## 4 HS-Type Conjugate Direction Method for IVMOPs
\]  
This implies that  
\[\mathcal{H}_{k}(u+tw)\ominus_{gH}\mathcal{H}_{k}(u)\leqq_{LU}(\gamma t)\odot D _{gH}\mathcal{H}_{k}(u,w),\quad\text{for all }t\in(0,\hat{t}_{k}). \tag{27}\]  
**Case 2:**  
\[ \begin{array}{l}\overline{\mathcal{H}}_{k}(u+tw)-\overline{\mathcal{H}}_{ k}(u)=t\underline{D}_{gH}\mathcal{H}_{k}(u,w)+t\underline{\varepsilon}_{k}(t), \\ \underline{\mathcal{H}}_{k}(u+tw)-\underline{\mathcal{H}}_{k}(u)=t\overline{D}_ {gH}\mathcal{H}_{k}(u,w)+t\overline{\varepsilon}_{k}(t).\end{array}  \tag{28}\]On the lines of the proof of Case 1, it can be shown that the inequality in (27) holds for all \(t\in(0,\hat{t}_{k})\) for some \(\hat{t}_{k}>0\). Since \(k\in\mathscr{I}_{m}\) was arbitrary, we conclude that for each \(k\in\mathscr{I}_{m}\), there exists \(\hat{t}_{k}>0\) such that (27) holds. Let us set \(\hat{t}:=\min\{\hat{t}_{1},\ldots,\hat{t}_{m}\}\), then we have  
\[\mathcal{H}_{k}(\mathsf{u}+t\mathsf{w})\ominus_{gH}\mathcal{H}_{k}(u)\leq_{LU }(\gamma t)\odot D_{gH}\mathcal{H}_{k}(\mathsf{u},\mathsf{w}),\quad\text{for all }k\in\mathscr{I}_{m},\text{ for all }t \in(0,\hat{t}).\]  
This completes the proof. **Remark 7**. : _From Lemma 7, it follows that for \(\mathsf{u}\in\mathbb{R}^{n}\), if \(D_{gH}\mathcal{H}_{k}(\mathsf{u},\mathsf{w})<_{LU}[0,0](k\in\mathscr{I}_{m})\), then there exists some \(\hat{t}>0\) such that (22) holds for all \(t\in(0,\hat{t})\). To compute the step length \(t\) numerically, we adopt the following backtracking process:_  
_We start with_ \(p=0\) _and check whether (_22_) holds for_ \(t=\frac{1}{2^{p}}\)_._  
1. _If the inequality in (_22_) is satisfied, we take_ \(t=\frac{1}{2^{p}}\) _as the step length._
2. _Otherwise, set_ \(p:=p+1\) _and update_ \(t=\frac{1}{2^{p}}\)_, repeating the process until the inequality in (_22_) is satisfied._  
_In view of the fact that, as some_ \(\hat{t}>0\) _exists such that (_22_) holds for all_ \(t\in(0,\hat{t})\)_, and the sequence_ \(\left(\frac{1}{2^{p}}\right)_{p\in\mathbb{N}\cup\{0\}}\) _converges to_ \(0\)_, the above process terminates after a finite number of iterations._  
_Thus, at any \(\mathsf{u}\in\mathbb{R}^{n}\) with \(D_{gH}\mathcal{H}_{k}(\mathsf{u},\mathsf{w})<_{LU}[0,0](k\in\mathscr{I}_{m})\), we can choose \(\eta\) as the largest \(t\) from the set_  
\[\bigg{\{}\frac{1}{2^{p}}\mid p\in\mathbb{N}\cup\{0\}\bigg{\}},\]  
_such that (22) is satisfied._  
Now, we present the HS-type conjugate direction algorithm for IVMOP. ```
1: Let \(\gamma\in(0,1)\), initial point \(\mathsf{u}^{(0)}\in\mathbb{R}^{n}\), \(\epsilon>0\), and set \(s=0\). 2: Solve the optimization problem (P)\({}_{\mathsf{u}^{(s)}}\) and obtain the values of \(\alpha(\mathsf{u}^{(s)})\) and \(d\Big{(}\mathsf{u}^{(s)}\Big{)}\). 3: If \(|\alpha(\mathsf{u}^{(s)})|<\epsilon\), then stop. Otherwise, proceed to the next step. 4: Calculate \(\mathsf{w}^{(s)}\) using (10). 5: Select \(\eta^{(s)}\) as the largest value of \(t\in\left\{\frac{1}{2^{p}}:p\in\mathbb{N}\cup\{0\}\right\}\) that satisfies (22). Update the iterate as follows: \[\mathsf{u}^{(s+1)}:=\mathsf{u}^{(s)}+\eta^{(s)}\mathsf{w}^{(s)}.\]
6: Set \(s:=s+1\), and go to Step 2. ```  
**Algorithm 1** HS-Type Conjugate Direction Algorithm for IVMOP  
**Remark 8**. : _It is worth noting that if \(\beta_{s}\) in (10) is set to zero and 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, \(\mathcal{H}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), then Algorithm 1 reduces to the steepest descent algorithm for MOPs, as proposed by [PERSON] and [PERSON] [5]._  
It is obvious that if Algorithm 1 has a finite iteration, then the last iterative point is an approximate critical point. Thus, it is relevant to consider the convergence analysis when Algorithm 1 generates an infinite sequence. That is, \(\alpha(\mathsf{u}^{(s)})\
eq 0\) for all \(s\in\mathbb{N}\cup\{0\}\).