## 3 Optimality Criteria
\begin{split}\sum_{t\in\mathbb{T}}\sigma_{t}^{\mathcal{G}} \mathcal{G}_{t}(a)&\geq\sum_{t\in\mathbb{T}}\sigma_{t}^{\mathcal{G }}\mathcal{G}_{t}(b)+\sum_{j\in\mathfrak{T}_{\Psi}}\sigma_{j}^{\Psi}\Psi_{j}(b, u_{j})\\ &+\sum_{s\in\mathbb{S}}[\sigma_{s}^{\Theta}\Theta_{s}(b,u_{s})+ \tau_{s}^{\Theta}(-\Theta_{s})(b,u_{s})]+\sum_{i\in\mathbb{I}}\sigma_{i}^{K}(- \mathbb{K}_{i})(b,u_{i})\\ &+\sum_{i\in\mathbb{I}}[\sigma_{i}^{K}K_{i}(b,u_{i})+\sigma_{i}^ {L}L_{i}(b,u_{i})].\end{split} \]  
Hence, from (25), a contradiction is reached, which concludes the proof of the weak duality theorem. In the subsequent theorem, a strong duality result is established, which relates UNMPVC with the associated WRD dual model. Theorem 4: _Let \(\overline{a}\in\mathbb{F}_{\mathcal{G}}\) be a robust local weak Pareto solution of UNMPVC. Suppose that \(\mathcal{G}_{t},\ t\in\mathbb{T},\ \Psi_{j},\ j\in\mathfrak{T}_{\Psi_{j}},\ \pm\Theta_{s},\ s\in \mathbb{S},\ -K_{i},\ i\in\mathfrak{T}_{0+}\cup\mathfrak{T}_{0-}\cup\mathfrak{T}_{00},K_{ i},\ i\in\mathfrak{T}_{0+},\) and \(L_{i},\ i\in\mathfrak{T}_{+0}\) are \(\mathfrak{d}^{*}\)-convex functions at \(\overline{a}\) and admit bounded USRC. Assume also that at \(\overline{a},\) GS-ACQ holds. Then, there exists \(\overline{\sigma}=(\overline{\sigma}^{\mathcal{G}},\overline{\sigma}^{ \mathcal{V}},\overline{\sigma}^{\Theta},\overline{\tau}^{\Theta},\overline{ \tau}^{\overline{\kappa}},\overline{\tau}^{\overline{\kappa}})\in\mathbb{R}^{ r}\times\mathbb{R}^{p}\times\mathbb{R}^{2\ell}\times\mathbb{R}^{3m},\) such that \((\overline{a},\overline{\sigma})\) becomes a robust local weak Pareto solution of the WRD. Moreover, the corresponding objective function values are equal._  
Proof: Given that GS-ACQ holds as a robust local weak Pareto solution \(\overline{a}\) of UNMPVC, then, in light of Theorem 3, there exists  
\[\overline{\sigma}=(\overline{\sigma}^{\mathcal{G}},\overline{\sigma}^{ \overline{\sigma}},\overline{\tau}^{\Theta},\overline{\tau}^{\Theta},\overline {\tau}^{\overline{\kappa}},\overline{\tau}^{\overline{\kappa}})\in\mathbb{R}^{ r}\times\mathbb{R}^{p}\times\mathbb{R}^{2\ell}\times\mathbb{R}^{3m},\]  
such that \(\overline{a}\) satisfies RS-stationary conditions of UNMPVC. Thus, there exist \(\overline{\xi}_{t}^{1}\in co\mathfrak{d}^{*}\mathcal{G}_{t}(\overline{a}),\)\(t\in\mathbb{T},\ \overline{\xi}_{j}^{2}\in co\mathfrak{d}^{*}\Psi_{j}(\overline{a},\overline{u}_{j}),\ j\in\mathfrak{T}_{\Psi},\overline{u}_{j}\in \Omega_{j}(\overline{a}),\ \overline{\xi}_{s}^{3}\in co\mathfrak{d}^{*}\Theta_{s}(\overline{a}, \overline{u}_{s}),\ \overline{\xi}_{s}^{4}\in co\mathfrak{d}^{*}(-\Theta_{s})(\overline{a}, \overline{u}_{s}),\ \mathfrak{s}\in\mathbb{S},\ \overline{u}_{s}\in\Omega_{s}(\overline{a}),\ \overline{\xi}_{i}^{5}\in co \mathfrak{d}^{*}(-\kappa_{i})(\overline{a},\overline{u}_{i}),\ i\in\mathbb{I},\ \overline{\xi}_{i}^{6}\in co \mathfrak{d}^{*}K_{i}(\overline{a},\overline{u}_{i}),i\in\mathbb{I},\) and \(\overline{\xi}_{i}^{7}\in co\mathfrak{d}^{*}L_{i}(\overline{a},\overline{u}_{ i}),i\in\mathbb{I},\ \overline{u}_{i}\in\Omega_{i}(\overline{a}),\) such that  
\[ \begin{split}\sum_{t\in\mathbb{T}}\overline{\sigma}_{t}^{ \overline{\kappa}}\mathbb{I}_{t}+\sum_{j\in\mathfrak{T}_{\Psi}}\overline{\sigma }_{j}^{\Psi}\overline{\xi}_{j}^{2}+\sum_{s\in\mathbb{S}}[\overline{\sigma}_{s }^{\Theta}\overline{\xi}_{s}^{3}+\overline{\tau}_{s}^{\Theta}\overline{\xi}_{ s}^{4}]+\sum_{i\in\mathbb{I}}\overline{\xi}_{i}^{\overline{\kappa}} \overline{\xi}_{i}^{5}+\sum_{i\in\mathbb{I}}[\overline{\tau}_{i}^{\overline{ \kappa}}\overline{\xi}_{i}^{6}+\overline{\tau}_{i}^{\overline{\kappa}}\overline{ \xi}_{i}^{7}]=0,\\ \overline{\sigma}_{t}^{\Theta}>0,\ t\in\mathbb{T},\ \overline{\sigma}_{j}^{ \Psi}\geq 0,\ j\in\mathfrak{T}_{\Psi},\ \overline{\sigma}_{s}^{\Theta}\geq 0,\overline{\tau}_{s}^{ \Theta}\geq 0,\ s\in\mathbb{S},\ \overline{\sigma}_{i}^{\overline{\kappa}}\geq 0,\overline{\tau}_{i}^{ \overline{\kappa}}\geq 0,\overline{\tau}_{i}^{\overline{\kappa}}\geq 0,\ i\in\mathbb{I}.\end{split}