A Decomposition-based Multi-modal Multi-objective Evolutionary Algorithm with Problem Transformation into Two-objective Subproblems Yusuke Nojima Osaka Metropolitan University Sakai, JAPAN nojima@omu.ac.jp Yuto Fujii Osaka Prefecture University Sakai, JAPAN yuto.fujii@ci.cs.osakafu-u.ac.jp Naoki Masuyama Osaka Metropolitan University Sakai, JAPAN masuyama@omu.ac.jp Yiping Liu Hunan University Changsha, CHINA ypliu@hnu.edu.cn Hisao Ishibuchi Southern University of Science and Technology Shenzhen, CHINA hisao@sustech.edu.cn ABSTRACT In some real-world multi-objective optimization problems, Pareto optimal solutions with different design parameter values are mapped to the same point with the same objective function values. Such problems are called multi-modal multi-objective optimization prob- lems (MMOPs). For MMOPs, multi-modal multi-objective evolution- ary algorithms (MMOEAs) have been developed for approximating both the Pareto front (PF) and the Pareto sets (PSs). However, most MMOEAs use population convergence in the objective space as the primary evaluation criterion. They do not necessarily have a high PS approximation ability. To better approximate both PF and PSs, we propose a decomposition-based MMOEA where an MMOP is transformed into a number of two-objective subproblems. One objective of each subproblem is a scalarizing function defined by a weight vector for the original MMOP, while the other is defined by a decision space diversity. Experimental results show a high approximation ability of the proposed method for both PF and PSs. CCS CONCEPTS • Computing methodologies → Search methodologies. KEYWORDS Multi-modal multi-objective evolutionary algorithm, MOEA/D ACM Reference Format: Yusuke Nojima, Yuto Fujii, Naoki Masuyama, Yiping Liu, and Hisao Ishibuchi. 2023. A Decomposition-based Multi-modal Multi-objective Evolutionary Algorithm with Problem Transformation into Two-objective Subproblems. In Genetic and Evolutionary Computation Conference Companion (GECCO ’23 Companion), July 15–19, 2023, Lisbon, Portugal. ACM, New York, NY, USA, 4 pages. https://doi.org/10.1145/3583133.3593950 Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). GECCO ’23 Companion, July 15–19, 2023, Lisbon, Portugal © 2023 Copyright held by the owner/author(s). ACM ISBN 979-8-4007-0120-7/23/07. https://doi.org/10.1145/3583133.3593950 1 INTRODUCTION Multi-objective optimization problems (MOPs) frequently appear in the real world [11, 13]. The set of optimal solutions in the decision space is called the Pareto set (PS). The image of the PS in the objective space is called the Pareto front (PF). In some applications, there also exist a particular type of MOPs called multi-modal multi- objective optimization problems (MMOPs) [6, 12]. In MMOPs, some Pareto optimal solutions with the same objective function values have different design parameter values. To offer more alternatives for the decision maker of MMOPs, it is important to find the Pareto optimal solutions on all the PSs. In general, multiobjective evolutionary algorithms (MOEAs) cannot effectively solve MMOPs since they do not consider the diversity of solutions in the decision space, but only the conver- gence and the diversity in the objective space. To address this is- sue, multi-modal multi-objective evolutionary algorithms (MMEAs) have been proposed such as DN-NSGA-II [6], DNEA [8], MMODE [15], MMODE_CSCD [4] for MMOPs. MMEAs need a specialized ability to approximate both the PF and the PSs better to find the Pareto optimal solutions on all the PSs. However, most MMEAs use population convergence in the objective space as the primary evaluation criterion [7]. As a result, they do not necessarily have a high approximation ability of the PSs in the decision space. To better approximate both the PF and PSs, we propose a de- composition based MMEA, called multi-modal multi-objective to two-objective (MM2T) in this paper. MM2T first transforms an MMOP into a number of single-objective subproblems using uni- formly distributed weight vectors like MOEA/D [17]. Then, each subproblem is transformed into a new two-objective problem de- fined by the minimization of a scalizing function based on a weight vector for the original MMOP and the maximization of a decision space diversity measure. We also propose a differential evolution (DE) [10] based crossover operator where parent selection is per- formed considering both the PF and PSs approximation. The rest of this paper is organized as follows. Section 2 explains the general background. Section 3 describes the proposed MM2T. Then, Section 4 compares MM2T with representative MMEAs on several MMOPs. Finally, Section 5 concludes this paper. 399