Multimodal Optimization Algorithm Research Articles

Differential evolution based on strategy adaptation and deep reinforcement learning for multimodal optimization problems

Multimodal optimization problems (MMOPs) include multiple optima, which are common in practical fields. However, the success of solving MMOPs requires the algorithm to have both exploration and exploitation performance. How to select a reasonable search strategy is a difficult problem facing MMOPs. In this research, a differential evolution based on strategy adaptation and deep reinforcement learning, termed SA-DQN-DE, is proposed to select mutation strategies reasonably and search the optima effectively, which mainly includes three aspects. First, strategy adaptation, which calculates selection probability based on the feedback of different mutation operators in previous evolution, and assigns mutation operations to each individual to provide guidance for the next stage of evolution is developed. Secondly, a new historical individual preservation method is designed to improve search efficiency. Thirdly, deep reinforcement learning is applied to select multiple local search operators to refine the accuracy of potential optimal solutions. The performance of SA-DQN-DE is tested on publicly acknowledged CEC2013 benchmark MMOP set. The experimental results demonstrate that the proposed SA-DQN-DE has competitive performance compared with some of its peer multimodal optimization 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.

Multi-modal Battle Royale optimizer

AbstractMultimodal optimization poses a challenging problem in the field of optimization as it entails the discovery of multiple local and global optima, unlike unimodal optimization, which seeks a single global solution. In recent years, the significance of addressing multimodal optimization challenges has grown due to the real-world complexity of many problems. While numerous optimization methods are available for unimodal problems, multimodal optimization techniques have garnered increased attention. However, these approaches often grapple with a common issue: the determination of the niching parameter, necessitating prior knowledge of the problem space. This paper introduces a novel multimodal optimization approach that circumvents the need for prior problem space knowledge and avoids the challenge of predefining the niching parameter. Building upon the Battle Royal Optimization (BRO) algorithm, this extended version formulates a multimodal solution by utilizing Coulomb's law to identify suitable neighbors. The incorporation of Coulomb's law serves the dual purpose of identifying potential local and global optima based on fitness values and establishing optimal distances from solution candidates. A comparison study was done between the MBRO and seven well-known multimodal optimization algorithms using 14 benchmark problems from the CEC 2013 and CEC 2015 competitions to see how well it worked. The experimental results underscore MBRO's proficiency in successfully identifying most, if not all, local and global optima, positioning it as a superior solution when compared to its competitors.

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

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

Parallelization of adaptive Bayesian cubature using multimodal optimization algorithms

PurposeBayesian cubature (BC) has emerged to be one of most competitive approach for estimating the multi-dimensional integral especially when the integrand is expensive to evaluate, and alternative acquisition functions, such as the Posterior Variance Contribution (PVC) function, have been developed for adaptive experiment design of the integration points. However, those sequential design strategies also prevent BC from being implemented in a parallel scheme. Therefore, this paper aims at developing a parallelized adaptive BC method to further improve the computational efficiency.Design/methodology/approachBy theoretically examining the multimodal behavior of the PVC function, it is concluded that the multiple local maxima all have important contribution to the integration accuracy as can be selected as design points, providing a practical way for parallelization of the adaptive BC. Inspired by the above finding, four multimodal optimization algorithms, including one newly developed in this work, are then introduced for finding multiple local maxima of the PVC function in one run, and further for parallel implementation of the adaptive BC.FindingsThe superiority of the parallel schemes and the performance of the four multimodal optimization algorithms are then demonstrated and compared with the k-means clustering method by using two numerical benchmarks and two engineering examples.Originality/valueMultimodal behavior of acquisition function for BC is comprehensively investigated. All the local maxima of the acquisition function contribute to adaptive BC accuracy. Parallelization of adaptive BC is realized with four multimodal optimization methods.

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.

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.

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.

A reinforcement learning-based neighborhood search operator for multi-modal optimization and its applications

In this paper, a reinforcement learning-based neighborhood search operator (RLNS) is proposed for multi-modal optimization problems where the main novelties lie in the reinforcement learning-based neighborhood range selection strategy, the neighborhood subpopulation generation strategy and the local vector encirclement model. The reinforcement learning-based neighborhood range selection strategy is proposed to dynamically adjust the subpopulation size to address the issue of too many parameters to be adjusted in the multi-modal optimization algorithm based on the niching methods, while the neighborhood subpopulation generation strategy and the local vector encirclement model are designed with the hope of enhancing the individual’s ability to local exploitation to obtain more accurate solutions. To verify the effectiveness of the proposed RLNS, SSA-RLNS, PSO-RLNS and EO-RLNS are proposed by integrating the proposed RLNS with the existing sparrow search algorithm, particle swarm optimization and equilibrium optimizer. The performances of the proposed SSA-RLNS, PSO-RLNS, EO-RLNS and existing multi-modal optimization algorithms are tested in CEC2015 multi-niche benchmark functions. The experimental results show that the SSA-RLNS, PSO-RLNS and EO-RLNS could locate multiple global optimal solutions with satisfactory accuracy, which illustrate that the proposed RLNS could be successfully used to deal with multi-modal optimization problems by integrating with common population-based optimization algorithms. Finally, the SSA-RLNS is successfully applied in the inverse kinematics of robot manipulator.

