## 4 Methodology
Using a different lens, Figure 4b, we encountered some inferences about \(\varpi\) and \(s\) when grouping the results by problem dimensionality. In general, it is worth noting the presence of two modes for the cardinality, near 30 and 70, when the two-dimensional problems are considered. Increasing dimensionality seems to slightly shift these modes to 20 and 60 when search domains reach 50 dimensions. This is an unexpected behaviour that can only be attributed to the capabilities of the proposed approach for dealing with high-dimensional problems. Plus, some particular facts to mention are as follows: for 5D domains, there are no low cardinalities; for 10D and 20D, uMHs of about 40 search operators are less frequent; and for 30D, the \(\varpi\) distribution is weighted to its extrema. Lastly, by looking at the steps required to design these metaheuristics, we can remark an evident transition from \(s\) values loaded about the first half of the studied range (i.e., from 1 to 100) for the 2D problems to a uniform-like distribution for those with 50D. In addition, it is somewhat understandable that for lower-dimensional domains, the hyper-heuristic implementation of Simulated \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**DU**} & \multirow{2}{*}{**Dim.**} & \multicolumn{3}{c}{**Experiment 1**} & \multicolumn{3}{c}{**Experiment 2**} \\  & & & **Perf.** & \(\varpi\) & \(s\) & **Perf.** & \(\varpi\) & \(s\) \\ \hline \multirow{4}{*}{Rastrigin} & \multirow{4}{*}{10} & 2 & 0.001 & 82 & 40 & \(\mathbf{3.9\times 10^{-9}}\) & 378 & 72 \\  & & 5 & 3.413 & 96 & 118 & \(\mathbf{1.405}\) & 498 & 190 \\  & & 10 & 40.683 & 100 & 142 & \(\mathbf{14.999}\) & 500 & 180 \\  & & 20 & 105.712 & 92 & 74 & \(\mathbf{69.849}\) & 500 & 360 \\  & & 30 & 196.954 & 89 & 192 & \(\mathbf{117.694}\) & 266 & 380 \\  & & 40 & 341.164 & 100 & 175 & \(\mathbf{194.557}\) & 465 & 345 \\  & & 50 & 463.786 & 94 & 162 & \(\mathbf{358.056}\) & 490 & 404 \\ \hline \multirow{4}{*}{Type I} & \multirow{4}{*}{01} & 2 & 1.3\(\times 10^{-4}\) & 68 & 24 & \(\mathbf{1.1\times 10^{-5}}\) & 311 & 127 \\  & & 5 & 0.008 & 100 & 45 & \(\mathbf{6.1\times 10^{-4}}\) & 500 & 203 \\  & & 10 & 0.046 & 89 & 164 & \(\mathbf{3.6\times 10^{-3}}\) & 499 & 145 \\  & & 20 & \(\mathbf{0.053}\) & 100 & 187 & 0.074 & 473 & 15 \\  & & 30 & 0.155 & 99 & 15 & \(\mathbf{0.087}\) & 499 & 491 \\  & & 40 & 0.155 & 100 & 170 & \(\mathbf{0.109}\) & 497 & 464 \\  & & 50 & 0.214 & 87 & 20 & \(\mathbf{0.094}\) & 497 & 392 \\ \hline \multirow{4}{*}{Schaffer N3} & \multirow{4}{*}{00} & 2 & 0.004 & 100 & 108 & \(\mathbf{1.6\times 10^{-3}}\) & 377 & 44 \\  & & 5 & 0.657 & 97 & 78 & \(\mathbf{0.236}\) & 483 & 247 \\ \cline{1-1}  & & 10 & 3.005 & 73 & 110 & \(\mathbf{2.305}\) & 496 & 318 \\ \cline{1-1}  & & 20 & 6.989 & 87 & 190 & \(\mathbf{6.578}\) & 471 & 455 \\ \cline{1-1}  & & 30 & 12.253 & 97 & 43 & \(\mathbf{11.029}\) & 495 & 310 \\ \cline{1-1}  & & 40 & 15.862 & 94 & 170 & \(\mathbf{15.697}\) & 500 & 202 \\ \cline{1-1}  & & 50 & 20.992 & 77 & 60 & \(\mathbf{20.093}\) & 497 & 158 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Annealing required 'few' steps for finding an optimal heuristic sequence. In such cases, we know the problem hardness is mainly related to its inherent characteristics, and the influence of the dimensionality _curse_ is not so relevant. The following analysis is intended to attain more information about this discussion. Figure 5 extends the results from Figure 4 by combining the dimensionalities and categories for distributions of both metaheuristic and hyper-heuristic iterations (i.e., cardinality and steps, respectively). Figure 5a shows that uMHs with more than 25 search operators are more prevalent when solving problems with extreme combinations of DU, i.e., 11 and 00. Although, there is an exception for each category. For the differentiable and unimodal problems, it appears when their dimensionality equals two, which is somewhat expected due to their inherent'simplicity'.