## 4 HS-Type Conjugate Direction Method for IVMOPs
Now, we introduce an Armijo-like line search method for the objective function \(\mathcal{H}\) of IVMOP. Consider \(u,w\in\mathbb{R}^{n}\) such that \(w\) is a descent direction at \(u\) for the function \(\mathcal{H}\). Let \(\gamma\in(0,1)\). A step length \(t\) is acceptable if it satisfies the following condition:  
\[\beta_{s}:=\begin{cases}0,&\text{if }(\beta_{s}^{\text{\rm{ss}}}<0\text{ or undefined})\text{ or }s=0;\\ 0,&\text{if }\widetilde{\Phi}\Big{(}\mathsf{u}^{(s-1)},d\Big{(}\mathsf{u}^{(s-1)} \Big{)}\Big{)}>\widetilde{\Phi}\Big{(}\mathsf{u}^{(s)},d\Big{(}  
Therefore, the inequality in (11) is satisfied.  
**Case 2:** Let \(\beta_{s}>0\). Our aim is to prove that  
\[\beta_{s}\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s-1)}) \leq_{LU}[0,0],\quad\text{ for all }k\in\mathcal{I}_{m}.\]8  
**Remark 6**. : _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 (22) reduces to the following Armijo-like line search, defined by [PERSON] and [PERSON] [5]:_  
\[\beta_{s}:=\begin{cases}0,&\text{if }(\beta_{s}^{\text{\rm{ss}}}<0\text{ or undefined})\text{ or }s=0;\\ 0,&\text{if }\widetilde{\Phi}\Big{(}\mathsf{u}^{(s-1)},d\Big{(}\mathsf{u}^{(s-1)} \Big{)}\Big{)}>\widetilde{\Phi}\Big{(}\mathsf{u}^{(s)},d\Big{(}  
Therefore, the inequality in (11) is satisfied.  
**Case 2:** Let \(\beta_{s}>0\). Our aim is to prove that  
\[\beta_{s}\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s-1)}) \leq_{LU}[0,0],\quad\text{ for all }k\in\mathcal{I}_{m}.\]9  
_where \(\mathcal{H}(u)\) represents the Jacobian of \(\mathcal{H}\) at \(u\)._  
In the next lemma, we prove the existence of such \(t\) which satisfies (22) for a given \(\gamma\in(0,1)\). **Lemma 7**. : _If \(\mathcal{H}\) is gH-differentiable and \(D_{gH}\mathcal{H}_{k}(u,w)<_{LU}[0,0]\) for each \(k\in\mathcal{I}_{m}\), then for the given \(\gamma\in(0,1)\), there exists \(\hat{t}>0\) such that_  
\[D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s-1)})\leq_{LU}[0,0], \quad\text{ for all }k\in\mathcal{I}_{m}.\]0  
Proof. : Let \(k\in\mathcal{I}_{m}\) be fixed. By the definition of the directional derivative of \(\mathcal{H}_{k}\), there exists a function \(\tau_{k}:\mathbb{R}^{n}\to\mathfrak{C}(\mathbb{R})\) such that  
\[D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s-1)})\leq_{LU}[0,0], \quad\text{ for all }k\in\mathcal{I}_{m}.\]1  
where \(\tau_{k}(t):=[\underline{\varepsilon}_{k}(t),\overline{\varepsilon}_{k}(t)] \to[0,0]\) as \(t\to 0\). From (23) and Definition 2, these possible two cases may arise:  
**Case 1:**  
\[ \begin{array}{l}\underline{\mathcal{H}}_{k}(u+tw)-\underline{\mathcal{H}}_{ k}(u)=t\underline{D}_{gH}\mathcal{H}_{k}(u,w)+t\underline{\varepsilon}_{k}(t), \\ \overline{\mathcal{H}}_{k}(u+tw)-\overline{\mathcal{H}}_{k}(u)=t\overline{D}_{ gH}\mathcal{H}_{k}(u,w)+t\overline{\varepsilon}_{k}(t).\end{array}  \tag{24}\]  
Since \(D_{gH}\mathcal{H}_{k}(u,w)<_{LU}[0,0]\), therefore \(\underline{D}_{gH}\mathcal{H}_{k}(u,w)<0\) and \(\overline{D}_{gH}\mathcal{H}_{k}(u,w)<0\). Define  
\[\epsilon:=\max\Bigl{\{}-(1-\gamma)\underline{D}_{gH}\mathcal{H}_{k}(u,w),-(1- \gamma)\overline{D}_{gH}\mathcal{H}_{k}(u,w)\Bigr{\}}>0. \tag{25}\]  
Since \(\tau_{k}(t)\to[0,0]\) as \(t\to 0\), there exists \(\hat{t}_{k}>0\) such that  
\[\|\tau_{k}(t)\|_{I}\leq\epsilon,\quad\text{for all }t\in(0,\hat{t}_{k}). \tag{26}\]  
Substituting (25) and (26) in (24), we have  
\[ \begin{array}{l}\underline{\mathcal{H}}_{k}(u+tw)-\underline{\mathcal{H}}_ {k}(u)\leq t\underline{D}_{gH}\mathcal{H}_{k}(u,w)-t(1-\gamma)\underline{D}_ {gH}\mathcal{H}_{k}(u,w),\quad\text{for all }t\in(0,\hat{t}_{k})\\ \overline{\mathcal{H}}_{k}(u+tw)-\overline{\mathcal{H}}_{k}(u)\leq t \overline{D}_{gH}\mathcal{H}_{k}(u,w)-t(1-\gamma)\overline{D}_{gH}\mathcal{H} _{k}(u,w),\quad\text{for all }t\in(0,\hat{t}_{k}).\end{array}