## 3 Optimality Criteria
Since \(\mathcal{G}_{t},\;t\in\mathbb{T}\) are \(\mathfrak{\partial}^{*}\)-pseudoconvex at \(\overline{\mathfrak{a}}\), and therefore we have \[\langle\mathfrak{z}_{t}^{1},\mathfrak{b}-\overline{\mathfrak{a}}\rangle<0,\; \;\forall\mathfrak{z}_{t}^{1}\in\mathfrak{\partial}^{*}\mathcal{G}_{t}( \overline{\mathfrak{a}}),\;\;\forall t\in\mathbb{T}.\] (15) From the feasibility conditions of UNMPVC, we have \[\Psi_{j}(\mathfrak{b},\mathfrak{u}_{j})\leq\Psi_{j}(\overline{\mathfrak{a}}, \overline{\mathfrak{u}}_{j}),\;\overline{\mathfrak{u}}_{j}\in\Omega_{j}( \overline{\mathfrak{a}}),\;\forall j\in\mathfrak{T}_{\Psi}.\] Since \(\Psi_{j},\;j\in\mathbb{J}\) are \(\mathfrak{\partial}^{*}\)-quasiconvex at \(\overline{\mathfrak{a}}\), we therefore have \[\langle\mathfrak{z}_{j}^{2},\mathfrak{b}-\overline{\mathfrak{a}}\rangle\leq 0,\;\; \forall\mathfrak{z}_{j}^{2}\in\mathfrak{\partial}^{*}\Psi_{j}(\overline{ \mathfrak{a}},\overline{\mathfrak{u}}_{j}),\;\;\forall j\in\mathfrak{T}_{ \overline{\Psi}}.\] (16) In a similar manner, we obtain \[\langle\mathfrak{z}_{s}^{3},\mathfrak{b}-\overline{\mathfrak{a}}\rangle\leq 0,\;\; \forall\mathfrak{z}_{s}^{3}\in\mathfrak{\partial}^{*}\Theta_{\mathfrak{s}}( \overline{\mathfrak{a}},\overline{\mathfrak{u}}_{s}),\;\overline{ \mathfrak{u}}_{s}\in\Omega_{\mathfrak{s}}(\overline{\mathfrak{a}}),\;\forall s \in\mathbb{S},\] (17) \[\langle\mathfrak{z}_{s}^{4},\mathfrak{b}-\overline{\mathfrak{a}}\rangle\leq 0,\;\;\forall\mathfrak{z}_{s}^{4}\in\mathfrak{\partial}^{*}(-\Theta_{s})( \overline{\mathfrak{a}},\overline{\mathfrak{u}}_{s}),\;\overline{\mathfrak{u}}_ {s}\in\Omega_{\mathfrak{s}}(\overline{\mathfrak{a}}),\;\forall s\in\mathbb{S}\] (18) \[\mathcal{K}_{00}^{-}=\mathcal{K}_{0-}^{-}=\{\mathfrak{a}:\mathfrak{ a}_{2}\geq 0\}.\]0 (19) \[\mathcal{K}_{00}^{-}=\mathcal{K}_{0-}^{-}=\{\mathfrak{a}:\mathfrak{ a}_{2}\geq 0\}.\]1 (20) \[\mathcal{K}_{00}^{-}=\mathcal{K}_{0-}^{-}=\{\mathfrak{a}:\mathfrak{ a}_{2}\geq 0\}.\]2 (21) From (15)-(21) and from the assumption \(\mathfrak{T}_{0-}^{\mathbb{K}}\cup\mathfrak{T}_{0+}^{\mathbb{K}}\cup\mathfrak{ T}_{+0}^{\mathbb{L}}=\emptyset\), there exist multipliers \(\sigma_{t}^{\mathbb{G}}>0,\;t\in\mathbb{T},\;\sigma_{t}^{\mathbb{G}}\geq 0,\;j\in \mathfrak{T}_{\Psi},\;\sigma_{s}^{\mathbb{G}}\geq 0,\;\mathfrak{\sigma}_{s}^{ \mathbb{G}}\geq 0,\;\mathfrak{s}\in\mathbb{S},\;\sigma_{i}^{\mathbb{K}}\geq 0,\;i\in \mathfrak{T}_{0+}\cup\mathfrak{T}_{0-}\cup\mathfrak{T}_{00},\)\(\tau_{i}^{\mathbb{L}}\geq 0,\;i\in\mathfrak{T}_{+0},\;\tau_{i}^{\mathbb{K}}\geq 0,\;i\in \mathfrak{T}_{0+},\) such that \[ \begin{split}\bigg{\langle}\sum_{t\in\mathbb{T}}\sigma_{t}^{ \mathbb{G}}\mathfrak{z}_{t}^{1}+\sum_{j\in\mathfrak{T}_{\Psi}}\sigma_{j}^{ \mathbb{K}}\mathfrak{z}_{j}^{2}&+\sum_{s\in\mathbb{S}}[\sigma_{i}^{ \mathbb{G}}\mathfrak{z}_{i}^{3}+\tau_{i}^{\mathbb{G}}\mathfrak{z}_{i}^{4}]\\ &+\sum_{i\in\mathbb{I}}\sigma_{i}^{\mathbb{K}}\mathfrak{z}_{i}^{5}+ \sum_{i\in\mathbb{I}}[\tau_{i}^{\mathbb{K}}\mathfrak{z}_{i}^{6}+\tau_{i}^{\mathbb{L}} \mathfrak{z}_{i}^{7}],\mathfrak{b}-\overline{\mathfrak{a}}\bigg{\rangle}<0,\end{split} \] (22) from which we obtain a contradiction to the fact that \(\overline{\mathfrak{a}}\) is a RS-stationary point of UNMPVC. 2. Using the definition of a strictly \(\mathfrak{\partial}^{*}\)-pseudoconvex function, the proof is similar to the first part. Hence, we omit it for brevity. Remark 2.: _1. Theorem 4 extends Theorem 3.4 derived in [55] from nonsmooth uncertain programming problems to a broader category of mathematical programming problems, namely UNMPVC. For \(\mathbb{I}=\emptyset\) and \(\mathbb{S}=\emptyset\) Theorem 4 simplifies to Theorem 3.4 derived in [55]._
2. _Theorem 4 extends Theorem 3.3 derived in [40] from nonsmooth MPVC to a wider category of optimization problems, UNMPVC.