# 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
**} & \multicolumn{2}{c}{**CMA-ES**} & \multicolumn{2}{c}{**MA-ES**} & \multicolumn{2}{c}{**CMA-N**} & \multicolumn{2}{c}{**CMA-NMO**} & \multicolumn{2}{c}{**CMA-NMM-MO**} & \multicolumn{2}{c}{**CMA-NMM-MO**} & \multicolumn{2}{c}{**MA-ESN-MO**} \\ \hline \multirow{3}{*}{\(f_{1}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 29.52(\(\approx\)) & 30.00(\(\approx\)) & 27.60(\(-\)) & 28.72(\(\approx\)) & 28.60(\(\approx\)) & 29.20 \\  & _PR_ & 0.02 & 0.02 & 0.82 & 0.83 & 0.76 & 0.79 & 0.79 & 0.82 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{8}\)} & _ANP_ & 1.08(\(-\)) & 0.68(\(-\)) & 42.60(\(-\)) & 51.60(\(\approx\)) & 44.04(\(-\)) & 52.04(\(\approx\)) & 43.36(\(-\)) & 48.88 \\  & _PR_ & 0.01 & 0.01 & 0.52 & 0.63 & 0.54 & 0.64 & 0.53 & 0.60 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{18}\)} & _ANP_ & 1(\(-\)) & 0.96(\(-\)) & 31.40(\(-\)) & 39.68(\(\approx\)) & 28.92(\(-\)) & 35.88(\(\approx\)) & 30.48(\(-\)) & 36.44 \\  & _PR_ & 0.004 & 0.004 & 0.14 & 0.18 & 0.13 & 0.16 & 0.14 & 0.16 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{19}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 9.92(\(-\)) & 11.40(\(\approx\)) & 8.64(\(-\)) & 11.44(\(\approx\)) & 9.68(\(-\)) & 11.20 \\  & _PR_ & 0.08 & 0.08 & 0.82 & 0.95 & 0.72 & 0.95 & 0.80 & 0.92 \\  & _SR_ & 0\% & 0\% & 0\% & 56\% & 4\% & 60\% & 0\% & 48\% \\ \hline \multirow{3}{*}{\(f_{20}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 2.64(\(-\)) & 6.00(\(\approx\)) & 2.80(\(-\)) & 5.40(\(-\)) & 2.40(\(-\)) & 5.96 \\  & _PR_ & 0.12 & 0.12 & 0.33 & 0.75 & 0.35 & 0.67 & 0.31 & 0.75 \\  & _SR_ & 0\% & 0\% & 0\% & 16\% & 0\% & 0\% & 0\% & 4\% \\ \hline \multirow{3}{*}{\(f_{3}\)} & _ANP_ & 1(\(-\)) & 1(\(-\)) & 3.20(\(-\)) & 3.80(\(-\)) & 3.44(\(-\)) & 3.96(\(\approx\)) & 3.20(\(-\)) & 3.92 \\  & _PR_ & 0.16 & 0.16 & 0.54 & 0.63 & 0.57 & 0.66 & 0.53 & 0.65 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{14}\)} & _ANP_ & 1(\(-\)) & 0.72(\(-\)) & 1.76(\(-\)) & 3.52(\(=\)) & 1.72(\(-\)) & 3.64(\(\approx\)) & 1.84(\(-\)) & 3.48 \\  & _PR_ & 0.16 & 0.12 & 0.29 & 0.58 & 0.28 & 0.60 & 0.30 & 0.58 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{15}\)} & _ANP_ & 1(\(-\)) & 0.76(\(-\)) & 1.20(\(-\)) & 2.04(\(\approx\)) & 1.12(\(-\)) & 2.40(\(\approx\)) & 1.32(\(-\)) & 2.12 \\  & _PR_ & 0.12 & 0.09 & 0.15 & 0.25 & 0.14 & 0.30 & 0.16 & 0.26 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{16}\)} & _ANP_ & 1(\(-\)) & 0(\(-\)) & 1.24(\(-\)) & 1.96(\(\approx\)) & 1.16(\(-\)) & 2.24(\(\approx\)) & 1.08(\(-\)) & 1.84 \\  & _PR_ & 0.16 & 0 & 0.20 & 0.33 & 0.19 & 0.37 & 0.18 & 0.32 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{17}\)} & _ANP_ & 0.84(\(-\)) & 0(\(-\)) & 1(\(-\)) & 1.48(\(\approx\)) & 1(\(-\)) & 1.44(\(\approx\)) & 1.04(\(-\)) & 1.48 \\  & _PR_ & 0.10 & 0 & 0.12 & 0.18 & 0.12 & 0.18 & 0.13 & 0.19 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{18}\)} & _ANP_ & 0.16(\(-\)) & 0(\(-\)) & 1(\(\approx\)) & 1.44(\(+\)) & 1(\(\approx\)) & 1.32(\(+\)) & 1(\(\approx\)) & 1.08 \\  & _PR_ & 0.02 & 0 & 0.16 & 0.24 & 0.16 & 0.22 & 0.16 & 0.18 \\  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{19}\)} & _ANP_ & 0.04(\(-\)) & 0(\(-\)) & 1(\(\approx\)) & 1.04(\(+\)) & 1(\(\approx\)) & 0.80(\(\approx\)) & 1(\(\approx\)) & 0.96 \\  & _PR_ & 0.005 & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \cline{1-1}  & _SR_ & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% & 0\% \\ \hline \multirow{3}{*}{\(f_{2 ranking sequences for the test problems are the CMA-NMO, CMA-NMM-MO, MA-ESN-MO, CMA-N, CMA-NMM, MA-ESN, CMA-ES, and MA-ES. The experimental results show that the improved algorithms CMA-NMO, CMA-NMM-MO, and MA-ESN-MO perform better than other original algorithms. Therefore, it can be concluded that the improvement strategies are effective.