## 5 Numerical Simulations  
In this section, we validate the performance of the proposed algorithm by numerical simulations because of the following reasons. If the optimization problem is strictly convex, then the global convergence is guaranteed for one dimension (1 missing comparison). For dimension two, it may converge to a non-stationary point, though the problem is strictly convex [50]. [PERSON] constructed a family of functions of two variables that are strictly convex with up to three-times continuously differentiable for which the algorithm converges to a non-stationary point. Furthermore, another version of the algorithm (adaptive Nelder-Mead algorithm, [48]) among several variants of the Nelder-Mead algorithm (see for instance, [60; 61]), has been studied for high dimensional problems, but its global convergence is not well examined.  
In general, the convergence properties of the proposed algorithm lack a precise statement. However, the numerical simulations could help clarify the performance of the algorithm with respect to the positive interval constraints [37]. The simulation results can provide information on how well the proposed algorithm fills an incomplete matrix, and validate its performance by considering the number of missing comparisons at large.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Iter** & **Min \(\lambda_{max}\)** & **Procedure** & **Iter** & **Min \(\lambda_{max}\)** & **Procedure** \\ \hline
8 & 7.45024 & reflect & 31 & 7.40973 & contract inside \\
9 & 7.45024 & reflect & 32 & 7.40973 & contract inside \\
10 & 7.45024 & reflect & 33 & 7.40973 & contract outside \\
11 & 7.44758 & contract inside & 34 & 7.40898 & reflect \\
12 & 7.44479 & contract outside & 35 & 7.40884 & reflect \\
13 & 7.44479 & reflect & 36 & 7.40875 & contract inside \\
14 & 7.44479 & reflect & 37 & 7.40875 & reflect \\
15 & 7.43308 & contract inside & 38 & 7.40868 & reflect \\
16 & 7.43308 & reflect & 39 & 7.4075 & reflect \\
17 & 7.43011 & contract inside & 40 & 7.4075 & reflect \\
18 & 7.42561 & contract inside & 41 & 7.4075 & contract outside \\
19 & 7.42561 & contract inside & 42 & 7.4075 & reflect \\
20 & 7.41755 & reflect & 43 & 7.4075 & reflect \\
21 & 7.41505 & contract inside & 44 & 7.40673 & reflect \\
22 & 7.41505 & reflect & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._  
Figure 2: The values of \(x_{1},\ldots,x_{6}\), and the first 44 iterations of the algorithm with the function value for minimization (13): (**a**) evolution of the variables with respect to iterations; and (**b**) function value vs. iterationIn the introduction, we specified that only incomplete PCMs corresponding to connected graphs are taken into account. In fact, it is also worth noting that, if the undirected graph associated with the incomplete matrix is connected, then the parameterized eigenvalue function becomes strictly convex, and therefore the optimal solution will be unique [15]. We stress again that we only consider the case of connected undirected graphs for examining the results of our simulation.  
Here, it should be recalled that the _Nelder-Mead algorithm_ directly provides the values to the missing entries. Finally, the values fill in the gap and then give the reconstructed (complete) PCMs. The simulation results obtained by the algorithm are measured by [PERSON]'s inconsistency ratio (_CR_).