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multimodal multi-objective optimization via determinantal point process-assisted evolutionary algorithm

NEURAL COMPUTING & APPLICATIONS 2024年第3期36卷 1381-1411页

作者： Cheng, Xinyu Gong, Wenyin Ming, Fei Zhu, Xiaofang China Univ Geosci Sch Comp Sci Wuhan 430074 Hubei Peoples R China Wuhan Univ Technol Sch Comp Sci & Artificial Intelligence Wuhan 430071 Hubei Peoples R China

multimodal multi-objective optimization problems (MMOPs) are widely present in real life. Due to the need of balancing between convergence and diversity in multi-objective optimization, as well as the need of balancing between diversities in objective and decision spaces, exploring the Pareto optimal front and Pareto optimal solution set becomes rather difficult in solving MMOPs. Recently, some multimodal multi-objective optimization algorithms have emerged. However, most of them are convergence-first, which may result in poor diversity of the solution set in decision space. To remedy this defect, in this paper, a determinantal point process (DPP)-assisted evolutionary algorithm is proposed to effectively solve MMOPs. In the proposed method, i) the DPPs are used to select subsets to consider convergence and diversity in both objective and decision spaces;ii) a kernel matrix is designed to retain solutions with poor convergence but good diversity in the decision space to explore the equivalent Pareto optimal solution sets;and iii) we propose a framework that combines the population and archive to better solve MMOPs. The results show that the proposed algorithm achieves the best performance in 18 of the 28 benchmark problems compared to six state-of-the-art algorithms.

关键词： multimodal multi-objective optimization Evolutionary algorithm Determinantal point processes Subset selection Decision space diversity.

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multimodal multi-objective optimization based on local optimal neighborhood crowding distance differential evolution algorithm

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NEURAL COMPUTING & APPLICATIONS 2024年第1期36卷 461-481页

作者： Gu, Qinghua Peng, Yifan Wang, Qian Jiang, Song Xian Univ Architecture & Technol Sch Resources Engn Xian 710055 Shaanxi Peoples R China Xian Univ Architecture & Technol Xian Key Lab Intelligent Ind Percept Calculat & De Xian 710055 Peoples R China Xian Univ Architecture & Technol Sch Management Xian 710055 Shaanxi Peoples R China

In practical applications, the optimal solutions of multi-objective optimization are not unique. Some problems exist different Pareto Sets (PSs) in the decision space mapped to the same Pareto Front (PF) in the objective space, which are called multimodal multi-objective problems (MMOPs). To tackle this issue, this paper proposes a multimodal multi-objective optimization based on a local optimal neighborhood crowding distance differential evolution algorithm. First, an adaptive partitioning strategy in the initialization phase is proposed by using the characteristics of the heuristic stochastic search. That ensures the local optimal solution is quickly found among multiple PSs. Second, opposition-based learning is combined with differential mutation to generate vectors, which accelerate the convergence of the population to the optimal solution. Finally, a method for neighborhood crowding distances on different Pareto ranks is designed. The distance is computed by a weighted sum of Euclidean distances for the nearest neighbors. While reducing computational complexity, this strategy reflects realistic crowding degree. With these methods, balances the diversity performance of the decision and the objective space, while improving the search capability. multiple PSs reveal the problem's potential characteristics and meet the needs of the decision-maker. The practical significance is verified by the application of actual distance minimization problem. According to experimental results, the proposed method can achieve a high level of comprehensive performance.

关键词： multimodal multi-objective optimization Local optimal Crowding distance Differential evolution

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multimodal multi-objective optimization with multi-stage-based evolutionary algorithm

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MEMETIC COMPUTING 2023年第4期15卷 377-389页

作者： Wu, Tianyong Ming, Fei Zhang, Hao Yang, Qiying Gong, Wenyin China Univ Geosci Coll Marine Sci & Technol Wuhan 430074 Hubei Peoples R China China Univ Geosci Sch Comp Sci Wuhan 430074 Hubei Peoples R China China Univ Geosci Fac Engn Wuhan 430074 Hubei Peoples R China China Univ Geosci Inst Adv Marine Res CUG Guangzhou 511466 Guangdong Peoples R China

