# The Nelder-Mead Simplex Algorithm Is Sixty Years Old: New Convergence Results and Open Questions
## 7 Computational Results and Conclusions  
Here, we consider the cardinality of set \(\mathbb{T}_{q}^{\ell}\) for both algorithms with and without shrinking operations. Define the quantities \(\widetilde{r}(n,\ell)=\left|\widetilde{\mathcal{T}}_{q}^{\ell}\right|/\left| \widetilde{\mathcal{T}}^{\ell}\right|\) and \(r(n,\ell)=\left|\mathcal{T}_{q}^{\ell}\right|/\left|\mathcal{T}^{\ell}\right|\). The larger the ratio \(\widetilde{r}(n,\ell)\) or \(r(n,\ell)\), the wider the convergence set. Table 1 and 2 contain the computed values \(\widetilde{r}(n,\ell)\) or \(r(n,\ell)\), respectively. Note that \(\widetilde{r}(n,\ell)\geq 1/5^{\ell}\) and \(r(n,\ell)\geq\left[\frac{(n+1)!}{3n+3+(n+1)!}\right]^{\ell}\). The ratio \(r(n,\ell)\) cannot achieve \(1\) (see the examples within Sections 1 and 6). For increasing \(n\), the \((n+1)!\) shrinking matrices somehow distort the real situation, since it is unlikely that all possible shrinking steps occur for one particular function. It is clear that Algorithm 2 has a wider convergence set than Algorithm 1. If we discard the shrinking operations (the case of strictly convex functions), then Theorem 5 also holds for the pair of sets  
\[\widetilde{\mathcal{T}}_{w}^{\ell}=\Big{\{}T_{i_{1}}\cdots T_{i_{\ell}}:T_{i_{ j}}\in\widetilde{\mathcal{T}}\setminus\mathcal{M}_{1},\;j=1,\ldots,n+1\Big{\}},\]  
and  
\[\mathcal{T}_{w}^{\ell}=\Big{\{}T_{i_{1}}\cdots T_{i_{\ell}}:T_{i_{j}}\in \mathcal{T}\setminus\mathcal{M}_{2},\;j=1,2,\ldots,\ell\Big{\}},\]  
instead of \(\mathbb{T}^{\ell}\) and \(\mathbb{T}_{q}^{\ell}\), respectively. Define again the ratios \(\widetilde{r}_{w}(n,\ell)=\left|\widetilde{\mathcal{T}}_{w}^{\ell}(q)\right| /\left|\widetilde{\mathcal{T}}_{w}^{\ell}\right|\) and \(r_{w}(n,\ell)=\left|\mathcal{T}_{w}^{\ell}(q)\right|/\left|\mathcal{T}_{w}^{ \ell}\right|\). Note that for \(n\leq 10\) and some \(0<q<1\), \(r_{w}(n,\ell)\geq 4^{\ell}/\left(3n+3\right)^{\ell}\). Table 3 and 4 contain the computed values of \(\widetilde{r}_{w}(n,\ell)\) and \(r_{w}(n,\ell)\), respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{5}{c}{\(\widetilde{\mathbf{r}}(\mathbf{n},\ell)\)} \\ \hline \(\mathbf{n}\setminus\ell\) & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
**2** & 0.39556 & 0.52563 & 0.59953 & 0.65873 & 0.70331 & 0.7403 \\
**3** & 0.28 & 0.3755 & 0.45766 & 0.52766 & 0.58397 & 0.63187 \\
**4** & 0.28 & 0.34304 & 0.40802 & 0.46877 & 0.52489 & 0.57442 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ratios for Algorithm 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{5}{c}{\(\widetilde{\mathbf{r}}(\mathbf{n},\ell)\)} \\ \hline \(\mathbf{n}\setminus\ell\) & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
**2** & 0.71111 & 0.83615 & 0.90204 & 0.94099 & 0.9641 & 0.97795 \\
**3** & 0.85185 & 0.93746 & 0.97389 & 0.98912 & 0.9956 & 0.99824 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ratios for Algorithm 2. The comparison of the last two tables shows again that Algorithm 1 has a smaller convergence set and a slower convergence rate than Algorithm 2. We can observe that the larger the \(\ell\), the larger is the ratio in all four tables. Although the ratios cannot achieve 1, it is conjectured that they might be quite close to 1 for a large \(\ell\). For both algorithms, we can observe the increase in convergence sets, if \(\ell\) is increasing. For larger \(n\) values, this increase is slower, especially if no shrinking is allowed. However, this requires an enormous computational time to check, even for modest pairs of \(n\) and \(\ell\). The reason for this behavior is unknown and yet to be investigated. However, we can establish the following conclusions. In case of the convergence of the simplex sequence, Algorithm 2 is better than Algorithm 1. Hence, the ordering of [PERSON] et al. [12] significantly improved the classical Nelder-Mead simplex method. Theorem 5 indicates that the generated simplex sequence has a strong tendency to converge to one common limit point.