Multimodal Multi-objective Optimization Problems Research Articles

An archive-assisted multi-modal multi-objective evolutionary algorithm

The multi-modal multi-objective optimization problems (MMOPs) pertain to characteristic of the decision space that exhibit multiple sets of Pareto optimal solutions that are either identical or similar. The resolution of these problems necessitates the utilization of optimization algorithms to locate multiple Pareto sets (PSs). However, existing multi-modal multi-objective evolutionary algorithms (MMOEAs) encounter difficulties in concurrently enhancing solution quality in both decision space and objective space. In order to deal with this predicament, this paper presents an Archive-assisted Multi-modal Multi-objective Evolutionary Algorithm, called A-MMOEA. This algorithm maintains a main population and an external archive, which is leveraged to improve the fault tolerance of individual screening. To augment the quality of solutions in the archive, an archive evolution mechanism (AEM) is formulated for updating the archive and an archive output mechanism (AOM) is used to output the final solutions. Both mechanisms incorporate a comprehensive crowding distance metric that employs objective space crowding distance to facilitate the calculation of decision space crowding distance. Besides, a data screening method is employed in the AOM to alleviate the negative impact on the final results arising from undesirable individuals resulting from diversity search. Finally, in order to enable individuals to effectively escape the limitation of niches and further enhance diversity of population, a diversity search method with level-based evolution mechanism (DSMLBEM) is proposed. The proposed algorithm’s performance is evaluated through extensive experiments conducted on two distinct test sets. Final results indicate that, in comparison to other commonly used algorithms, this approach exhibits favorable performance.

Dual-space distribution metric-based evolutionary algorithm for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are characterized by having multiple global or local Pareto solution sets. In most of multimodal multi-objective evolutionary algorithms, the distribution of population in objective space and decision space cannot be taken into account, simultaneously. In this paper, an innovative evolutionary algorithm based on dual-space distribution metric for multimodal multi-objective optimization (DsDMEA) is designed to solve this problem. Specifically, the two enhanced performance metrics are coupled together as a single dual-space distribution metric (DsDM) serving as the criterion for environmental selection. DsDMEA explores solutions on Pareto front and Pareto solutions sets that are not associated with the reference point, which show limited contributions to the distribution in dual spaces. Then DsDMEA removes any bi-noncontributing solutions using DsDM during environmental selection until the population size is met. This achieves a balance between the distribution of solutions and convergence in decision space. At the same time, this paper introduces a novel fitness computation method, enabling the precise localization of local Pareto fronts within the population. Finally, eight state-of-the-art multimodal multi-objective evolutionary algorithms are adopted to make a comparison on two types of benchmark to verify the performance of the DsDMEA. The experimental results demonstrated the DsDMEA effectiveness in solving MMOPs with local problems and traditional MMOPs.

A grid self-adaptive exploration-based algorithm for multimodal multiobjective optimization

In multimodal multiobjective optimization, the key is to find as many equivalent Pareto optimal solutions as possible through broad exploration in the decision space. The grid search strategy can achieve quick convergence by guiding the evolution with historical information in each grid while enabling broad exploration. However, inaccurate utilization of information in grids may lead to losing numerous potential solutions, especially on imbalanced problems. In order to resolve this issue, a grid self-adaptive exploration-based algorithm (GSEA) is proposed in this paper. In GSEA, the historical information in the grid is accurately utilized through grid-based self-adaptive exploration and niche clearing methods, which retain a large number of solutions with potential and effectively handle multimodal multiobjective optimization problems (MMOPs). Experimental results show that the proposed algorithm outperforms seven other state-of-the-art multimodal multiobjective evolutionary algorithms (MMEAs) on two types of MMOPs, and the approach can effectively deal with the MMOPs with middle-scale decision variables as memory allows.

