# 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 5 shows that the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO perform better than other algorithms on\(f_{11}\)-\(f_{17}\). The CMA-NMO performs the best on\(f_{18}\). For function \(f_{19}\), the performance of all the algorithms is basically the same except for the CMA-ES and MA-ES. None of the algorithms can find multiple optimal solutions on\(f_{20}\). For \(f_{11}\)-\(f_{20}\), the MA-ESN-MO works better than the CMA-ES, MA-ES, CMA-N, CMA-NMO, CMA-NMM-MO, and MA-ESN on nine, nine, seven, zero, seven, one, and seven test problems, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Fun.** & \(\mathbf{\varepsilon}\) & **CMA-ES** & **MA-ES** & **CMA-N** & **CMA-NMO** & **CMA-NMM** & **CMA-NMM-MO** & **MA-ESN** & **MA-ESN-MO** \\ \hline \multirow{4}{*}{\(f_{1}\)} & 0.1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\  & 0.01 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\  & 0.001 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\  & 0.0001 & 2 & 1.96 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline \multirow{4}{*}{\(f_{2}\)} & 0.1 & 1 & 1 & 5 & 5 & 5 & 5 & 5 & 5 \\  & 0.01 & 1 & 1 & 5 & 5 & 5 & 5 & 4.92 & 5 \\  & 0.001 & 1 & 1 & 4.52 & 5 & 5 & 5 & 4.32 & 5 \\  & 0.00001 & 1 & 1 & 3.20 & 3.68 & 5 & 1.96 & 2.16 & 1.44 \\ \hline \multirow{4}{*}{\(f_{3}\)} & 0.1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  & 0.01 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  & 0.0001 & 1 & 1 & 1 & 1 & 1 & 1 & 0.96 & 1 \\  & 0.00001 & 1 & 1 & 0.60 & 0.28 & 1 & 0.48 & 0.56 & 0.60 \\ \hline \multirow{4}{*}{\(f_{4}\)} & 0.1 & 1 & 1 & 4 & 4 & 2.68 & 4 & 4 & 4 \\  & 0.01 & 1 & 1 & 4 & 4 & 2.68 & 4 & 4 & 4 \\  & 0.0001 & 1 & 1 & 2.64 & 4 & 2.68 & 4 & 2.56 & 4 \\  & 0.00001 & 1 & 1 & 2.56 & 3.96 & 2.56 & 3.80 & 2.36 & 3.96 \\ \hline \multirow{4}{*}{\(f_{5}\)} & 0.1 & 1.12 & 1.04 & 2 & 2 & 2 & 2 & 2 & 2 \\  & 0.01 & 1.12 & 1.04 & 1.80 & 2 & 1.64 & 2 & 1.80 & 2 \\  & 0.001 & 1.12 & 1.04 & 1.52 & 2 & 1.48 & 2 & 1.60 & 2 \\  & 0.00001 & 1.12 & 1.04 & 1.40 & 1.80 & 1.32 & 1.72 & 1.32 & 1.80 \\ \hline \multirow{4}{*}{\(f_{6}\)} & 0.1 & 1 & 1 & 13.56 & 17.56 & 12.72 & 17.52 & 14.12 & 17.40 \\  & 0.01 & 1 & 1 & 13.48 & 17.48 & 12.72 & 17.48 & 14.04 & 17.36 \\  & 0.001 & 1 & 1 & 13.44 & 17.40 & 12.72 & 17.36 & 14.00 & 17.36 \\  & 0.00001 & 1 & 0.96 & 13.40 & 17.20 & 12.72 & 17.24 & 14.00 & 17.04 \\ \hline \multirow{4}{*}{\(f_{7}\)} & 0.1 & 1 & 1 & 36.00 & 35.28 & 27.68 & 35.04 & 36.00 & 35.20 \\  & 0.01 & 1 & 1 & 29.68 & 30.44 & 27.68 & 29.00 & 28.84 & 30.00 \\  & 0.0001 & 1 & 1 & 29.40 & 30.28 & 27.68 & 28.96 & 28.84 & 29.64 \\  & 0.00001 & 1 & 1 & 29.52 & 29.92 & 27.16 & 28.32 & 28.56 & 29.12 \\ \hline \multirow{4}{*}{\(f_{8}\)} & 0.1 & 1.08 & 0.96 & 42.64 & 53.40 & 44.04 & 54.04 & 43.44 & 50.84 \\  & 0.01 & 1.08 & 0.92 & 42.64 & 52.96 & 44.04 & 53.48 & 43.40 & 50.20 \\  & 0.001 & 1.08 & 0.76 & 42.64 & 52.40 & 44.04 & 52.92 & 43.40 & 49.68 \\  & 0.00001 & 1.08 & 0.64 & 42.60 & 50.80 & 44.04 & 51.28 & 43.36 & 47.72 \\ \hline \multirow{4}{*}{\(f_{9}\)} & 0.1 & 1 & 1.04 & 216 & 89.20 & 28.92 & 73.32 & 216 & 81.48 \\  & 0.01 & 1 & 1.00 & 31.40 & 57.32 & 28.92 & 52.16 & 30.48 & 54.40 \\  & 0.001 & 1 & 1.00 & 31.40 & 46.92 & 28.92 & 41.60 & 30.48 & 43.72 \\  & 0.00001 & 1 & 0.96 & 31.40 & 33.56 & 28.92 & 31.00 & 30.48 & 31.00 \\ \hline \multirow{4}{*}{\(f_{10}\)} & 0.1 & 1 & 1 & 9.92 & 11.68 & 8.64 & 11.72 & 9.68 & 11.72 \\  & 0.01 & 1 & 1 & 9.92 & 11.60 & 8.64 & 11.72 & 9.68 & 11.56 \\ \cline{1-1}  & 0.001 & 1 & 1 & 9.92 & 11.40 & 8.64 & 11.64 & 9.68 & 11.28 \\ \cline{1-1}  & 0.00001 & 1 & 1 & 9.92 & 11.32 & 8.64 & 11.24 & 9.68 & 10.72 \\ \hline \hline \end{tabular}
\end{table}