# Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization
## 1 Introduction  
Many problems from the real world are classified as optimization problems. Some optimization problems that have one global solution are called single modal optimization problems, while others that have multiple global and local optima are known as multimodal optimization problems. Traditional evolutionary algorithms (EAs) are effective at converging to a single global optimum because of the global selection strategy used. However, it is inappropriate for EAs to solve multimodal optimization problems. In order to overcome the weakness, niching techniques are incorporated into EAs, such as differential evolution [1; 2], particle swarm optimization [3], the covariance matrix adaptation evolution strategy (CMA-ES) [4], self-adaptive niching CMA-ES [4], and genetic algorithm [5], to solve multimodal optimization problems. The representative niching strategies include crowing [6], restricted tournament selection [7], fitness sharing [8], clearing [9], and speciation [5]. The covariance matrix adaptation evolution strategy (CMA-ES), which was proposed by [PERSON] and [PERSON] [10], is one of the popular optimization algorithms for solving unconstrainedreal-parameter optimization. Different from other optimization algorithms, CMA-ES makes use of two evolution paths to realize exploitation and exploration during the search process. The two evolution paths are the learning of the mutation strength and the rank-1 update of the covariance matrix, respectively. The self-adaptively updated covariance matrix, which uses evolution path information, can be considered a time series prediction of the evolution of the parent [11]. Since CMA-ES employs the covariance matrix to exploit the information from the previous and current generations, it has attracted broad investigation in recent years. However, compared with other classical algorithms, such as DE or PSO, CMA-ES is slightly more complicated, because it has two evolution paths and an update of the covariance matrix. To simplify the standard CMA-ES, [PERSON] and [PERSON] proposed the matrix adaptation evolution strategy (MA-ES) [11], in which one of the evolution paths (namely the p-evolution path) is dropped, and the covariance matrix (namely the C matrix) is discarded. The experimental results in [11] show that the MA-ES exhibits a similar performance as the CMA-ES, which considers both standard population sizes (\(\lambda<N\)) and large population sizes (\(\lambda=\text{O}\left(N^{2}\right)\)). The MA-ES only simplifies the CMA-ES. The performance of the MA-ES is basically the same as that of the CMA-ES. Therefore, the MA-ES is a robust local search strategy that efficiently solves unimodal optimization problems. It is unable to find multiple solutions in multimodal problems because of the designed parameters and updating rules [12]. At present, to the best of our knowledge, no work has been reported on utilizing the MA-ES to solve the multimodal problems. There has been an effort to provide two versions of CMA-ES, which are called the niching covariance matrix adaptation evolution strategy and the self-adaptive niching CMA-ES, respectively, for solving multimodal problems [13; 14]. The two improved versions of the CMA-ES introduced niching strategies that can maintain the population diversity and realize parallel convergence within some subpopulations to obtain multiple good solutions. However, the performance of the two improved versions of CMA-ES is highly sensitive to niching parameters. So far, some works have been done to convert a multimodal optimization problem (MMOP) to a multi-objective optimization problem (MOP) [15; 16; 17; 18; 19]. The advantage of transforming an MMOP into an MOP is that it is unnecessary to use the problem-dependent niching parameters. However, the prerequisite for MOP is objective confliction, which makes it difficult to transform an MMOP into a multi-objective optimization problem [17].