A key issue in tackling multimodal multi-objective optimization problems (MMOPs) is achieving the balance between objective space diversity and decision space diversity to obtain multiple Pareto sets (PSs) while guaranteeing convergence to the Pareto front (PF). However, most of the existing methods for MMOPs face the following two shortages: (i) they put insufficient emphasis on improving decision space diversity, resulting in missing some PSs or PS segments;and (ii) they lack the utilization of promising historical individuals which may help search the PSs. To alleviate these limitations, this paper proposes a novel multi-stage evolutionary algorithm with two improved optimization strategies. Specifically, the proposed method decomposes solving MMOP into two tasks, i.e., the Exploration task and the Exploitation task. The Exploration task first aims to explore the decision space to detect the multiple PSs, then, the Exploitation task aims to enhance the diversities on both objective and decision spaces (i.e., exploiting the PF and PSs). To better search PSs, historical individuals that are well-distributed in the decision space are stored as the evolutionary experience, and then used to generate offspring individuals. Moreover, a new differential evolution is designed to force crowded individuals to move to sparse and undetected regions on the PSs to enhance the diversity of PSs. Extensive experimental studies compare the proposed method with five state-of-the-art methods tailored for MMOPs on two benchmark test suites. The results demonstrate that the proposed method can outperform others in terms of three performance indicators.

关键词： multimodal multi-objective optimization Evolutionary algorithms multi-stage strategy Historical evolutionary experience Differential evolution Exploration and exploitation

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Exploring a Promising Region and Enhancing Decision Space Diversity for multimodal multi-objective optimization

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Tsinghua Science and Technology 2024年第2期29卷 325-342页

作者： Fei Ming Wenyin Gong School of Computer Science China University of GeosciencesWuhan 430074China

During the past decade,research efforts have been gradually directed to the widely existing yet less noticed multimodal multi-objective optimization problems(MMOPs)in the multi-objective optimization ***,researchers have begun to investigate enhancing the decision space diversity and preserving valuable dominated solutions to overcome the shortage caused by a preference for objective space ***,many existing methods still have limitations,such as giving unduly high priorities to convergence and insufficient ability to enhance decision space *** overcome these shortcomings,this article aims to explore a promising region(PR)and enhance the decision space diversity for handling *** traditional methods,we propose the use of non-dominated solutions to determine a limited region in the PR in the decision space,where the Pareto sets(PSs)are included,and explore this region to assist in solving ***,we develop a novel neighbor distance measure that is more suitable for the complex geometry of PSs in the decision space than the crowding *** on the above methods,we propose a novel dual-population-based coevolutionary *** studies on three benchmark test suites demonstrates that our proposed methods can achieve promising performance and versatility on different *** effectiveness of the proposed neighbor distance has also been justified through comparisons with crowding distance methods.

关键词： multimodal multi-objective optimization evolutionary algorithms promising region neighbor distance decision space coevolution

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Hybrid Operator and Strengthened Diversity Improving for multimodal multi-objective optimization

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Tsinghua Science and Technology 2024年第5期29卷 1409-1421页

作者： Guoting Zhang Yonghao Du Xiaobin Zhu Xiaolu Liu Beijing Institute of Tracking and Telecommunication Technology Beijing 100094China College of Systems Engineering National University of Defense TechnologyChangsha 410073China

multimodal multi-objective optimization problems(MMOPs)contain multiple equivalent Pareto subsets(PSs)corresponding to a single Pareto front(PF),resulting in difficulty in maintaining promising diversities in both objective and decision spaces to find these *** used to solve MMOPs,evolutionary algorithms mainly consist of evolutionary operators that generate new solutions and fitness evaluations of the *** enhance performance in solving MMOPs,this paper proposes a multimodal multi-objective optimization evolutionary algorithm based on a hybrid operator and strengthened diversity ***,a hybrid operator mechanism is devised to ensure the exploration of the decision space in the early stage and approximation to the optima in the latter ***,an elitist-assisted differential evolution mechanism is designed for the early exploration *** addition,a new fitness function is proposed and used in environmental and mating selections to simultaneously evaluate diversities for PF and *** studies on 11 widely used benchmark instances from a test suite verify the superiority or at least competitiveness of the proposed methods compared to five state-of-the-art algorithms tailored for MMOPs.