Multi-objective sand cat swarm optimization based on adaptive clustering for solving multimodal multi-objective optimization problems

Multimodal multi-objective optimization problems (MMOPs) represent a highly challenging class of complex problems, characterized by the presence of several Pareto solution sets in the decision space which map to the identical Pareto-optimal front. The goal of solving MMOPs is to find multiple distinct Pareto sets to sustain a balance between good convergence and diversification of populations. In this paper, a multi-objective sand cat swarm optimization algorithm (MOSCSO) is developed to address MMOPs. In the MOSCSO algorithm, an adaptive clustering-based specific congestion distance technique is introduced to compute the level of crowdedness. This ensures an even distribution of individuals, avoiding excessive crowding in the local area. Subsequently, enhanced search-and-attack prey updating mechanisms are designed to effectively increase not only the exploration and exploitation capabilities of the algorithm but also to enhance the diversity of the swarm in both the decision space and the objective space. To verify the effectiveness of the proposed algorithm, the MOSCSO is applied to solve the CEC2019 complex multimodal benchmark function. The experimental outcomes illustrate that the proposed approach possesses excellent performance in searching for Pareto solutions compared with other algorithms. Meanwhile, the method is also employed to address the map-based distance minimization problem, which further validates the usefulness of the MOSCSO.

Clustering-based evolutionary algorithm for constrained multimodal multi-objective optimization

Handling constrained multimodal multi-objective optimization problems (CMMOPs) is a tremendous challenge as it involves the discovery of multiple equivalent constrained Pareto sets (CPSs) with the identical constrained Pareto front (CPF). However, the existing constrained multi-objective evolutionary algorithms are rarely suitable for solving CMMOPs due to the fact that they focus solely on locating CPF and do not intend to search for multiple equivalent CPSs. To address this issue, this paper proposes a framework of clustering-based constrained multimodal multi-objective evolutionary algorithm, termed FCCMMEA. In the proposed FCCMMEA, we adopt a clustering method to separate the population into multiple subpopulations for locating diverse CPSs and maintaining population diversity. Subsequently, each subpopulation evolves independently to produce offspring by an evolutionary algorithm. To balance the convergence and feasibility, we develop a quality evaluation metric in the classification strategy that considers the local convergence quality and constraint violation values, and it divides the populations into superior and inferior populations according to the quality evaluation of individuals. Furthermore, we also employ a diversity maintenance methodology in environmental selection to maintain the diverse population. The proposed FCCMMEA algorithm is compared with seven state-of-the-art competing algorithms on a standard CMMOP test suite, and the experimental results validate that the proposed FCCMMEA enables to find multiple CPSs and is suitable for handling CMMOPs. Also, the proposed FCCMMEA won the first place in the 2023 IEEE Congress on Evolutionary Computation competition on CMMOPs.

Multimodal multiobjective differential evolution algorithm based on enhanced decision space search

Multimodal multiobjective optimization problems (MMOPs) have attracted extensive research interest. These problems are characterized by the presence of multiple equivalent optimal solutions in the decision space, all corresponding to the same optimal values in the objective space. However, effectively finding a high-quality and evenly distributed Pareto sets (PSs) remains a challenge for researchers. This paper introduces a multimodal multiobjective differential evolution algorithm based on enhanced decision space search (MMODE_EDSS). By adopting two types of strategies to enhance the decision space search capability, the algorithm generates multiple high-quality non-dominated solutions. In the early stages of evolution, neighborhood information is used to enhance search capabilities, while in the later stages, data interpolation methods following clustering are employed for searching. Moreover, to improve the overall population distribution, an environmental selection mechanism based on dual-space crowding distance is adopted. The effectiveness of the proposed algorithm, MMODE_EDSS, is evaluated by comparing it with eight state-of-the-art multimodal multiobjective evolutionary algorithms (MMOEAs). Experimental results confirm the significant advantages of MMODE_EDSS.

