## 4 HS-Type Conjugate Direction Method for IVMOPs
: Since \((\alpha(\bar{u}),d(\bar{u})))\in\mathcal{K}_{\bar{u}}\), therefore we have  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]3  
From (4), we obtain  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]4  
Consequently,  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]5  
Let us define \(\alpha_{\bar{u}}:=\widetilde{\Phi}(\bar{u},d(\bar{u}))\). Then, we have  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]6  
Therefore, we obtain  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]7  
This implies that \((\alpha_{\bar{u}},d(\bar{u}))\in\mathcal{K}_{\bar{u}}\). Since \((\alpha(\bar{u}),d(\bar{u})))\) is the optimal solution of the problem (P)\({}_{\bar{u}}\), we obtain  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]8  
Combining (5) and (6), we conclude  
\[\zeta_{k}(u):=\overline{\mathcal{H}}_{k}(u)+\underline{\mathcal{H}}_{k}(u), \quad\text{for all }u\in\mathbb{R}^{n}.\]9  
This completes the proof. Let \(s\in\{0,1,2,\ldots\}\) be fixed and let \(\{\mathsf{u}^{(r)}\}_{0\leq r\leq s}\subset\mathbb{R}^{n}\). Now, we introduce a Hestenes-Stiefel-type direction (HS-type direction) \(\mathsf{w}_{s}^{\text{\rm{ss}}}\) at \(\mathsf{u}^{(s)}\). \[\mathsf{w}_{s}^{\text{\rm{ss}}}:= \begin{cases}d(\mathsf{u}^{(s)});&\text{if }\,s=0,\\ d(\mathsf{u}^{(s)})+\beta_{s}^{\text{\rm{ss}}}\mathsf{w}_{(s-1)}^{\text{\rm{ss}} };&\text{if }\,s\geq 1,\end{cases}  \tag{7}\]  
where \(\mathsf{w}_{(s-1)}^{\text{\rm{ss}}}\) represents the HS-type direction at \((s-1)\)-th step, and for \(s\geq 1\), \(\beta_{s}^{\text{\rm{ss}}}\) is defined as follows:  
\[\beta_{s}^{\text{\rm{ss}}}:=\frac{-\widetilde{\[PERSON]}\big{(}\mathsf{u}^{(s)},d( \mathsf{u}^{(s)})\big{)}+\widetilde{\[PERSON]}\big{(}\mathsf{u}^{(s-1)},d(\mathsf{ u}^{(s-1)})\big{)}}{\widetilde{\[PERSON]}\big{(}\mathsf{u}^{(s)},\mathsf{w}_{(s-1)} ^{\text{\rm{ss}}}\big{)}-\widetilde{\[PERSON]}\big{(}\mathsf{u}^{(s-1)},\mathsf{ w}_{(s-1)}^{\text{\rm{ss}}}\big{)}}. \tag{8}\]  
**Remark 5**. : _If every component of the objective function \(\mathcal{H}\) of the IVMOP is a real-valued function rather than an interval-valued function, that is, \(\mathcal{H}:\mathbb{R}^{n}\to\mathbb{R}^{m}\), then Equation (7) reduces to the HS-type direction defined for vector-valued functions, as considered by [PERSON] and [PERSON] [18]. As a result, the parameter \(\beta_{s}^{\text{\rm{ss}}}\) introduced in (8) extends the HS-type direction from MOPs to IVMOPs, which belong to a broader class of optimization problems. Moreover, when \(m=1\), Equation (7) further reduces to the classical HS-type direction for a real-valued function, defined by [PERSON] and [PERSON] [17]._  
It can be observed that \(\beta_{s}^{\text{\rm{ss}}}\), defined in (8), becomes undefined when  
\[\widetilde{\Phi}\Big{(}\mathsf{u}^{(s)},\mathsf{w}_{(s-1)}^{\text{\rm{ss}}} \big{)}=\widetilde{\Phi}\Big{(}\mathsf{u}^{(s-1)},\mathsf{w}_{(s-1)}^{\text{ \rm{ss}}}\Big{)},\]  
and the direction \(\mathsf{w}_{s}^{\text{\rm{ss}}}\) defined in Equation (7) may not provide a descent direction. Therefore, to address this issue, we adopt an approach similar to that proposed by [PERSON] [40] and [PERSON] and [PERSON] [18].