# Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization
## 4 The MA-ESN-MO Algorithm
### Matrix Adaptation Evolution Strategy with Multi-Objective Optimization Algorithm

Table 2: Experimental results of _ANP_ (average peaks found) obtained by the covariance matrix adaptation evolution strategy (CMA-ES), matrix adaptation evolution strategy (MA-ES), CMA-ES-Niching (CMA-N) CMA-ES-Niching with the multi-objective optimization algorithm (CMA-NMO), CMA-ES Niching with Mahalanobis Metric (CMA-NMM), CMA-NMM with multi-objective optimization algorithm (CMA-NMM-MO), MA-ES-Niching (MA-ESN), and matrix adaptation evolution strategy with multi-objective optimization algorithm (MA-ESN-MO) for functions\(f_{1}\)-\(f_{10}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Fun.** & **CMA-ES** & **MA-ES** & **MA-N** & **CMA-NMO** & **CMA-NMM** & **CMA-NMM-MO** & **MA-ESN** & **MA-ESN-MO** \\ \hline \multirow{3}{*}{\(f_{1}\)} & _ANP_ & 2(\%) & 1.96(\%) & 2(\%) & 2(\%) & 2(\%) & 2(\%) & 2(\%) & 2 \\  & _PR_ & 1 & 0.98 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  & _SR_ & 100\% & 96\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ \hline \multirow{3}{*}{\(f_{2}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 3.64(\(-\)) & 4.32(\(\%\)) & 5(\(+\)) & 4.28(\(\%\)) & 3.04(\(-\)) & 4.40 \\  & _PR_ & 0.2 & 0.2 & 0.72 & 0.86 & 1 & 8.56 & 0.68 & 0.88 \\  & _SR_ & 0\% & 0\% & 28\% & 40\% & 100\% & 40\% & 4\% & 44\% \\ \hline \multirow{3}{*}{\(f_{3}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 0.72(\(-\)) & 1(\(\times\)) & 1(\(\times\)) & 1(\(\times\)) & 0.72(\(-\)) & 1 \\  & _SR_ & 1 & 1 & 0.72 & 1 & 1 & 1 & 0.72 & 1 \\  & _SR_ & 100\% & 100\% & 72\% & 100\% & 100\% & 100\% & 72\% & 100\% \\ \hline \multirow{3}{*}{\(f_{4}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 2.60(\(-\)) & 3.96(\(\approx\)) & 2.56(\(-\)) & 4(\(\%\)) & 2.48(\(-\)) & 4 \\  & _PR_ & 0.25 & 0.25 & 0.65 & 0.99 & 0.64 & 1 & 0.62 & 1 \\  & _SR_ & 0\% & 0\% & 8\% & 96\% & 8\% & 100\% & 4\% & 100\% \\ \hline \multirow{3}{*}{\(f_{5}\)} & _ANP_ & 1.12(\(-\)) & 1.04(\(-\)) & 1.52(\(-\)) & 1.92(\(-\)) & 1.36(\(-\)) & 1.92(\(\approx\)) & 1.48(\(-\)) & 1.92 \\  & _PR_ & 0.56 & 0.52 & 0.76 & 0.96 & 0.68 & 0.96 & 0.74 & 0.96 \\  & _SR_ & 12\% & 4\% & 60\% & 92\% & 48\% & 92\% & 52\% & 92\% \\ \hline \multirow{3}{*}{\(f_{6}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 13.40(\(-\)) & 17.36(\(\approx\)) & 12.72(\(-\)) & 17.36(\(\approx\)) & 14.00(\(-\)) & 17.20 \\  & _PR_ & 0.05 & 0.05 & 0.74 & 0.96 & 0.70 & 0.96 & 0.77 & 0.95 \\  & _SR_ & 0\% & 0\% & 0\% & 52\% & 4\% & 60\% & 0\% & 44\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental results of ANP, Peak ratio (PR) and Success Rate (SR) obtained by the CMA-ES, MA-ES, CMA-N, CMA-NMO, CMA-NMM, CMA-NMM-MO, MA-ESN, and MA-ESN-MO on \(\varepsilon\) = 0.0001 for functions \(f_{1}\) to \(f_{10}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Fun. ** & \(\varepsilon\) & **CMA-ES** & **MA-ES** & **CMA-N** & **CMA-NMO** & **CMA-NMM** & **CMA-NMM-MO** & **MA-ESN-MO** \\ \hline \multirow{3}{*}{\(f_{11}\)} & 0.1 & 1 & 1.04 & 6.00



In order to test the statistical significance of the eight compared algorithms, the [PERSON]'s test at the 5% significance level, which is implemented by using KEEL software [42], is employed based on the _PR_ values. Table 6 summarizes the statistical test results. It can be seen from Table 6 that the MA-ESN-MO provides higher R+ values than R\(-\) values compared with the CMA-ES, MA-ES, CMA-N, CMA-NMM, and MA-ESN. Furthermore, the \(p\) values of the CMA-ES, MA-ES, CMA-N, CMA-NMO, and MA-ESN are less than 0.05, which means that the MA-ESN-MO is significantly better than these competitors. The \(p\) values of the CMA-NMO and CMA-NMM-MO are equal to one, which means that the performance of the MA-ESN-MO is not different from that of the CMA-NMO and CMA-NMM-MO. To further determine the ranking of the eight compared algorithms, the [PERSON]'s test, which is also implemented by using KEEL software, is conducted. As shown in Table 7, the overall

\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Fun.