A multimodal multi-objective differential evolution with series-parallel combination and dynamic neighbor strategy

Multimodal multi-objective optimization problems (MMOPs), which aim to identify as many optimal solutions as possible and exhibit multiple equivalent Pareto optimal solution sets (PSs) that correspond to the same Pareto optimal front (PF), commonly arise in a wide range of optimization problems in the real world. However, some dominated solutions that exhibit greater diversity in the decision space may be substituted by non-dominated solutions with a higher level of decision space crowding. To tackle this issue, this paper proposes a multimodal multi-objective differential evolution with series-parallel combination and dynamic neighbor strategy (MMODE_SPDN), which can balance convergence, objective space diversity and decision space diversity. Specifically, two archives are initially updated serially followed by the overall update of the parallel structure, in which the serial-first approach can enhance population diversity and the parallel structure can greatly reduce the amount of calculation. In addition, a dynamic neighbor strategy which utilizes adaptive selection among neighbors to generate difference vectors in the decision space and objective space and then adopts the main and auxiliary parent method during the mutation process is proposed. Furthermore, the utilization of an auxiliary archive and the clustering-based special crowding distance (CSCD) method are employed to facilitate the updating of the archive, thereby enhancing diversity. MMODE_SPDN is compared with other multimodal multi-objective optimization evolutionary algorithms (MMOEAs) on numerous test problems and the experimental results demonstrate that MMODE_SPDN exhibits superior performance.

A dynamic-speciation-based differential evolution with ring topology for constrained multimodal multi-objective optimization

Constrained multimodal multi-objective optimization problems (CMMOPs) consist of multiple equivalent constrained Pareto sets (CPSs) that have the identical constrained Pareto front (CPF). It is challenging for primary multimodal multi-objective evolutionary algorithms (MMEAs) to solve CMMOPs since they do not consider constraints. To tackle this challenge, a dynamic speciation-based differential evolution with ring topology, termed DSRDE, for solving CMMOPs is developed in this paper. To search for multiple equivalent CPSs in CMMOPs, the dynamic speciation-based niche strategy is developed. The dynamic speciation-based niche strategy divides the population into multiple species, each of which searches for diverse and equivalent CPSs in different regions. Particularly, the species number is dynamically decreased to explore the equivalent CPSs with good convergence. Then, a ring topology is constructed among multiple species and their neighbors to balance the diversity, convergence, and feasibility of solutions. Continuously, each species interacts information with its neighbors and searches for equivalent CPSs. DSRDE adopts the popular constrained dominance principle to handle constraints and uses the differential evolutionary algorithm to locate diverse CPSs in the ring topology. It is compared with several state-of-the-art algorithms in two CMMOPs test suites for evaluating the performance of the proposed DSRDE. The experimental results confirm that DSRDE is competitive and has the ability to find multiple CPSs when tackling CMMOPs. DSRDE is also implemented in a real-world CMMOP and obtains superior performance.

Integration of preferences in multimodal multi-objective optimization

Various existing multimodal multi-objective evolutionary algorithms (MMEAs) efficiently search for an approximation to the Pareto optimal front (PF), which consists of multiple equivalent Pareto optimal sets (PSs), when tackling multimodal multi-objective optimization problems (MMOPs). However, in practical applications, the Decision-maker (DM) often seeks specific portions of the PF, known as the Region of Interest (ROI), corresponding to particular instances rather than the whole PF. Existing MMEAs only focus on the whole PF and lack exploration of the ROI. Consequently, we propose an innovative MMEA that incorporates the DM’s preferences and local convergence quality (MMEA-PLC). The proposed algorithm integrates the DM’s preferences into the traditional Pareto-dominance relationship, enabling the search for multiple equivalent Pareto optimal set subsets (PSSs) within the ROI. Due to the influence of local PFs, accurately representing the ROI can be challenging when expressed across different PFs. Therefore, we design a preference-based Pareto-dominance relationship to accurately convey the preferences on different PFs by mapping solutions onto a hyperplane in the objective space. Then, we further develop a scheme for assessing local convergence quality (LCQ) to ensure that the algorithm can identify multiple equivalent PSSs within the same ROI. To verify the effectiveness of the proposed algorithm, MMEA-PLC was experimentally compared with eight state-of-the-art MMEAs on 28 benchmark functions and a real-life application problem. Experimental results show that MMEA-PLC has significant competitive advantages in different preference scenarios.

