## 6 Experiments and Discussion  
In this section, we furnish several numerical examples to illustrate the effectiveness of Algorithm 1 and solve them by employing MATLAB R2024a. To solve problems (P1), and (P2) presented in Examples 1 and 2, respectively, we employ Algorithm 1, implemented on a system equipped with an Intel(R) Core(TM) i7-8700 [EMAIL_ADDRESS] GB processor with 8 GB of RAM. However, the above system fails to solve problem (P3) presented in Example 3, which is a large-scale IVMOP \((n\geq 100)\). Therefore, to solve problem (P3), we employ Algorithm 1, implemented on a high-performance computational system running Ubuntu, with the following specifications: Memory: 128.0 GiB, Processor: Intel(r) Xeon(r) Gold 5415+ (32 cores), and OS type: 64-bit. Now, to execute Algorithm 1, we employ the following steps to find \(u^{(s+1)}\) for a given \(u^{(s)}\):  
1. Choose the parameter \(\gamma\), such that \(\gamma\in(0,1)\). 2. To find the values of \(\alpha(u^{(s)})\) and \(d(u^{(s)})\), we solve the following optimization problem: \[(\text{P})_{u^{(s)}} \text{Minimize}\quad\varphi(\alpha,\mathfrak{o}):=\alpha,\] subject to \[D_{gH}\mathcal{H}_{k}(u^{(s)},\mathfrak{o})+\frac{1}{2}\Big{[} \|\mathfrak{v}\|^{2},\|\mathfrak{v}\|^{2}\Big{]}\leq_{LU}[\alpha,\alpha],\] \[k\in\mathscr{I}_{m},\] by employing a numerical method (for example, using optimvar functions in the tools of MATLAB software). 3. Compute the values of \(\beta_{s}\) and the direction \(\mathsf{w}^{(s)}\), which satisfy Equations (9) and (10), respectively. 4. Compute the step size \(\eta^{(s)}\) as per the Armijo-like line search, that is, the inequality in (22), equipped with the backtracking as discussed in Remark 7. Equivalently, \[\eta^{(s)}:=\max\biggl{\{}\frac{1}{2^{p}}:p\in\mathbb{N}\cup\{0\}\biggr{\}}, \text{ when }\] \[\mathcal{H}_{k}\Big{(}u^{(s)}+\tfrac{1}{2^{p}}\mathsf{w}^{(s)} \Big{)}\ominus_{gH}\mathcal{H}_{k}(u^{(s)})\leq_{LU}\big{(}\tfrac{\gamma}{2} \big{)}\odot D_{gH}\mathcal{H}_{k}(u^{(s)},\mathsf{w}^{(s)}),\text{ for all }k\in\mathscr{I}_{m}.\]
5. Finally, one can find the values of \(u^{(s+1)}\) from the given condition: \[u^{(s+1)}=u^{(s)}+\eta^{(s)}\mathsf{w}^{(s)}.\]  
In the following example, we consider a locally convex IVMOP to demonstrate the effectiveness of Algorithm 1. **Example 1**. : _Consider the following problem (P1) which belongs to the class of IVMOPs._  
\[(\text{P1})\quad\text{Minimize}\quad(\mathcal{H}_{1}(u_{1},u_{2}), \mathcal{H}_{2}(u_{1},u_{2})),\] \[\text{subject to}\quad(u_{1},u_{2})\in\mathbb{R}^{2},\]  
_where \(\mathcal{H}_{k}:\mathbb{R}^{2}\to\mathfrak{C}(\mathbb{R})\)\((k=1,2)\) are defined as follows:_  
\[\mathcal{H}_{1}(u_{1},u_{2}):=\biggl{[}\frac{1}{2}(u_{1}-1)^{2}+(u _{2}-1)^{2}-\frac{1}{2}(u_{1}-1)^{3}\;\vee\;(u_{1}-u_{2})^{2}+e^{u_{1}+u_{2}} \biggr{]},\] \[\mathcal{H}_{2}(u_{1},u_{2}):=\Bigl{[}(u_{1}-1)^{2}+(u_{2}-1)^{2} \;\vee\;(u_{1}+1)^{2}+(u_{2}+1)^{2}\Bigr{]}.\]_It is evident that \((1,1)\) is a critical point of the objective function of (P1). Since the components of the objective function in (P1) are locally convex at \((1,1)\), it follows from Lemma 3 that \((1,1)\) is a locally weak effective solution of (P1)._  
_Now, we employ Algorithm 1 to solve (P1), with an initial point \((-4,12)\). The stopping criterion is defined as \(|\alpha(\mathbf{u}^{(s)})|<\epsilon=10^{-4}\). The numerical results for Algorithm 1 are shown in \[D_{gH}\mathcal{H}_{k}(u^{(s)},\mathfrak{o})+\frac{1}{2}\Big{[} \|\mathfrak{v}\|^{2},\|\mathfrak{v}\|^{2}\Big{]}\leq_{LU}[\alpha,\alpha],\]0_  
From Step 3 of Table 1, we conclude that the sequence converges to a locally weak effective solution \((1,1)\) of (P1). It is worth noting that the locally weak effective solution of an IVMOP is not an isolated point. However, applying Algorithm 1 with a given initial point can lead to one such locally weak effective solution.