## 1 Introduction  
Multimodal multi-objective optimization problems (MMOPs) have become a research hot spot in recent years. As we know, there are at least two conflicting objectives in multi-objective optimization problems (MOPs) [1; 2; 3]. However, for actual problems, some of them have multiple equivalent Pareto optimal solutions in the decision space that are corresponding to the same objective value. These special MOPs are termed as MMOPs [4]. Clearly, how to maintain the population diversity in the decision space and find the high-quality Pareto front (PF) approximation in the objective space are two challenging tasks.  
In order to effectively solve MMOPs, the niching methods [5; 6; 7; 8], the diversity maintenance technique [9], and multiple population strategies [10; 11] have been incorporated into multi-objective evolutionary algorithms (MOEAs), such as the decision space-based niching NSGAII (DN-NSGAII) [4] and the multi-objective particle swarm optimization (PSO) based on the self-organizing mechanism (SMPSO-MM) [12]. Besides the above \"soft isolation\" methods, the \"hard isolation\" methods (i.e., Zoning search (ZS)) [13] is employed to reduce the solution complexity of MMOPs. Subsequently, the ZS with an adaptive resource-allocating method is proposed by [PERSON] et al. [14], which can further improve the solution efficiency of the ZS. From the above studies, it is clear that the previous studies focused on how to preserve the equivalent Pareto optimal solutions. Additionally, search engineering is also important for solving MMOPs as it can greatly affect the population diversity throughout the evolution. Like other meta-heuristics algorithms, brain storm optimization (BSO) [15; 16] has good global exploration ability and is suitable for solving different types of MOPs. Moreover, existing studies [17; 18; 19; 20; 21] show that the BSO is a competitive method to solve multimodal optimization problems. Therefore, the BSO may be an effective method to solve MMOPs.  
In the present study, a zoning search-based multimodal multi-objective brain storm optimization (ZS-MMBSO) is proposed. In the ZS-MMBSO, the ZS [13] is used to divide the decision space, which can reduce the search complexity and improve the population diversity. In addition, a novel individual generation strategy is proposed which can enhance the exploration capability in the early stage and the exploitation ability in the later stage. Note that a \"soft isolation\" method (i.e., K-means) is used in the BSO. Therefore, the proposed algorithm uses both \"soft isolation\" and \"hard isolation\" to locate and preserve the equivalent solutions. In this paper, five famous multimodal multi-objective evolutionary algorithms (MMOEAs) are chosen to compare with the ZS-MMBSO on 22 MMOPs introduced in IEEE CEC2019 [22]. The experimental results indicate that compared with other well-known MMOEAs, the ZS-MMBSO is competitive in solving MMOPs.  
The remaining parts of this article are as follows: The basic concepts of the MOPs, the performance metric, and the BSO are introduced in Section 2. In Section 3, the relevant work in the field of MMO is reviewed. Section 4 shows the details of the ZS-MMBSO. And in Section 5, the experimental results and analyses are reported. Section 6 summarizes the conclusions.