A similarity-detection-based evolutionary algorithm for large-scale multimodal multi-objective optimization

In recent years, there has been a surge in the development of evolutionary algorithms tailored for multimodal multi-objective optimization problems (MMOPs). These algorithms aim to find multiple equivalent Pareto optimal solution sets (PSs). However, little work has been done on MMOPs with large-scale decision variables, especially when the Pareto optimal solutions are sparse. These problems pose significant challenges due to the dimension curse, the unknown sparsity, and the unknown number of equivalent PSs. In this paper, we propose an evolutionary algorithm based on similarity detection called SD-MMEA to solve large-scale MMOPs with sparse Pareto-optimal solutions. Specifically, it employs a multi-population independent evolution to explore multiple PSs and distinguishes different PSs by double detection of the similarity between subpopulations. Simultaneously, develop online scoring mechanisms for decision variables to guide the subpopulations to explore in different directions. In addition, during the latter stage of evolution, the decision variables of individuals are further optimized by a double-layer grouping process. The proposed algorithm is compared with six state-of-the-art algorithms. Experimental results show that SD-MMEA has significant advantages in solving large-scale MMOPs with sparse solutions.

Growing Neural Gas Network-based surrogate-assisted Pareto set learning for multimodal multi-objective optimization

The key issue in handling multimodal multi-objective optimization problems (MMOPs) is to find multiple Pareto sets (PSs) corresponding to one Pareto front (PF). Therefore, learning the PSs is critical to facilitate solving MMOPs while unfortunately, current research only focuses on PF learning which is helpless in finding multiple PSs by the information of one PF. Moreover, since the PSs of an MMOP are usually non-functional, traditional approximative function model-based PF learning is inapplicable. Consequently, developing new PS learning techniques is desired. Inspired by data-driven evolutionary algorithms, data can be used to train surrogate models to assist the algorithm. This article proposes an online data-driven PSs learning technique that aims to learn the topologies of PSs through a surrogate model to facilitate the search for PSs. Specifically, the Growing Neural Gas network is trained using non-dominated solutions to learn the topologies of PSs during the evolutionary process. Then, the nodes of the network are used to generate new solutions and adopted as reference points for environmental selection. A new algorithm is developed based on the PS learning technique for MMOPs. Experimental studies on three benchmark test suites and two different real-world applications demonstrate the superiority of our method over six state-of-the-art algorithms dedicated to MMOPs. The PSs learning technique can obtain the topologies of PSs and facilitate the search for them.

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

A Bi-Objective Evolutionary Algorithm for Multimodal Multiobjective Optimization

Multimodal multi-objective optimization problems (MMOPs) possess multiple Pareto optimal sets corresponding to the identical Pareto optimal front (PF). To handle MMOPs, we propose a bi-objective evolutionary algorithm (BOEA), which transforms an MMOP into a bi-objective optimization problem. This problem is constructed by the penalty boundary intersection technique and a diversity indicator to obtain multiple Pareto optimal sets. The first objective reflects the population convergence and factors in the population diversity in the objective space, while the other objective concentrates more on the population diversity in the decision space. Furthermore, an environmental selection strategy is designed to choose the offspring solutions, which consists of non-dominated sorting based on the transformed optimization problem and hierarchical clustering for selecting promising solutions. Experiments on 34 MMOPs demonstrate that BOEA performs better than selected state-of-the-art representatives, including 22 MMOPs from CEC2019 and 12 imbalanced MMOPs. In addition, the effectiveness of BOEA is further validated by six feature selection problemsin real-world applications.