关键词： multimodal multi-objective optimization evolutionary algorithm hybrid operator strengthened diversity

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Variational graph autoencoder-driven balancing strategy for multimodal multi-objective optimization

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INFORMATION SCIENCES 2025年 712卷

作者： Yang, Lei Zhang, Erlei Dang, Qianlong Northwest A&F Univ Coll Informat Engn Yangling 712100 Peoples R China Northwest A&F Univ Coll Sci Yangling 712100 Peoples R China

multimodal multi-objective optimization aims to balance the diversity and the convergence to obtain multiple complete and uniform Pareto optimal solution sets. In recent years, using machine learning models to improve the performance of evolutionary algorithms has become a hot topic. However, few studies utilize machine learning models to solve the imbalance problem between the diversity and the convergence in multimodal multi-objective optimization. Therefore, this paper proposes a multimodal multi-objective evolutionary algorithm driven by variational graph autoencoder (VGAE), which can reproduce diversified offspring with good convergence by reconstructing the parent population. In reproduction, the parent population is constructed into graph data, and the VGAE is adopted to map the graph data to the latent space, obtaining the distribution information represented by the low-dimensional vector. By sampling the distribution, the VGAE can generate the offspring with the similar distribution to the parent, which can fill the less dense regions in the decision space and improve the exploitation ability. In archive updating, the convergence state based on the inverted generation distance between the non-dominated solutions and the worst dominated solutions is defined, and the state information of the convergence archive is transferred to the diversity archive to determine the dynamic niche. This niche comprehensively considers the distribution state and convergence degree of solutions in the diversity and convergence archives, which is employed to calculate the local convergence quality, retaining more promising solutions. The results of 48 benchmark problems and a practical application show that the proposed algorithm outperforms eight competitive algorithms.

关键词： multimodal multi-objective optimization Imbalance problem Variational graph autoencoder Inverted generation distance Dynamic niche

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A niching differential evolution with Hilbert curve for multimodal multi-objective optimization

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 95卷

作者： Li, Guosen Li, Wenfeng He, Lijun Gao, Cong Wuhan Univ Technol Sch Transportat & Logist Engn Wuhan 430063 Peoples R China Wuhan Univ Technol Sanya Sci & Educ Innovat Pk Hainan 572000 Peoples R China Wuhan Univ Technol State Key Lab Maritime Technol & Safety Wuhan 430063 Peoples R China

multimodal multi-objective optimization problems have a many-to-one relationship between the decision space and the objective space. That is, distinct solutions in the decision space share the same objective value. How to coordinate population convergence and diversity while locating multimodal solutions is a challenging research topic. Some evolutionary algorithms using niching techniques have been reported in the literature. These algorithms prefer to induce multiple niches based on population information. Owing to the impact of convergence-first principle, the population tends to gather in easier-to-search regions, making it tough to yield more dispersed solutions in different niches. To remedy this situation, this paper proposes a niching differential evolution with Hilbert curve. First, a neighborhood-driven reproduction method is presented based on Hilbert curve, which features a two-layer architecture to capture promising regions and identify multimodal solutions. Second, a convergence-based density indicator is designed as a selection criterion to distinguish between convergence solutions and diversity solutions in the decision space. Moreover, fifteen intricate multimodal multi-objective test functions are devised. The experiments are performed on a series of test functions and a map-based practical problem. Empirical results attest that the proposed algorithm is competitive in dealing with multimodality compared with ten multimodal multi-objective algorithms.

关键词： Evolutionary algorithm multimodal multi-objective optimization Niching Hilbert curve

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A multi-population competitive evolutionary algorithm based on genotype preference for multimodal multi-objective optimization

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 92卷

作者： Zhong, Keyu Xiao, Fen Gao, Xieping Hunan Normal Univ Minist Educ Key Lab Comp & Stochast Math Changsha 410081 Peoples R China Hunan Normal Univ Sch Math & Stat Changsha 410081 Peoples R China Xiangtan Univ Sch Comp Sci Xiangtan 411105 Peoples R China Hunan Normal Univ Coll Informat Sci & Engn Changsha 410081 Peoples R China

