## 3 Optimality Criteria  
In this section, we consider a nonsmooth MOP with vanishing constraints under data uncertainty. We introduce GS-ACQ at a feasible point of UNMPVC. Moreover, we introduce RW-stationary, RM-stationary and RS-stationary points for UNMPVC. By employing GS-ACQ, we establish that the RS-stationary is a necessary optimality criterion for UNMPVC. Subsequently, we establish sufficient optimality criteria for UNMPVC. In the remaining portion of this article, we examine the following UNMPVC:  
\[\text{(UNMPVC) Minimize}\ \ \mathcal{G}(\mathfrak{a}):=( \mathcal{G}_{1}(\mathfrak{a}),\cdots,\mathcal{G}_{r}(\mathfrak{a})),\] subject to \[\ \ \mathbb{V}_{j}(\mathfrak{a},\ u_{j})\leq 0,\ j=1,\cdots,p,\] \[\ \ \Theta_{s}(\mathfrak{a},\mathfrak{u}_{s})=0,\ \mathfrak{s}=1, \ldots,\ell,\] \[\ \ \mathbb{K}_{i}(\mathfrak{a},\mathfrak{u}_{i})\geq 0,\ i=1, \ldots,m,\] \[\ \ \ \mathbb{L}_{i}(\mathfrak{a},\mathfrak{u}_{i})\mathbb{K}_{i}( \mathfrak{a},\mathfrak{u}_{i})\leq 0,\ i=1,\ldots,m,\]  
where \(\mathcal{G}_{t}:\mathbb{R}^{n}\to\mathbb{R}\), \(t\in\mathbb{T}:=\{1,\ldots,r\}\), \(\mathbb{V}_{j}:\mathbb{R}^{n}\times\mathbb{R}^{n_{j}}\to\mathbb{R}\), \(j\in\mathbb{J}:=\{1,\cdots,p\}\), \(\Theta_{s}:\mathbb{R}^{n}\times\mathbb{R}^{n_{s}}\to\mathbb{R}\), \(\mathfrak{s}\in\mathbb{S}:=\{1,\cdots,\ell\}\), \(\mathbb{K}_{i}:\mathbb{R}^{n}\times\mathbb{R}^{n_{i}}\to\mathbb{R}\), \(\mathfrak{i}\in\mathbb{I}:=\{1,\cdots,m\}\), and \(\mathbb{L}_{i}:\mathbb{R}^{n}\times\mathbb{R}^{n_{i}}\to\mathbb{R}\), \(\mathfrak{i}\in\mathbb{I}:=\{1,\cdots,m\}\) are extended real-valued functions and admit bounded USRC; \(u_{j}\in\Omega_{j}\), \(j\in\mathbb{J}\), \(u_{s}\in\Omega_{s}\), \(\mathfrak{s}\in\mathbb{S}\) and \(\mathfrak{u}_{i}\in\Omega_{i}\), \(\mathfrak{i}\in\mathbb{I}\) are uncertain parameters; and \(\Omega_{j}\), \(\Omega_{s}\), \(\Omega_{i}\) are nonempty convex subsets of \(\mathbb{R}^{n_{j}}\), \(\mathbb{R}^{n_{s}}\) and \(\mathbb{R}^{n_{i}}\), respectively. Associated with UNMPVC, the robust multiobjective programming problem (RNMPVC) is defined as  
\[\text{(RNMPVC) Minimize}\ \ \mathcal{G}(\mathfrak{a}):=( \mathcal{G}_{1}(\mathfrak{a}),\cdots,\mathcal{G}_{r}(\mathfrak{a})),\] subject to \[\ \ \mathbb{V}_{j}(\mathfrak{a},\ u_{j})\leq 0,\ \forall\mathfrak{u}_{j}\in\Omega_{j},\ j \in\mathbb{J},\] \[\ \ \Theta_{s}(\mathfrak{a},\mathfrak{u}_{s})=0,\ \forall \mathfrak{u}_{s}\in\Omega_{s},\ \mathfrak{s}\in\mathbb{S},\] \[\ \ \mathbb{K}_{i}(\mathfrak{a},\mathfrak{u}_{i})\geq 0,\ \forall \mathfrak{u}_{i}\in\Omega_{i},\ i\in\mathbb{I},\] \[\ \ \mathbb{L}_{i}(\mathfrak{a},\mathfrak{u}_{i})\mathbb{K}_{i}( \mathfrak{a},\mathfrak{u}_{i})\leq 0,\ \forall\mathfrak{u}_{i}\in\Omega_{i},\mathfrak{i}\in\mathbb{I}.\]  
\(\overline{u}_{j}\in\Omega_{j}(\overline{a}),\ j\in\mathbb{J},\ \overline{u}_{s}\in\Omega_{s}( \overline{a}),\ s\in\mathbb{S},\ \overline{u}_{i}\in\Omega_{i}(\overline{a}),\ i\in \mathbb{I},\) such that they satisfy conditions (1)-(3) along with the subsequent condition:_  
\[\ \ \mathbb{V}_{j}(\mathfrak{a},\ u_{j})\leq 0,\ j=1,\cdots,p,\]0  
**Definition 11**. : _Let \(\overline{a}\in\mathbb{F}_{\mathcal{G}}\). \(\overline{a}\) is a generalized robust strong stationary (RS-stationary) point if there exist \(\sigma=(\sigma^{\mathcal{G}},\sigma^{\
abla},\sigma^{\Theta},\sigma^{\mathbb{K }})\in\mathbb{R}^{r+p+\ell+m},\ \tau=(\tau^{\Theta},\tau^{\mathbb{K}},\tau^{\mathbb{L}})\in \mathbb{R}^{\ell+2m}\) and \(\overline{u}_{j}\in\Omega_{j}(\overline{a}),\ j\in\mathbb{J},\ \overline{u}_{s}\in\Omega_{s}( \overline{a}),\ s\in\mathbb{S},\ \overline{u}_{i}\in\Omega_{i}(\overline{a}),\ i\in \mathbb{I},\) such that they satisfy conditions (1)-(3) along with the subsequent condition:_  
\[\ \ \mathbb{V}_{j}(\mathfrak{a},\ u_{j})\leq 0,\ j=1,\cdots,p,\]1  
From the above definitions, we conclude that  
\[\ \ \mathbb{V}_{j}(\mathfrak{a},\ u_{j})\leq 0,\ j=1,\cdots,p,\]2  
In the following theorem, by employing GS-ACQ, we establish that the RS-stationary conditions are necessary for a local weak Pareto efficiency of UNMPVC. **Theorem 3**.