Multimodal Multi-objective Optimization Research Articles

A dual-population coevolutionary algorithm for balancing convergence and diversity in the decision space in multimodal multi-objective optimization

Many multimodal multi-objective evolutionary algorithms (MMEAs) are effective in solving multimodal multi-objective problems (MMOPs), which have multiple equivalent Pareto optimal sets (PSs) mapping to the same Pareto optimal front (PF). Due to the existence of the global convergence-first mechanism, these MMEAs will remove the solutions that can improve the diversity of the decision space but have poor convergence and even lead to the loss of PS when encountering MMOPs with an imbalance between convergence and diversity in the decision space (MMOP-ICD) or an MMOP with a local PS (MMOPL). We propose a new dual-population coevolutionary algorithm to address these issues. The auxiliary population helps the main population locate areas where equivalent PSs may exist, and the main population focuses on balancing convergence and diversity in the decision space. When updating the auxiliary population, a strength local convergence quality (SLCQ) is used to explore the distribution of the equivalent PSs. When updating the main population, the new niche-based truncation strategy first deletes the solutions that contribute less to convergence. Then, a distance-based subset selection method balances the diversity between the decision and objective spaces. The comparison results show the overall performance of the proposed algorithm is significantly better than other state-of-the-art algorithms.

Differential evolutionary using opposition-based learning for multimodal multi-objective optimization

The correspondence between the decision space and the objective space is often many-to-one in multi-objective optimization problems. Therefore, a class of problems with such mapping relationships is defined as a MMOPs. For these problems, how to ensure the final solution converges to each Pareto solution set and guarantees the diversity of the algorithm is an urgent problem. The method of the paper with opposition-based strategy, a multimodal multi-objective optimization algorithm, is proposed. The algorithm proposed is called MMODE_OP, which is framed by a differential evolutionary algorithm, and opposition-based learning is applied to the initialization phase and generation-hopping phase to filter out the more promising individuals in the population for iteration to enhance the global search capability and the diversity of population. In addition, different Gaussian perturbation strategies are adopted with iteration to achieve the search of the neighborhood, which can further not only improve the quality of the Pareto solution set but also enable the convergence of the Pareto solution set quickly. This method improves the algorithm’s local and global search ability, and enables multiple the Pareto solution set and improving the convergence. In the meantime, adaptive scaling factors and crossover factors are designed in this paper to enhance the improved search capability. Finally, the experiment results of MMODE_OP and other excellent algorithms on 13 test problems corroborate the proposed methods have superior performance.

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

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

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.

Exploring a Promising Region and Enhancing Decision Space Diversity for Multimodal Multi-Objective Optimization

Exploring a Promising Region and Enhancing Decision Space Diversity for Multimodal Multi-Objective Optimization

An improved differential evolution algorithm for multi-modal multi-objective optimization.

Multi-modal multi-objective problems (MMOPs) have gained much attention during the last decade. These problems have two or more global or local Pareto optimal sets (PSs), some of which map to the same Pareto front (PF). This article presents a new affinity propagation clustering (APC) method based on the Multi-modal multi-objective differential evolution (MMODE) algorithm, called MMODE_AP, for the suit of CEC'2020 benchmark functions. First, two adaptive mutation strategies are adopted to balance exploration and exploitation and improve the diversity in the evolution process. Then, the affinity propagation clustering method is adopted to define the crowding degree in decision space (DS) and objective space (OS). Meanwhile, the non-dominated sorting scheme incorporates a particular crowding distance to truncate the population during the environmental selection process, which can obtain well-distributed solutions in both DS and OS. Moreover, the local PF membership of the solution is defined, and a predefined parameter is introduced to maintain of the local PSs and solutions around the global PS. Finally, the proposed algorithm is implemented on the suit of CEC'2020 benchmark functions for comparison with some MMODE algorithms. According to the experimental study results, the proposed MMODE_AP algorithm has about 20 better performance results on benchmark functions compared to its competitors in terms of reciprocal of Pareto sets proximity (rPSP), inverted generational distances (IGD) in the decision (IGDX) and objective (IGDF). The proposed algorithm can efficiently achieve the two goals, i.e., the convergence to the true local and global Pareto fronts along with better distributed Pareto solutions on the Pareto fronts.

Evolutionary multimodal multiobjective optimization guided by growing neural gas

Evolutionary multimodal multiobjective optimization aims to search for a set of Pareto optimal solutions that are well distributed in both the objective and decision spaces. In recent years, there has been a growing interest in enhancing the search ability of evolutionary algorithms using machine learning techniques. However, there are few studies on machine learning-assisted evolutionary multimodal multiobjective optimization. In this paper, we propose an evolutionary multimodal multiobjective algorithm guided by growing neural gas, which learns the topological structure of the obtained good solutions during the evolution process. These good solutions are well-converged and well-distributed in the decision space, which indicate the shape of the Pareto optimal solution set. The proposed algorithm employs three novel mating selection operators based on the topological structure to efficiently and effectively search for Pareto optimal solutions. The first operator exploits Pareto optimal solutions within the topological structure, while the others explore Pareto optimal solutions around and far away from the topological structure, respectively. The proposed algorithm is compared with eight state-of-the-art evolutionary multimodal multiobjective algorithms on 20 test problems. Experimental results demonstrate that the proposed algorithm is competitive in solving these problems.

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.

Multimodal Multiobjective Differential Evolutionary Optimization With Species Conservation

Multimodal Multiobjective Differential Evolutionary Optimization With Species Conservation

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.

Enhanced Multiobjective Optimization Algorithm for Intelligent Grid Management of Renewable Energy Sources

Optimal scheduling of microgrids (MGs) is a crucial component of smart grid optimization, playing a vital role in minimizing energy consumption and environmental degradation. However, existing methods tend to consider only a single optimization and do not consider the multiobjective optimization problem of MGs in a comprehensive and integrated way. This study proposes a comprehensive multiobjective optimal scheduling methodology for renewable energy MGs, incorporating demand-side management (DSM) considerations. Initially, a DSM multiobjective optimization model is formulated, focusing on the load shifting of controllable devices within the MG to refine the electricity consumption structure. This model contemplates the renewable energy consumption of the MG, customer electricity purchase costs, and load smoothness. Subsequently, a multiobjective optimization model for grid-connected MGs, encompassing wind and photovoltaic power generation, is constructed with the dual objectives of economic and environmental optimization for the MG. Ultimately, a multimodal multiobjective optimization algorithm, amalgamating a local convergence index and an environment selection strategy, is proposed to solve the model. The experimental results show that compared with other methods, the proposed method in this paper can reduce the integrated cost by 32.6% and 38.9% in summer and 19.4% and 40.2% in winter. This stands out as a unique contribution in the field of MG optimization, as it integrates DSM considerations into a multiobjective optimization model. This methodology achieves a balance between minimizing energy consumption and environmental degradation while also enhancing economic efficiency.

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

A Multitask-Assisted Evolutionary Algorithm for Constrained Multimodal Multiobjective Optimization

A Multitask-Assisted Evolutionary Algorithm for Constrained Multimodal Multiobjective Optimization

A Generative Adversarial Networks Model Based Evolutionary Algorithm for Multimodal Multi-Objective Optimization

A Generative Adversarial Networks Model Based Evolutionary Algorithm for Multimodal Multi-Objective Optimization

Weak relationship indicator-based evolutionary algorithm for multimodal multi-objective optimization

Weak relationship indicator-based evolutionary algorithm for multimodal multi-objective optimization

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

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