## 2 Preliminaries and Notations  
The following abbreviations are used in this article:  
\[\text{MOP}:\text{Multiobjective programming problems}\] \[\text{CQ}:\text{Constraint qualifications}\] \[\text{UNMPVC}:\text{Nonsmooth multiobjective programming problems with vanishing}\] \[\text{constraints under data uncertainty}\] \[\text{GS-ACQ}:\text{Generalized standard Abadie3. constraint qualification}\] \[\text{RW-staionary}:\text{Robust weakly stationary point}\] \[\text{RM-staionary}:\text{Robust Mordukhovich stationary point}\] \[\text{RS-staionary}:\text{Robust strong stationary point}\] \[\text{WRD}:\text{Wolfe-type robust dual model}\] \[\text{MWRD}:\text{Mond-Weir-type robust dual model}\]  
Throughout this article, we use the notation \(\mathbb{R}^{n}\) to denote the \(n\)-dimensional Euclidean space and \(\mathbb{N}\) to denote the set of all natural numbers. Let \(\mathsf{b},\mathsf{d}\in\mathbb{R}^{n}\). The notations given below will be employed:  
\[\text{CQ}:\text{Constraint qualifications}\]0 \[\mathsf{b}\preceq\mathsf{d} \Longleftrightarrow \begin{cases}\mathsf{b}_{\ell}\leq \mathsf{d}_{\ell},\ \forall\ell\in\{1,\cdots,n\},\\ \mathsf{b}_{k}<\mathsf{d}_{k},\text{ for at least one }k\in\{1,\cdots,n\}.\end{cases} \]  
Let \(\mathcal{D}\) be a nonempty subset of \(\mathbb{R}^{n}\). We denote the closure of \(\mathcal{D}\) as \(\mathsf{cl}\,\mathcal{D}\), its convex hull as \(\mathsf{co}\,\mathcal{D}\), and the convex cone generated by \(\mathcal{D}\) (including the origin) as \(\mathsf{cone}\,\mathcal{D}\). Let \(\mathcal{G}:\mathbb{R}^{n}\to\overline{\mathbb{R}}\) be an extended real-valued function. Then, the domain of function \(\mathcal{G}\) is denoted by \(\text{dom}(\mathcal{G})\) and is defined as follows:\[\text{dom}(\mathcal{G}):=\{\mathfrak{a}\in\mathbb{R}^{n}:\;\mathcal{G}(\mathfrak{a}) \
eq+\infty\}.\]  
We now define the following sets:  
\[\mathcal{D}^{-}:=\{\mathfrak{a}\in\mathbb{R}^{n}:\;\langle\mathfrak{a},\; \mathfrak{b}\rangle\leq 0,\;\forall\mathfrak{b}\in\mathcal{D}\},\]  
\[\mathcal{D}^{s}:=\{\mathfrak{a}\in\mathbb{R}^{n}:\;\langle\mathfrak{a},\; \mathfrak{b}\rangle<0,\;\forall\mathfrak{b}\in\mathcal{D}\}.\]  
The following definitions are used in this article:  
**Definition 1** ([56]). : _The contingent cone \(\mathcal{T}(\mathcal{D},\mathfrak{a})\) of a nonempty subset \(\mathcal{D}\) of \(\mathbb{R}^{n}\) at \(\mathfrak{a}\in\operatorname{cl}\mathcal{D}\) is defined as_  
\[\mathcal{T}(\mathcal{D},\mathfrak{a}):=\{\mathfrak{d}\in\mathbb{R}^{n}:\exists \mathfrak{a}_{n}\downarrow 0\;\&\;\mathfrak{d}_{n}\to\mathfrak{d}\text{ such that }\mathfrak{a}+\mathfrak{a}_{n}\mathfrak{d}_{n}\in \mathcal{D},\;\forall n\in\mathbb{N}\}.\]  
**Definition 2** ([56]). : _The lower and upper Dini derivatives of function \(\mathcal{G}\) at \(\mathfrak{a}\) in some direction \(\mathfrak{d}\in\mathbb{R}^{n}\) are defined, respectively, as_  
\[\mathcal{G}^{-}(\mathfrak{a};\mathfrak{d}):=\liminf_{\mu\downarrow 0}\frac{ \mathcal{G}(\mathfrak{a}+\mu\mathfrak{d})-\mathcal{G}(\mathfrak{a})}{\mu},\]  
\[\mathcal{G}^{+}(\mathfrak{a};\mathfrak{d}):=\limsup_{\mu\downarrow 0}\frac{ \mathcal{G}(\mathfrak{a}+\mu\mathfrak{d})-\mathcal{G}(\mathfrak{a})}{\mu}.\]  
**Definition 3** ([56]). : \(\mathcal{G}:\mathbb{R}^{n}\to\overline{\mathbb{R}}\) _is said to have an upper semi-regular convexificator (USRC) at \(\mathfrak{a}\in\text{dom}(\mathcal{G})\), denoted as \(\partial^{*}\mathcal{G}(\mathfrak{a})\), provided that the set \(\partial^{*}\mathcal{G}(\mathfrak{a})\) is a closed set and for every \(\mathfrak{d}\in\mathbb{R}^{n}\), one has_  
\[\mathcal{G}^{+}(\mathfrak{a};\mathfrak{d})\leq\sup_{\kappa\in\partial^{*} \mathcal{G}(\mathfrak{a})}\langle\kappa,\;\mathfrak{d}\rangle.\]  
**Definition 4** ([57]). : _Let \(\mathcal{G}:\mathbb{R}^{n}\to\overline{\mathbb{R}}\) and admit an bounded USRC at \(\mathfrak{a}\in\text{dom}(\mathcal{G})\), denoted by \(\partial^{*}\mathcal{G}(\mathfrak{a})\).