## 5 Main Results  
In this section, we establish the convergence of the sequence generated by Algorithm 1. Moreover, we deduce that the sequence exhibits linear order convergence under appropriate assumptions. Furthermore, we investigate the worst-case complexity of the sequence generated by Algorithm 1. In the following theorem, we establish the convergence of the sequence generated by Algorithm 1. **Theorem 3**. : _Let \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) be an infinite sequence generated by Algorithm 1. Suppose that the set_  
\[\mathcal{T}:=\{\mathfrak{u}\in\mathbb{R}^{n}:\mathcal{H}(\mathfrak{u}) \preceq_{m}\mathcal{H}(\mathfrak{u}^{(0)})\}, \tag{29}\]  
_is bounded. Under these assumptions, every accumulation point of the sequence \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) is a critical point of the objective function of IVMOP._  
Proof. : From (22), for all \(s\in\{0,1,2,\ldots\}\), we have  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]  
Using Remark 2, for all \(s\in\{0,1,2,\ldots\}\), we obtain  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{u }^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{ (s)},\mathfrak{w}^{(s)})\leq_{LU}[0,0],\quad\text{ for all }k\in\mathcal{I}_{m}. \tag{30}\]  
This implies that  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\leq_{LU}\mathcal{H}_{k}(\mathfrak{u}^{( s)}),\quad\text{ for all }s\in\{0,1,2,\ldots\}\text{ and }\text{ for all }k\in\mathcal{I}_{m}. \tag{31}\]  
From (31), the sequence \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) lies in \(\mathcal{T}\) which is a bounded subset of \(\mathbb{R}^{n}\). As a result, the sequence \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) is also bounded. Hence, it possesses at least one accumulation point, say \(\mathfrak{h}\). We claim that \(\mathfrak{h}\) is a critical point of the objective function of IVMOP. Indeed, as \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) is a bounded sequence in \(\mathbb{R}^{n}\) and for all \(k\in\mathcal{I}_{m}\), \(\mathcal{H}_{k}\) is gH-continuous on \(\mathbb{R}^{n}\); therefore, using (31), we conclude that, for all \(k\in\mathcal{I}_{m}\), the sequence \(\left(\mathcal{H}_{k}(\mathfrak{u}^{(s)})\right)_{s\in\mathbb{N}\cup\{0\}}\) is non-increasing and bounded. Consequently, from (30), it follows that  
\[\lim_{s\to\infty}\eta^{(s)}\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)}, \mathfrak{w}^{(s)})=[0,0],\quad\text{ }k\in\mathcal{I}_{m}. \tag{32}\]  
Since \(\eta^{(s)}\in(0,1]\) for all \(s\in\mathbb{N}\cup\{0\}\), therefore, the value \(\limsup_{s\to\infty}\eta^{(s)}\) exists. Therefore, the following two possible cases may arise:  
**Case 1:** Let \(\limsup_{s\to\infty}\eta^{(s)}>0\). Hence, employing (32) and taking into account the fact that \(\mathfrak{u}\) is an accumulated point of the sequence \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\), there exist subsequence \((\mathfrak{u}^{(s)})_{j\in\mathbb{N}\cup\{0\}}\) and \((\eta^{(s)})_{j\in\mathbb{N}\cup\{0\}}\) of \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) and \((\eta^{(s)})_{s\in\mathbb{N}\cup\{0\}}\), respectively, such that  
\[\lim_{j\to\infty}\mathfrak{u}^{(s)}=\mathfrak{\mathfrak{u}},\quad\lim_{j\to \infty}\eta^{(s)}=\limsup_{s\to\infty}\eta^{(s)}>0,\]  
and  
\[\lim_{j\to\infty}D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)} )=[0,0],\quad\text{ for all }k\in\mathcal{I}_{m}. \tag{33}\]Our aim is to show that \(\bar{u}\) is a critical point of \(\mathcal{H}\). On the contrary, assume that \(\bar{u}\) is a noncritical point of \(\mathcal{H}\).