# 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
Figure 8 shows the results of ANP obtained in 25 independent runs by each algorithm for functions \(f_{1}\)-\(f_{20}\) on \(\varepsilon=0.0001\). In order to show these clearly in Figure 8, the CMA-ES, MA-ES, CMA-N, CMA-NMO, CMA-NMM, CMA-NMM-MO, MA-ESN and MA-ESN-MO are abbreviated to CMA, MA, CN, CO, CMM, CMO, MAN, and MMO, respectively. Table 8 shows the success rate of all the algorithms in finding all the global optimal solutions from \(f_{1}\) to \(f_{10}\). It can be observed that the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO are able to achieve the success rate of 100% on \(f_{1}\)-\(f_{5}\). Moreover, the CMA-NMO generates a higher success rate than the CMA-N from \(f_{6}\)-\(f_{10}\). Similarly, the CMA-NMM-MO and MA-ESN-MO generate higher success rates than the CMA-NMM and MA-ESN from \(f_{6}\)-\(f_{10}\), respectively. The CMA-ES and MA-ES obtain the success rate of 100% on \(f_{1}\) and \(f_{3}\). However, the CMA-ES and MA-ES generate relatively low success rates on other eight test functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**VS** & **R\({}^{+}\)** & **R\({}^{-}\)** & **Exact _p_-Value** & **Asymptotic _p_-Value** \\ \hline CMA-ES & 188.5 & 1.5 & \(\geq\)0.2 & 0.000144 \\ \hline MA-ES & 208.5 & 1.5 & \(\geq\)0.2 & 0.000103 \\ \hline CMA-N & 182.0 & 8.0 & \(\geq\)0.2 & 0.00043 \\ \hline CMA-NMO & 52. Figure 8: Cont. Figure 8: Box-plot of peaks found by the CMA-ES, MA-ES, CMA-N, CMA-NMO, CMA-NMM, CMA-NMM-MO, MA-ESN, and MA-ESN-MO on 20 test problems. Figure 9 displays the average number of peaks in terms of the mean value achieved by each of eight algorithms on \(\varepsilon\) = 0.0001 for CEC2013 multimodal problems versus the number of _FES_. It can be seen that the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO perform better than other algorithms, which suggests that the mechanism of multi-objective optimization can help the algorithm find multiple global optimal solutions. In addition, the curves of ANP achieved by CMA-ES, MA-ES, CMA-N, CMA-NMM, and MA-ESN are ups and downs instead of steady growth. The reason is that the better parents are discarded after producing the offspring. However, sometimes, the performance of the parent is better than that of the offspring. The curves of the CMA-NMO, CMA-NMM-MO, and MA-ESN-MO show a gradual upward trend. The reason is that the archive is introduced in these algorithms, which is helpful to ensure the convergence of the algorithm. Figure 9: _Cont._

## 6 Conclusions

The CMA-ES has received considerable attention as an efficient optimization algorithm. The MA-ES might be more attractive because of a simpler operator compared to the CMA-ES. Although niching techniques have been introduced into the CMA-ES, the performance of the CMA-ES is unsatisfactory for solving multimodal optimization problems. This paper proposed a matrix adaptation evolution strategy with the multi-objective optimization algorithm to solve multimodal optimization problems. The strategy of the multi-objective optimization is used to ensure the diversity of population. The archive is employed to maintain the peak found by the algorithm until the end of the run. The population is divided into multiple subpopulations, which are used to explore and exploit in parallel to find multiple optimal solutions. The experimental results suggest that the proposed strategies can achieve a better performance than the original algorithm on CEC2013 test problems. This research is partly supported by the Doctoral Foundation of Xi'an University of Technology (112-451116017). **Acknowledgments:** Thanks to [PERSON] for providing the source code of CMA-N and CMA-NMM. The author declares no conflict of interest. ## References

* (1) [PERSON]; [PERSON] Comparison of multimodal optimization algorithms based on evolutionary algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference, Seattle, WA, USA, 8-12 July 2006; pp. 1305-1312. * (2) [PERSON] Efficient differential evolution using speciation for multimodal function optimization.