{nN}-(1_{N\times N}\bigotimes L))\hat{\mathbf{x}}\). Since \(\omega^{*}\) is GNE, there exists \(\hat{\mathbf{x}}^{*}=\hat{\mathbf{x}}^{*^{*}}\). Assuming that Assumptions 4 holds, then \(\langle\hat{\mathbf{x}}-\hat{\mathbf{x}}^{*},\frac{1}{2}c\hat{\mathbf{L}}(\hat{\mathbf{x}}- \hat{\mathbf{x}}^{*})\rangle\geq\zeta^{y}\|\hat{\mathbf{x}}^{y}\|^{2}\) and

\[\langle\hat{\mathbf{x}}-\hat{\mathbf{x}}^{*},\mathbf{R}^{T}(\widehat{\mathbb{F }}(\hat{\mathbf{x}})-\widehat{\mathbb{F}}(\hat{\mathbf{x}}^{*}))\rangle\] \[=\langle\hat{\mathbf{x}}^{x}+\hat{\mathbf{x}}^{y}-\hat{\mathbf{x}}^{*},\mathbf{R} ^{T}(\widehat{\mathbb{F}}(\hat{\mathbf{x}})-\widehat{\mathbb{F}}(\hat{\mathbf{x}}^{*} ))\rangle\] \[\geq\frac{\theta_{1}}{N}\|\hat{\mathbf{x}}^{x}-\hat{\mathbf{x}}^{y}\|^{2} -\frac{\theta_{2}+\theta_{3}}{N^{0.5}}\|\hat{\mathbf{x}}^{y}\|\|\hat{\mathbf{x}}^{x}- \hat{\mathbf{x}}^{*}\|-\theta_{3}\|\hat{\mathbf{x}}^{y}\|^{2}.\]

Combining the above two inequalities, when \(c\geq\frac{2}{\sigma}\big{(}\frac{(\theta_{2}+\theta_{3})^{2}+\theta_{1}\theta _{3}}{4\theta_{1}}\big{)}\) holds, \(\widetilde{\Lambda}_{p}\) is restricted monotone on \(\mathbb{Q}\). \(\square\)

**Lemma 5**: _Assume that Lemma 4 holds and that the monotonicity of the operator \(\widetilde{\Lambda}\) on the inner product space \(\mathbb{Q}\) is restricted. In addition, \(\mathbf{O}_{\widetilde{\Lambda}}\) is restricted non-expansive on \(\mathbb{Q}\)._

**Proof. ** Assume that \(\widetilde{\Lambda}_{p}\) is a restricted monotone operator. By a similar process as in Lemma 2, Lemma 5 can also be proved. \(\square\)

**Remark 3**: _The operator \(\widehat{\mathbb{B}}\) is still maximally monotonic since Assumption 1 holds. Furthermore, \(\mathbf{P}_{\widetilde{\Lambda}}\) is still single-valued and well-defined, even though \(\widetilde{\Lambda}\) is a restricted monotone operator._

**Theorem 2**: _Suppose \(\lambda^{*}\) satisfies (6) and \(x^{*}\) is the \(v\)-GNE of Problem (3). When Lemma 5 is valid. The sequences \(\{\mathbf{\hat{x}}_{k}\}\) and \(\{\mathbf{\lambda}_{k}\}\) generated by Algorithm 1 satisfy \(\lim_{k\to\infty}\mathbf{\hat{x}}_{k}=(1_{N}\otimes x^{*})\) and \(\lim_{k\to\infty}\Lambda_{k}=(1_{N}\otimes\lambda^{*})\)._

**Proof.