## 4 HS-Type Conjugate Direction Method for IVMOPs  
In this section, we present an HS-type conjugate direction method for IVMOPs. Moreover, we establish the convergence of the sequence generated by this method. Consider the following IVMOP:  
\[ \begin{array}{ll}\text{(IVMOP)}&\text{Minimize}&\mathcal{H}(u):=(\mathcal{ H}_{1}(u),\ldots,\mathcal{H}_{m}(u)),\\ &\text{subject to}&u\in\mathbb{R}^{n},\end{array} \]  
where the functions \(\mathcal{H}_{k}:\mathbb{R}^{n}\to\mathfrak{C}(\mathbb{R})\ (k\in\mathcal{I}_{m})\) are defined as  
\[\mathcal{H}_{k}(u):=\big{[}\underline{\mathcal{H}}_{k}(u),\ \overline{ \mathcal{H}}_{k}(u)\big{]},\quad\text{ for all }u\in\mathbb{R}^{n}.\]  
The functions \(\mathcal{H}_{k}\ (k\in\mathcal{I}_{m})\) are assumed to be continuously gH-differentiable unless otherwise specified. The notions of effective and weak effective solutions for IVMOP are recalled in the subsequent definition [28]. **Definition 12**. : _A point \(\bar{u}\in\mathbb{R}^{n}\) is said to be an effective solution of the IVMOP if there is no other point \(u\in\mathbb{R}^{n}\) such that_  
\[\mathcal{H}(u)\preceq_{m}\mathcal{H}(\bar{u})\quad\text{and}\quad\mathcal{H} (u)\
eq\mathcal{H}(\bar{u}).\]  
_Similarly, a point \(\bar{u}\in\mathbb{R}^{n}\) is said to be a weak effective solution of the IVMOP provided that there is no other point \(u\in\mathbb{R}^{n}\) for which:_  
\[\mathcal{H}(u)\prec_{m}\mathcal{H}(\bar{u}).\]  
In the rest of the article, we employ \(\mathcal{P}\) to represent the set of all critical points of \(\mathcal{H}\). Let \(\bar{u}\in\mathbb{R}^{n}\). In order to determine the descent direction for the objective function \(\mathcal{H}\) of IVMOP, we consider the following scalar optimization problem with interval-valued constraints [32]:  
\[ \begin{array}{ll}\text{(P)}_{\bar{u}}&\text{Minimize}&\varphi(\alpha, \mathfrak{d}):=\alpha,\\ &\text{subject to}&D_{gH}\mathcal{H}_{k}(\bar{u},\mathfrak{d})+\frac{1}{2} \Big{[}\|\mathfrak{d}\|^{2},\|\mathfrak{d}\|^{2}\Big{]}\leq_{LU}[\alpha, \alpha],\\ &k\in\mathcal{I}_{m},\end{array} \]  
where \(\varphi:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}\) is a real-valued function. It can be shown that the problem \(\text{(P)}_{\bar{u}}\) has a unique solution. Any feasible point of the problem \(\text{(P)}_{\bar{u}}\) is represented as \((\alpha_{\bar{u}},\mathfrak{d}_{\bar{u}})\), where \(\alpha_{\bar{u}}\in\mathbb{R}\) and \(\mathfrak{d}_{\bar{u}}\in\mathbb{R}^{n}\). Let \(\mathcal{K}_{\bar{u}}\subseteq\mathbb{R}^{n+1}\) denote the feasible set of \(\text{(P)}_{\bar{u}}\). We consider the functions \(d:\mathbb{R}^{n}\to\mathbb{R}^{n}\) and \(\alpha:\mathbb{R}^{n}\to\mathbb{R}\), which are defined as follows:  
\[(\alpha(\bar{u}),d(\bar{u})):=\operatorname*{arg\,min}_{(\alpha_{\bar{u}}, \mathfrak{d}_{\bar{u}})\in\mathcal{K}_{\bar{u}}}\varphi(\alpha_{\bar{u}}, \mathfrak{d}_{\bar{u}}).\]From now onwards, for any \(u\in\mathbb{R}^{n}\), the notation \((\alpha(u),d(u))\) will be used to represent the optimal solution of the problem (P)\({}_{u}\). Now, for every \(k\in\mathscr{I}_{m}\), we consider a function \(\zeta_{k}:\mathbb{R}^{n}\to\mathbb{R}\), defined as follows:  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]  
**Remark 3**. : _Since \((\alpha(\bar{u}),d(\bar{u}))\) is a solution of the problem (P)\({}_{\bar{u}}\), therefore for all \(k\in\mathscr{I}_{m}\), it follows that:_  
\[D_{gH}\mathcal{H}_{k}(\bar{u},d(\bar{u}))+\frac{1}{2}\Big{[}\|d(\bar{u})^{2}, \|d(\bar{u})\|^{2}\Big{]}\leq_{LU}[\alpha(\bar{u}),\alpha(\bar{u})].\]  
_This implies that the function \(\zeta_{k}:\mathbb{R}^{n}\to\mathbb{R}\) satisfies the following inequality:_  
\[\
abla\zeta_{k}(\bar{u})^{T}d(\bar{u})\leq 2\alpha(\bar{u}),\quad\text{for all }k \in\mathscr{I}_{m}.\]  
In the subsequent discussions, we utilize the following lemmas from [PERSON] et al. [32]. **Lemma 1**. : _Let \(\bar{u}\in\mathbb{R}^{n}\).