# Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization
## 4 The MA-ESN-MO Algorithm
### Transforming an MMOP into an MOP  
The premise of employing multi-objective techniques is that different objectives should conflict with each other. Therefore, two strongly conflicting objectives are constructed as follows:  
\[\left\{\begin{array}{l}\boldsymbol{f_{1}}(\mathbf{x})=\boldsymbol{\alpha} \times f(\mathbf{x})_{norm}+\mathbf{x}\\ \boldsymbol{f_{2}}(\mathbf{x})=\boldsymbol{\alpha}\times f(\mathbf{x})_{norm} -\mathbf{x}\end{array}\right. \tag{7}\]  
where \(\alpha\) is a scaling factor that gradually increases during evolution, \(f(\mathbf{x})_{norm}\) denotes the normalized objective function values, and \(\mathbf{x}\) is the decision variable. An MMOP is transformed into \(D\) bi-objective optimization problems, where \(D\) is the dimension of an MMOP.  
In Equation (7), if the value of \(\mathbf{x}\) increases, the value of \(-\mathbf{x}\) will decrease, and vice versa. Moreover, \(\alpha\) and \(f(\mathbf{x})_{norm}\) are positive, which brings the same change for \(\mathbf{x}\) and \(-\mathbf{x}\). It can be concluded that \(f_{1}(\mathbf{x})\) conflicts with \(f_{2}(\mathbf{x})\). Therefore, the multi-objective optimization methods can be used. \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\) are mainly influenced by \(\mathbf{x}\) and \(-\mathbf{x}\) at the early stage of evolution. Population distribution can influence the diversity of the population. Gradually, \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\) are mainly influenced by the fitness at the later stage of evolution. Therefore, the algorithm can quickly converge to find the multiple optimal solutions. Similar to the mapping relationship between the decision space and objective space in a MOP, an example of the mapping relationship between an MMOP and an MOP is shown in Figure 1 (\(D=2\)). \(\alpha\) is designed as follows:  
\[\alpha=D\times(b-a)\times(\frac{FES}{MaxFES})^{D} \tag{8}\]  
where \(D\) denotes the dimension of the problem, [a, b]D denotes the range of decision space, and \(FES\) and \(MaxFES\) denote the number of function evaluations and the maximum number of function evaluations, respectively.  
Figure 1: Relationship between multimodal optimization in a decision space and multi-objective optimization in an objective space. (**a**) Three optimal solutions of a multimodal function in a decision space. (**b**) Three optimal solutions of a transformed multi-objective function in an objective space.  
Figure 2 shows a transformation example of the equal maxima function. The equal maxima function has five global optima. Figure 2a shows the distribution of the population (x) and their fitness (f(x)) when \(FES=1000\). Figure 2b shows the results of the transformation from multimodal optimization to bi-objective optimization. Figure 2c shows the result of the non-dominated sorting procedure, which is employed to find a set of representative Pareto optimal solutions. The representative Pareto optimal solutions will be used as the parents of the next generation. Figure 2d shows the distribution of the individuals and their fitness corresponding to the Pareto optimal solutions.