# 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
#### 5.1.2 Success Rate

The success rate denotes the percentage of successfully detecting all the global optima out of NR runs for each function. #### 5.1.3 Average Number of Peaks Found

The average number of peaks (\(ANP\)) denotes the average number of peaks found by an algorithm over NR runs. \[ANP=\frac{\sum_{run=1}^{NR}NGF_{run}}{NR} \tag{10}\]

### Experimental Results of Eight Optimization Algorithms

The experimental results and analyses are shown in the following. The best solution of every single function in different algorithms is highlighted with boldface. The results of the test functions from \(f_{1}\) to \(f_{10}\) are shown in Table 2. As can be seen, the performance of the CMA-ES is the same as MA-ES on\(f_{2}\)-\(f_{5}\),\(f_{7}\), and \(f_{10}\). The CMA-ES performs slightly better than the MA-ES on\(f_{1}\),\(f_{6}\),\(f_{8}\), and \(f_{9}\). As mentioned earlier, the MA-ES is a simplified version of the CMA-ES; therefore, their performance is similar. The CMA-NMO performs better than the CMA-N on\(f_{2}\),\(f_{4}\),\(f_{5}\),\(f_{6}\),\(f_{8}\), and \(f_{10}\). The CMA-N beats the CMA-NMO on\(f_{7}\) and \(f_{9}\) when \(\varepsilon\) = 0.1. The CMA-NMM-MO performs better than the CMA-NMM on\(f_{4}\)-\(f_{10}\). The CMA-NMM outperforms the CMA-NMM-MO on\(f_{2}\) and \(f_{3}\). The MA-ESN-MO performs better than the MA-ESN on\(f_{2}\)-\(f_{6}\),\(f_{8}\), and \(f_{10}\). The MA-ESN beats the MA-ESN-MO on\(f_{7}\) and \(f_{9}\) when \(\varepsilon\) = 0.1. The experimental results of the test functions from \(f_{1}\)-\(f_{10}\) show that the performance of the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO has been greatly improved because of the introduction of the multi-objective optimization strategy, the dynamic peak identification strategy, and the archive. Table 3 shows the results of the test function from \(f_{11}\) to \(f_{20}\). As can be seen, the CMA-ES performs slightly better than the MA-ES. This suggests that the two evolution paths are more effective in solving complex problems. The CMA-NMO performs better than the CAM-N on \(f_{12}\), but worse on \(f_{20}\). For functions \(f_{11}\)-\(f_{19}\), the CMA-NMO performs better than the CMA-N on different \(\varepsilon\) except for \(\varepsilon\) = 0.1. For function \(f_{11}\)-\(f_{20}\), the CMA-NMM-MO performs better than the CMA-NMM except for function \(f_{19}\) and \(f_{20}\). The MA-ESN-MO performs better than the MA-ESN on\(f_{12}\) but worse on \(f_{19}\) and \(f_{20}\) than the MA-ESN. For functions \(f_{11}\),\(f_{13}\)-\(f_{18}\), the MA-ESN-MO performs better than the MA-ESN on different \(\varepsilon\) except \(\varepsilon\) = 0.1. The experimental results show that the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO can find better solutions than the CMA-ES, MA-ES, CMA-N, CMA-N, CMA-NMM, and MA-ESN. Tables 4 and 5 shows the results of ANP in terms of the mean value (Mean), Peak ratio (\(PR\)), and Success Rate (\(SR\)) obtained in 25 independent runs by each algorithm for functions \(f_{1}\) to \(f_{10}\) and \(f_{11}\) to \(f_{20}\), respectively. In view of statistics, the Wilcoxon signed-rank test [41] at the 5% significance level is used to compare the MA-ESN-MO with other compared algorithms. \"\(\approx\)\", \"+\", and \"-\" are applied to express the performance of the MA-ESN-MO as similar to, worse than, and better than that of the compared algorithm, respectively. The statistical results are reported in Tables 4 and 5. Table 4 shows that all the algorithms can accurately find multiple optimal solutions on \(f_{1}\) except for the MA-ES. The CMA-NMM performs better than the other algorithms on function \(f_{2}\). For function \(f_{3}\), all of the algorithms can accurately find optimal solutions except for CMA-N andMA-ESN. The CMA-NMO, CMA-NMM-MO, and MA-ESN-MO perform better than other algorithms on\(f_{4}\)-\(f_{10}\). For \(f_{1}\)-\(f_{10}\), the MA-ESN-MO works better than the CMA-ES, MA-ES, CMA-N, CMA-NMO, CMA-NMM, CMA-NMM-MO, and MA-ESN on eight, eight, eight, zero, seven, zero, and eight test problems, respectively.