# 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 exploration efficiency is dependent on the distribution of individuals, namely the population diversity. The exploitation efficiency is associated with the fitness of individuals. Then, diversity and fitness can be used to indirectly measure the exploration and exploitation, respectively. Therefore, diversity and fitness can be employed to achieve a trade-off between exploration and exploitation [39]. In view of this idea, the diversity and the fitness are considered in Equation (7). In the proposed algorithm, the non-dominated sorting mechanism is used to find the non-dominated solutions that are helpful for exploring the solution space efficiently and exhaustively. The non-dominated solutions, which take into account fitness and diversity, can be used as the seeds of niching. The non-dominated sorting procedure comes from the non-dominated sorting genetic algorithm II (NSGA-II) [40], which is a classical multi-objective optimization algorithm. Details of the fast non-dominated sorting algorithm can be found in [40]. In MA-ES, the offspring is yielded by the parent that is mutated according to a normal distribution. Then the parent will be discarded, while the excellent offspring will be preserved as the parent for the next generation. However, sometimes, the offspring may achieve worse performance than their  
Figure 2: Transformation example of the equal maxima function. parents. Take \(f_{2}\) equal maxima (1D) as an example; \(f_{2}\) has five global optima. Figure 3 shows three independent runs of MA-ES on \(f_{2}\) (the equal maxima 1D function). It can be seen that the number of global optima is not stable during the evolution. Originally, the parents have found the global optima. However, the parents are not preserved. The offspring are only near the optimal solution. Therefore, the number of the global optima found by the algorithm is variable during the evolution. In order to solve the problem, the archive is employed to save the individuals that have better performance. Specifically speaking, if the offspring performs worse, the individuals in the archive will be used as the parents for the next generation. Based on the above explanation, the pseudo-code of the MA-ESN-MO algorithm is illustrated in Algorithm 2, from which we can see that the MA-ESN-MO generation cycle is performed within the while-end while-loop (line 3 to line 26). \(\textbf{C}^{(g)}\) denotes the number of global optima found at generation \(g\). At each generation, a number of \(\lambda\) offspring is generated according to Equation (1), where the parental stage \(\textbf{x}\) is mutated according to a normal distribution yielding offspring \(\textbf{y}^{g}\). For a given individual \(i\), \(f(\textbf{y}_{i})\) can be calculated from the original objective function of MMOP. Then, the population is sorted according to the individual fitness in line (6). Afterward, the two objective values of the transformed bi-objective optimization problem can be obtained according to Equations (7) and (8). Then, the seeds of niching (also referred to as the parent of the next generation) are generated by the non-dominated sorting procedure or dynamic peak identification algorithm according to the condition in line (8). Then, the path cumulation \(s\), matrix **M**, and the step size \(\sigma\) (also referred to as mutation strength) are updated according to Equations (2), (5) and (6), respectively. At each generation, the excellent individual will be preserved in the archive as the parent for the next generation (line 18, line 23). Therefore, if \(\textbf{C}^{(g)}\) is less than \(\textbf{C}^{(g-1)}\), this shows that the performance of offspring is worse than that of their parents. Then, the individuals in the archive will be used as the parents for the next generation (line 21). Figure 3: Number of global optima found by the matrix adaptation evolution strategy (MA-ES) over three independent runs on \(f_{2}\).