## 5 Main Results
Therefore, from (45), we have  
\[\frac{a}{2}\|\mathfrak{u}^{(s+1)}-\bar{\mathfrak{u}}\|^{2}\leq 2\mathcal{L} \|\mathfrak{u}^{(s)}-\bar{\mathfrak{u}}\|^{2}+\
abla\zeta_{k}(\bar{ \mathfrak{u}})^{T}\Big{(}\mathfrak{u}^{(s)}-\mathfrak{u}^{(s+1)}\Big{)},\quad \text{for all}\;k\in\mathscr{I}_{m}. \tag{46}\]  
Since \(\bar{\mathfrak{u}}\) is a critical point, therefore, there exists some \(k\in\mathscr{I}_{m}\) such that the following inequality holds:  
\[\
abla\zeta_{k}(\bar{\mathfrak{u}})^{T}\Big{(}\mathfrak{u}^{(s)}-\mathfrak{u }^{(s+1)}\Big{)}\leq 0.\]  
Hence, from (46), we get  
\[\|\mathfrak{u}^{(s+1)}-\bar{\mathfrak{u}}\|\leq\rho\|\mathfrak{u}^{(s)}-\bar{ \mathfrak{u}}\|,\]  
where \(\rho:=\sqrt{\frac{4\mathcal{L}}{a}}\). From the hypothesis, it follows that \(\rho<1\). Hence, the sequence \((\mathfrak{u}^{(s)})_{s\in\mathbb{N}\cup\{0\}}\) converges linearly to the critical point of the objective function of IVMOP. This completes the proof. **Remark 9**. : _Let \(\mathfrak{u},\mathfrak{w}\in\mathbb{R}^{n}\) such that \(\mathfrak{w}\) is a descent direction at \(\mathfrak{u}\) for the function \(\mathcal{H}\). Let \(\gamma\in(0,1)\). Then, in an Armijo-like line search strategy, a step length \(t\) is considered acceptable if it satisfies the following condition:_  
\[\mathcal{H}_{k}(\mathfrak{u}+\mathfrak{tw})\ominus_{gH}\mathcal{H}_{k}( \mathfrak{u})\leq_{LU}(\gamma t)\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}, \mathfrak{w}),\quad\text{for all}\;k\in\mathscr{I}_{m}.\]  
_This implies that the function \(\zeta_{k}:\mathbb{R}^{n}\to\mathbb{R}\) satisfies the following Armijo-like line search strategy:_  
\[\zeta_{k}(\mathfrak{u}+\mathfrak{tw})-\zeta_{k}(\mathfrak{u})\leq(\gamma t) \
abla\zeta_{k}(\mathfrak{u})^{T}\mathfrak{w},\quad\text{for all}\;k\in \mathscr{I}_{m}. \tag{47}\]  
The following lemma will play a crucial role to investigate the worst-case complexity of the sequence generated by Algorithm 1. **Lemma 9**. : _Let us assume that the gH-gradient of the functions \(\mathcal{H}_{k}\), that is, \(\
abla_{gH}\mathcal{H}_{k}\)\((k\in\mathscr{I}_{m})\) are gH-Lipschitz continuous with Lipschitz constant \(\mathcal{L}\). Moreover, if there exists some \(c>0\) such that the following inequality holds:_  
\[\|\mathfrak{w}^{(s)}\|\leq c\|d(\mathfrak{u}^{(s)}\|,\]  
_then the step size \(\eta^{(s)}\) in Algorithm 1 always satisfies the following inequality:_  
\[\eta^{(s)}>\frac{1-\gamma}{4\mathcal{L}\mathcal{L}^{2}},\]  
_for every \(s\in\mathbb{N}\cup\{0\}\)._Proof. : Let \(s\in\mathbb{N}\cup\{0\}\) be fixed. Since \(\eta^{(s)}\) is a step size in Algorithm 1, therefore, in view of Remark 9, there exists some \(k\in\mathscr{I}_{m}\), such that:  
\[2\gamma\eta^{(s)}\
abla\zeta_{k}(\mathfrak{u}^{(s)})^{T}\mathsf{w}^{(s)}<\zeta_{ k}\Big{(}\mathfrak{u}^{(s)}+2\eta^{(s)}\mathsf{w}^{(s)}\Big{)}-\zeta_{k}( \mathfrak{u}^{(s)}). \tag{48}\]  
Now,  
\[\zeta_{k}\Big{(}\mathfrak{u}^{(s)}+2\eta^{(s)}\mathsf{w}^{(s)} \Big{)}-\zeta_{k}(\mathfrak{u}^{(s)})\] \[=\int\limits_{0}^{2\eta^{(s)}}\frac{d}{dt}\zeta_{k}\Big{(} \mathfrak{u}^{(s)}+\mathsf{tw}^{(s)}\Big{)}dt\] \[\
abla\zeta_{k}(\bar{\mathfrak{u}})^{T}\Big{(}\mathfrak{u}^{(s)}-\mathfrak{u }^{(s+1)}\Big{)}\leq 0.\]0 \[\
abla\zeta_{k}(\bar{\mathfrak{u}})^{T}\Big{(}\mathfrak{u}^{(s)}-\mathfrak{u }^{(s+1)}\Big{)}\leq 0.\]1  
Since the functions \(\
abla_{gH}\mathcal{H}_{k}\) (\(k\in\mathscr{I}_{m}\)) are gH-Lipschitz continuous with Lipschitz constant \(\mathcal{L}>0\), it follows from Lemma 8 that the functions \(\
abla\zeta_{k}\) (\(k\in\mathscr{I}_{m}\)) are Lipschitz continuous with Lipschitz constant \(2\mathcal{L}\).