Two-stage evolutionary algorithm with fuzzy preference indicator for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are multi-objective optimization problems (MOPs) involving multiple equivalent global or local Pareto optimal solution sets (PSs). For decision-makers, not only the global optimal solution sets need to be found, but also the value of local optimal solution sets cannot be ignored. However, most multimodal multi-objective evolutionary algorithms (MMOEAs) tend to select solutions with better convergence, and it is difficult to obtain the global PSs and local PSs at the same time. Therefore, we propose a fuzzy preference indicator-based two-stage evolutionary algorithm (FPITSEA) in this paper. To evaluate more comprehensively the potential of each solution in the population for locating the global and local PS during the evolutionary process, a fuzzy preference indicator is designed in FPITSEA. The fuzzy preference indicator is used to guide the evolution of the population in the first stage to find the global and local Pareto optimal regions. Subsequently, an independent evolution strategy is implemented in the second stage to distinguish different PSs as accurately as possible while also ensuring the convergence quality of the solution set. In addition, an improved distance-based subset selection method is proposed, aiming to simultaneously improve the distribution of the solution set in the decision space and objective space. Experimental results on several test sets of MMOPs show the advantages of FPITSEA over several state-of-the-art MMOEAs.

A particle swarm optimization algorithm based on modified crowding distance for multimodal multi-objective problems

Multimodal multi-objective optimization problems (MMOPs) are commonly encountered in practice. The difficulty in solving MMOPs is obtaining all of the Pareto optimal sets without degrading the performance of the Pareto optimal front. To address this challenge, this study proposes a particle swarm optimization algorithm based on modified crowding distance (MOPSO_MCD). In MOPSO_MCD, a modified method for calculating the crowding distance (MCD) is devised, which allows for a more comprehensive assessment of the crowding relationship between individuals in the decision space and the objective space. Moreover, a cosine similarity-based elite selection mechanism is designed to identify the neighborhood optimal individuals of the individuals in the population and improve the decision space diversity. Additionally, an offspring competition mechanism is proposed to keep the population from trapping in the local optimum and enhance the global search ability of the MOPSO_MCD algorithm. Experimental results and statistical analysis show that MOPSO_MCD performs better than the other comparison algorithms on sixteen test functions and a map-based practical problem.

A multi-modal multi-objective evolutionary algorithm based on scaled niche distance

Multi-modal multi-objective optimization problems (MMOPs) refer to several solutions in the decision space that share the same or similar objective value. Balancing the diversity of the objective space and decision space while maintaining the convergence of the population is a challenging and important problem. To address this issue, a novel multi-modal multi-objective evolutionary algorithm (MMEA) named MMEA-SND is proposed in this study. In the MMEA-SND, to locate Pareto-optimal solutions, and improve the diversity of solutions in the decision space, a diversity fitness is designed by the niche method to calculate the fitness of solutions in the diversity archive. In order to balance the diversity of solutions in the objective space and decision space, a scaled niche distance (SND) method is proposed in environmental selection. In this context, SND are utilized to measure the distances between each solution in the objective space and decision space. Furthermore, a parameter is implemented to avoid disregarding locally optimal solutions. To verify the performance of MMEA-SND, six state-of-the-art MMEAs are adopted to make a comparison on 42 benchmark problems. The experimental results show that the proposed MMEA-SND achieves a competitive performance in solving MMOPs.

