## 3 Optimality Criteria
As a result, there exist non-negative multipliers \(\sigma_{t}^{G}>0,\ t\in\mathbb{T}\), \(\sigma_{t}^{\overline{\mathfrak{v}}}\geq 0\), \(j\in\mathfrak{T}_{\overline{\mathfrak{v}}}\), \(\sigma_{s}^{\Theta}\geq 0\), \(\tau_{s}^{\Theta}\geq 0\), \(s\in\mathbb{S}\), \(\tau_{i}^{\mathbb{K}}\geq 0\), \(i\in\mathfrak{T}_{0+}\)\(\sigma_{i}^{\mathbb{K}}\geq 0\), \(i\in\mathfrak{T}_{0+}\cup\mathfrak{T}_{0-}\cup\mathfrak{T}_{00}\), \(\tau_{i}^{\mathbb{L}}\geq 0,i\in\mathfrak{T}_{+0}\), such that  
\[ \begin{split} 0\in\sum_{t\in\mathbb{T}}\sigma_{t}^{G} \operatorname{co}\partial^{*}\mathcal{G}_{t}(\overline{\mathfrak{a}})& +\sum_{j\in\mathfrak{T}_{\overline{\mathfrak{v}}}}\sigma_{j}^{ \overline{\mathfrak{v}}}\operatorname{co}\partial^{*}\mathcal{F}_{j}^{*}( \overline{\mathfrak{a}})\\ &+\sum_{s\in\mathbb{S}}\tau_{s}^{\Theta}\operatorname{co} \partial^{*}(-\Theta_{s}^{*})(\overline{\mathfrak{a}})+\sum_{i\in\mathfrak{T}_{ 0+}\cup\mathfrak{T}_{0-}\cup\mathfrak{T}_{00}}\sigma_{i}^{\mathbb{K}} \operatorname{co}\partial^{*}(-\mathbf{K}_{i}^{*})(\overline{\mathfrak{a}})\\ &+\sum_{i\in\mathfrak{T}_{0+}}\tau_{i}^{\mathbb{K}} \operatorname{co}\partial^{*}\mathbf{K}_{i}^{*}(\overline{\mathfrak{a}})+\sum_ {i\in\mathfrak{T}_{+0}}\tau_{i}^{\mathbb{L}}\operatorname{co}\partial^{*} \mathbf{L}_{i}^{*}(\overline{\mathfrak{a}}).\end{split}  \tag{14}\]  
Taking \(\sigma_{\mathfrak{T}_{\pm 0}\cup\mathfrak{T}_{+-}}^{\mathbb{K}}=0,\ \tau_{\mathfrak{T}_{00}\cup\mathfrak{T}_{0-}\cup\mathfrak{T}_{+-}\cup \mathfrak{T}_{+0}}^{\mathbb{K}}=0,\ \tau_{\mathfrak{T}_{\pm 0}\cup\mathfrak{T}_{0+}\cup\mathfrak{T}_{0-}\cup \mathfrak{T}_{00}}=0,\) from (14), we obtain  
\[ \begin{split} 0\in\sum_{t\in\mathbb{T}}\sigma_{t}^{G} \operatorname{co}\partial^{*}\mathcal{G}_{t}(\overline{\mathfrak{a}})& +\sum_{j\in\mathfrak{T}_{\overline{\mathfrak{v}}}}\sigma_{j}^{ \overline{\mathfrak{v}}}\operatorname{co}\partial^{*}\mathcal{F}_{j}^{*}( \overline{\mathfrak{a}})\\ &+\sum_{s\in\mathbb{S}}[\sigma_{s}^{\Theta}\operatorname{co} \partial^{*}\mathcal{G}_{s}^{*}(\overline{\mathfrak{a}})+\tau_{s}^{\Theta} \operatorname{co}\partial^{*}(-\Theta_{s}^{*})(\overline{\mathfrak{a}})]\\ &+\sum_{i\in\mathbb{I}}\sigma_{i}^{\mathbb{K}}\operatorname{co} \partial^{*}(-\mathbf{K}_{i}^{*})(\overline{\mathfrak{a}})\\ &+\sum_{i\in\mathbb{I}}[\tau_{i}^{\mathbb{K}}\operatorname{co} \partial^{*}\mathbf{K}_{i}^{*}(\overline{\mathfrak{a}})+\tau_{i}^{\mathbb{L}} \operatorname{co}\partial^{*}\mathbf{L}_{i}^{*}(\overline{\mathfrak{a}})],\end{split} \]  
\[\sigma_{t}^{G}>0,\ t\in\mathbb{T},\ \sigma_{t}^{\overline{\mathfrak{v}}}\geq 0,\ j\in \mathfrak{T}_{\overline{\mathfrak{v}}},\ \sigma_{s}^{\Theta}\geq 0,\ \tau_{s}^{\Theta}\geq 0,\ s\in\mathbb{S},\ \tau_{i}^{\mathbb{K}}\geq 0,\ \sigma_{i}^{\mathbb{K}}\geq 0,\ \tau_{i}^{\mathbb{L}}\geq 0,\ i\in\mathbb{I}\]  
\[\sigma_{\mathfrak{T}_{\pm 0}\cup\mathfrak{T}_{\pm-}}^{\mathbb{K}}=0,\ \tau_{\mathfrak{T}_{\pm 0}\cup \mathfrak{T}_{\pm-}\cup\mathfrak{T}_{\pm-}}^{\mathbb{K}}=0,\ \tau_{\mathfrak{T}_{\pm-}\cup\mathfrak{T}_{\pm 0}\cup \mathfrak{T}_{0-}}^{\mathbb{K}}=0,\]\[\forall i\in\mathfrak{T}_{00},\ \ \tau_{1}^{L}=0,\ (\sigma_{1}^{K}-\tau_{1}^{K})= \sigma_{1}^{K}\geq 0.\]  
Hence, \(\overline{a}\) is an RS-stationary point of NMPVC\({}^{*}\). From Theorem 2, there exist \(\bar{u}_{j}\in\Omega_{j}(\overline{a})\), \(\bar{u}_{s}\in\Omega_{s}(\overline{a})\), and \(\bar{u}_{i}\in\Omega_{i}(\overline{a})\), such that \(\overline{a}\) is a RS-stationary point of UNMPVC. Remark 1:
1. _It is worthwhile to note that Theorem 3 extends the scope of Theorem 3.1 established by [PERSON] et al. [40] for nonsmooth MPVC to a broader class of programming problems involving data uncertainty. If \(\Omega_{j},\ j\in\mathbb{J},\ \Omega_{s},\ s\in\mathbb{S}\), and \(\Omega_{i},\ i\in\mathbb{I}\) are singleton sets, then Theorem 3 reduces to Theorem 3.1 established in [40]._
2. _In the domain of robust optimization, Theorem 3 extends the robust first-order optimality conditions established by [PERSON] et al.