# 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
The CMA-ES and MA-ES perform well in solving unimodal optimization problems because of their strong exploratory ability. In multimodal optimization problems, the parent is responsible for finding the area where the potential optimal solution existed, while the offspring are responsible for exploitation. However, sometimes the offspring perform worse than their parents. As a result, the offspring may fail to find the optimal solution. As the evolution continues, the area where the potential optimal solution existed may be abandoned. Therefore, it will be difficult for the population to find all the global optimal solutions. To alleviate this issue, the archive is introduced in the MA-ESN-MO to save better individuals. At each generation, the individuals in the archive will be used as the parents for the next generation. If the optimal individual from offspring performs better than its parent, the individual will be saved in the archive. Otherwise, the parent will be saved in the archive. Then, it ensures the population to find all the global optimal solutions. 3. The non-dominated solutions include all the multiple optima of an MMOP. However, they also include some inferior solutions. Then, the best niche may occupy the population's resources during the evolution. To address this issue, the dynamic peak identification strategy is used to avoid converging toward a single global optimum. However, the performance of the dynamic peak identification strategy is highly sensitive to the niching radius. In other words, the performance of the algorithm deteriorates with an inappropriate niching radius. Therefore, the transformation strategy and the dynamic peak identification strategy are dynamically employed in MA-ESN-MO to improve the performance of the algorithm. ## 5 Experiments and Discussions

To verify the effectiveness of the improvement strategies proposed in this paper, 20 test problems from CEC2013 [21] multimodal benchmarks are used. The algorithms for testing include CMA-ES [4], MA-ES [11], CMA-N [14], CMA-NMO, CMA-NMM [20], CMA-NMM-MO, MA-ESN [11; 15], and MA-ESN-MO. CMA-NMO, CMA-NMM-MO, and MA-ESN-MO are produced by introducing the improvement strategies into CMA-ES, CMA-NMM, and MA-ES, respectively. To obtain an unbiased comparison, all the experiments are run on a PC with an Intel Core i7-3770 3.40 GHz CPU and 4 GB memory. All of the experiments are run 25 times, and the codes are implemented in Matlab R2013a (MathWorks, Natick, MA, USA). ### Parameter Settings and Performance Criteria

Table 1 shows a brief description of testing problems. \(f_{1}\)-\(f_{10}\) are simple and low-dimensional multimodal problems, while \(f_{11}\)-\(f_{20}\) are composition multimodal problems composed of several basic problems with different characteristics. \(f_{1}\)-\(f_{5}\) have a small number of global optima, and \(f_{6}\)-\(f_{10}\) have a large number of global optima. For each algorithm, the subpopulation size is set to 10. Some functions from the CEC2013 are drawn in the following. The equal maxima (\(f_{2}\)) has five global optima. There are no local optima, as shown in Figure 4. The Himmelblau function (\(f_{4}\)) has four global optima, as shown in Figure 5. Figure 6 shows an example of the Shubert 2D function (\(f_{6}\)), where there are 18 global optima in nine pairs. Figure 7 shows the 2D version of Composition Function 2 (CF\({}_{2}\)). CF\({}_{2}\) (\(f_{12}\)) is constructed



In this experiment, three criteria [6] are employed to measure the performance of different multimodal optimization algorithms on each function. Figure 5: Himmelblau. Figure 6: Shubert 2D function. Figure 7: Composition Function 2. #### 5.1.1 Peak Ratio

The peak ratio (\(PR\)) is used to measure the average percentage of all known global optima found over \(NR\) independent runs:

\[PR=\frac{\sum_{run=1}^{NR}NGF_{run}}{NGO*NR} \tag{9}\]

where \(NGF_{run}\) is the number of global optima found in the \(run\)-th run, \(NGO\) is the number of true global optima, and \(NR\) is the number of runs (\(NR\) = 25).