Many existing multimodal multi-objective evolutionary algorithms (MMOEAs) exhibit poor performance in addressing multimodal multi-objective optimization problems (MMOPs), mainly due to limited genetic diversity in environmental selection. In this paper, we propose a multi-population competitive evolutionary algorithm based on genotype preference (MPCEA-GP) to solve MMOPs. Firstly, we propose a population selection strategy based on genotype preference to maintain the genetic diversity of the population. This strategy utilizes the spectral radius to assess the overall convergence quality of the population, rather than evaluating each individual separately, and favors selecting the population with the minimum spectral radius, thereby preserving the genotypes of both optimal and suboptimal individuals. Secondly, to address the challenge of diminished genetic diversity during the evolutionary process, we incorporate historical survival population with substantial genetic diversity into the competition between parent and offspring, and preferentially select individuals with significant genotype differences to recombine into a new population. By merging two selected populations, a joint population with sufficient genetic diversity is constructed. Finally, a genotype-phenotype-based fitness criterion is devised to evaluate the fitness of individuals. This criterion not only compares genotypes using the Pareto dominance principle but also concurrently considers both genotype and phenotype diversity, aiding the population in more precisely identifying individuals with both good convergence and diversity. Empirical results show that MPCEA-GP outperforms state-of-the-art MMOEAs for 40 chosen benchmark functions and two complex real- world applications.

关键词： multimodal multi-objective optimization Genotype Phenotype Historical survival population

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Learnable self-supervised support vector machine based individual selection strategy for multimodal multi-objective optimization

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INFORMATION SCIENCES 2025年 690卷

作者： Gao, Xiaochuan Bai, Weiting Dang, Qianlong Yang, Shuai Zhang, Guanghui Northwest A&F Univ Coll Sci Yangling 712100 Shaanxi Peoples R China Anhui Agr Univ Sch Informat & Artificial Intelligence Hefei 230036 Peoples R China Hebei Agr Univ Sch Informat Sci & Technol Baoding 071001 Peoples R China

multimodal multi-objective optimization problem (MMOP) is a frontier research problem, which can provide decision makers with more choices without making trade-offs. Many multimodal multi-objective evolutionary algorithms (MMOEAs) have been proposed to solve MMOP. However, most MMOEAs tend to prioritize the objective dominance of individuals in the process of individual selection, and only individuals with the same objective dominance will be considered the diversity, which leads to the loss of many promising solutions. To solve the above problem, this paper proposes a learnable self-supervised support vector machine (SVM) based multimodal multi-objective optimization algorithm (SVMEA). Support vector machine can learn the knowledge about distinguishing the advantages and disadvantages of individuals from the data in the existing training set and select individuals, in which the objective dominance of individuals is as important as diversity. Moreover, a crowding distance calculation method based on Manhattan distance is designed. Compared with the traditional method using Euclidean distance to calculate crowding distance, it can better evaluate the diversity of individuals in the decision space and assist the selection of elite solutions. Experimental results show that the proposed SVMEA is competitive with seven other advanced MMOEAs on 34 benchmark problems and a practical application problem.

关键词： multimodal multi-objective optimization Learnable self-supervised support vector machine Individual selection Manhattan distance Crowding distance

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A neighborhood-assisted evolutionary algorithm for multimodal multi-objective optimization

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MEMETIC COMPUTING 2024年第2期16卷 139-158页

作者： Zhang, Weiwei Li, Jiaqiang Li, Guoqing Zhang, Weizheng Zhengzhou Univ Light Ind Sch Comp Sci & Technol Zhengzhou 450000 Peoples R China Ningbo Univ Fac Elect Engn & Comp Sci Ningbo 315211 Peoples R China

multi-modal multi-objective optimization problems (MMOPs) involve multiple Pareto sets (PSs) in decision space corresponding to the same Pareto front (PF) in objective space. The difficulty lies in locating multiple equivalent PSs while ensuring a well-converged and well-distributed PF. To address this, a neighborhood-assisted reproduction strategy is proposed. Through interactions with non-dominated solutions, the generated offspring could spread out along the PF, while ineractions with neighbors could improve the convergence ability. Importantly, individuals can participate in multiple neighborhoods, reducing the computational burden. Additionally, a neighborhood-assisted environmental selection strategy is prposed to encourage exploration of diverse solution regions, ensuring a balanced distribution of the population and preservation of multiple PSs. Comparative experiments are implemented on the CEC 2019 MMOPs test suite, and the superior performance of the proposed algorithm is demonstrated in comparison to several state-of-the-art approaches.

关键词： multimodal multi-objective optimization Neighborhood Evolutionary algorithm Niching based algorithm

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