Zoning Search With Adaptive Resource Allocating Method for Balanced and Imbalanced Multimodal Multi-Objective Optimization Qinqin Fan and Okan K. Ersoy, Fellow, IEEE Abstract—Maintaining population diversity is an important task in the multimodal multi-objective optimization. Although the zoning search (ZS) can improve the diversity in the decision space, assigning the same computational costs to each search subspace may be wasteful when computational resources are limited, especially on imbalanced problems. To alleviate the above-mentioned issue, a zoning search with adaptive resource allocating (ZS-ARA) method is proposed in the current study. In the proposed ZS-ARA, the entire search space is divided into many subspaces to preserve the diversity in the decision space and to reduce the problem complexity. Moreover, the computational resources can be automatically allocated among all the subspaces. The ZS-ARA is compared with seven algorithms on two different types of multimodal multi-objective problems (MMOPs), namely, balanced and imbalanced MMOPs. The results indicate that, similarly to the ZS, the ZS-ARA achieves high performance with the balanced MMOPs. Also, it can greatly assist a “regular” algorithm in improving its performance on the imbalanced MMOPs, and is capable of allocating the limited computational resources dynamically. Index Terms—Computational resource allocation, decision space decomposition, evolutionary computation, multimodal multi-objective optimization.    I. Introduction M ULTIMODAL multi-objective optimization problems (MMOPs) are commonly observed in the multi- objective optimization (MO) [1], [2]. Unlike general multi- objective optimization and multimodal optimization [3]–[9], the objectives of the multimodal multi-objective optimization (MMO) must not only achieve a good approximation in the objective space, but also locate enough equivalent Pareto opti- mal solutions in the decision space [10]. Therefore, maintain- ing the diversity in these two spaces is vital in the MMO [11], [12]. Until now, various multimodal multi-objective evolutionary algorithms (MMOEAs), which are called “regular” algorithms in this study, have been proposed and performed well on benchmark test suites. However, Liu et al. [13] have stated that existing “regular” MMOEAs may perform poor with the imbalanced MMOPs in which search complexities of Pareto optimal sets (PSs) are not the same in different regions. In other words, most proposed MMOEAs belong to convergence-first search methods, thus they may trap into easy regions and cannot find enough equivalent Pareto optimal solutions when these MMOPs are imbalanced. To alleviate the above-mentioned problem, Liu et al. [13] used a convergence-penalized density method to maintain the diversity in the decision space. Moreover, both exploitation and exploration strategies are used to improve the search efficiency. The results show that their proposed algorithm, called the “special” algorithm, performs better on these imbalanced MMOPs when compared with other “regular” algorithms. Recently, Fan and Yan [11] proposed a zoning search (ZS) method considered as a decision space decomposition strategy to solve the balanced MMOPs. Intuitively, such a search strategy may also be employed to solve the imbalanced MMOPs since the ZS can reduce the search difficulty of problems in each subspace, i.e., the degree of imbalance in each subspace can be reduced, and help MMOEAs improve the diversity in the decision space. However, the same amount of computational resource is allocated in each search subregion in [11], which may reduce the search efficiency and waste computational resources, especially when the distribution of PSs is unsymmetrical in the decision space. To maintain the diversity in the decision space and take full use of limited computational resources, a zoning search with the adaptive resource allocating method (ZS-ARA) is proposed to solve the imbalanced and balanced MMOPs in the current study. In the ZS-ARA, the decision search space is divided into many subspaces for reducing the complexity of PSs in each decision subspace, and computational resources can be automatically assigned among all the subspaces to improve their use efficiency. Results indicate that the ZS- ARA can perform well on both the balanced MMOPs and the imbalanced MMOPs. More importantly, the proposed algorithm can help a “regular” MMOEA (i.e., MO_Ring_ PSO_SCD proposed by Yue et al. [14]) enhance its performance in solving the imbalanced MMOPs. Therefore, it Manuscript received December 1, 2020; revised January 18, 2021; accepted February 13, 2021. This work was partially supported by the Shandong Joint Fund of the National Nature Science Foundation of China (U2006228), and the National Nature Science Foundation of China (61603244). Recommended by Associate Editor MengChu Zhou. (Corresponding author: Qinqin Fan.) Citation: Q. Q. Fan and O. K. Ersoy, “Zoning search with adaptive resource allocating method for balanced and imbalanced multimodal multi-objective optimization,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 6, pp. 1163–1176, Jun. 2021. Q. Q. Fan is with the Logistics Research Center, Shanghai Maritime University, Shanghai 201306, China (e-mail: forever123fan@163.com). O. K. Ersoy is with the School of Electronic and Computer Engineering, Purdue University, West Lafayette, IN 47906 USA (e-mail: ersoy@purdue. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2021.1004027 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 8, NO. 6, JUNE 2021 1163