An Immune-Inspired Approach with Interval Allocation in Solving Multimodal Multi-Objective Optimization Problems with Local Pareto Sets

An Immune-Inspired Approach with Interval Allocation in Solving Multimodal Multi-Objective Optimization Problems with Local Pareto Sets

An enhanced multimodal multi-objective genetic algorithm with a novel adaptive crossover mechanism for feature selection

Nowadays, multimodal multi-objective optimization problems (MMOPs) have received increasing attention from many researchers. In such problems, there are situations where two or more Pareto Sets (PSs) correspond to the same Pareto Front (PF). It is crucial to obtain as many PSs as possible without compromising the performance of the objective space. Therefore, this paper proposes an enhanced multimodal multi-objective genetic algorithm with a novel adaptive crossover mechanism, named AEDN_NSGAII. In the AEDN_NSGAII, the special crowding distance strategy can provide potential development opportunities for individuals with a larger crowding distance. An adaptive crossover mechanism is established by combining the simulated binary crossover (SBX) operator and the Laplace crossover (LP) operator, which adaptively improves the ability to obtain Pareto optimal solutions. Meanwhile, an elite selection mechanism can efficiently get more excellent individuals as parents to enhance the diversity of the decision space. Then, the proposed algorithm is evaluated on the CEC2019 test suite by the Friedman method and discussed for its feasibility through ablation experiments and boxplot analysis of PSP indicators. Experimental results show that AEDN_NSGAII can effectively search for more PSs without weakening the diversity and convergence of objective space. Finally, the performance of AEDN_NSGAII on the multimodal feature selection problem is compared with that of the other four algorithms. The statistical analysis demonstrates that the proposed algorithm has great potential for resolving this issue.

Competitive swarm optimization with subset selection based manifold learning for multimodal multi-objective optimization

There are two major challenges when handling the multimodal multi-objective optimization (MMO) problems. One is the loss of diversity since most of the evolutionary algorithms designed for MMO prefer the base algorithm with rapid convergence. Therefore, the extra niching or other diversity preserving mechanism is inevitable. The other is the distribution of the Pareto optimal solutions with imbalanced density. Since the Pareto optimal sets may show different characteristics in the decision space, it is tough to converge the solutions to the Pareto front uniformly. To address these issues, subset selection and manifold learning based competitive swarm optimization algorithm, namely MMO_CSO, is proposed. Competitive swarm optimizer which has well balance on both diversity and convergence is adopted. Moreover, a subset selection strategy is applied to select diversified individuals for learning the manifold structure of the Pareto set. Thereby, the subset selection based manifold learning mechanism is designed to generate the promising solutions which could approach the real Pareto solutions and fill the sparse Pareto subregion. Compared against six state-of-the-art peer algorithms, the proposed MMO_CSO has a better performance to search for the Pareto optimal solutions both in decision space and objective space on CEC2019 MMO benchmarks.

A decomposition and dynamic niching distance-based dual elite subpopulation evolutionary algorithm for multimodal multiobjective optimization

Solving multimodal multiobjective optimization problems(MMOPs) require obtaining multiple Pareto optimal solution sets (PSs). In recent years, researchers have proposed some multimodal multiobjective evolutionary algorithms (MMEAs), however, many existing MMEAs perform unsatisfactorily in dealing with complicated MMOPs, and even fail to find all Pareto solution sets in a stable manner. Many existing algorithms rely on globally optimal individuals to guide population evolution, which leads to the population rapidly converging to a few easy-to-find Pareto solution sets (PSs) and losing hard-to-find PSs. MMEAs that cannot consistently find all PSs are difficult to handle the increasing complexity of MMOPs. In order to address these problems, this research proposes a decomposition and dynamic niching distance-based dual elite subpopulation evolutionary algorithm, named MMEA-DES. In the proposed algorithm, decomposition-based methods are employed to maintain the diversity of the objective space of MMOPs, and a dynamic niching distance (DND) is proposed to calculate the crowding of the decision space. In addition, the dual elite subpopulations (DES) based on decomposition and DND are implemented to support the algorithm in obtaining well-distributed solutions in both decision space and objective space. To validate the performance of the proposed algorithm, a comparison with state-of-the-art algorithms on 23 test problems shows that the algorithm proposed in this research can stably solve various types of complicated MMOPs.