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.

Design of a novel multimodal optimization algorithm and its application in logistics optimization

<abstract> <p>This study was purposed to design a multimodal continuous optimization algorithm based on a scheme agent to address the multidimensional complexity of optimization. An evolutionary sampling method of subarea exploration and multiple exploitations was developed by employing the scheme with variable population size so as to obtain higher optimization speed and accuracy. Second, the distribution plan was quantified into high-dimensional variable parameters based on the characteristics of logistics distribution optimization problems, and a high-dimensional discrete optimization model was constructed. Then, we identified and addressed the prominent issues and malignant virtual changes in the application of continuous algorithms to discrete problems. We have introduced a reasonable mutation mechanism during the optimization sampling process to mitigate this issue. Continuous real coordinate points were transformed across the neighborhood to standard discrete integer coordinate points by normalizing and logicizing the optimization sampling coordinates; also, the discretization of the continuous algorithm was realized. This approach could effectively prevent the algorithm from searching for targets in continuous optimization space, thereby fully reducing the complexity of the objective function distribution after conversion. The experiments showed that the transformed multimodal discrete optimization algorithm effectively addressed the optimization design problem of logistics distribution.</p> </abstract>

A Multitask-Assisted Evolutionary Algorithm for Constrained Multimodal Multiobjective Optimization

A Multitask-Assisted Evolutionary Algorithm for Constrained Multimodal Multiobjective Optimization

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

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

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.

A coarse- and fine-grained niching-based differential evolution for multimodal optimization problems and its application in multirobot task allocation

Multimodal optimization problems (MOPs) need to locate the global optima as many as possible and refine the accuracy of the identified optima as high as possible. How to divide niches reasonably and search solutions effectively are two difficult problems facing MOPs. In this paper, a coarse- and fine-grained niching-based differential evolution (CFNDE) algorithm with micro-search strategy is proposed to divide population reasonably and search the optima effectively, which mainly includes four contributions. Firstly, a framework of coarse- and fine-grained niching (CFN) strategy is proposed, where the coarse-grained clustering method provides the information of clusters to the fine-grained clustering method to accurately divide the population into niches. Secondly, a micro-search-based mutation (MSM) strategy is proposed to accelerate the convergence of population, which searches for the optimal solution in the range between the current individual and the nearest individual. Thirdly, a multi-level local search (MLLS) strategy that incorporates the cluster-level and population-level search mechanisms is proposed to refine the accuracy of solutions. Finally, a stagnation detection and re-search (SDR) strategy is proposed to re-search the local optima in evolution. We compare CFNDE with the-state-of-art multimodal optimization algorithms on the widely used benchmark CEC’2013. The experimental results show that CFNDE performs better than or similar to some recently proposed algorithms in 17 out of 20 tested functions. Besides, the proposed CFNDE is also applied to the multirobot task allocation problems and CFNDE provides multiple completely different optimal scheduling schemes to deal with emergencies.

A Niching Regression Adaptive Memetic Algorithm for Multimodal Optimization of the Euclidean Traveling Salesman Problem

The traveling salesman problem (TSP) has been studied for many years. In particular, the multimodal optimization of the TSP is important for practical applications, because decision-makers can select the best candidate based on current conditions and requirements. In the Euclidean Traveling Salesman Problem, there are n points in Rd space with Euclidean distance between any two points, that is, d(x,y)=||x-y||2. The goal is to find a tour of minimum length visiting each point. In this paper, we only focus on the case that d=2. Recently, in order to efficiently handle the multimodal optimization of the TSP, some methods have been developed to deal with it. Nevertheless, these methods usually perform poorly for large-scale cases. In this paper, we propose a niching regression adaptive memetic algorithm to handle the multimodal optimization of the Euclidean TSP. We use the memetic algorithm as the baseline algorithm and incorporate the neighborhood strategy to maintain the population diversity. In addition, we design a novel regression partition initialization and adaptive parameter control to enhance our algorithm, and propose the concept of the redundant individual to improve the search efficiency. To validate performance of the proposed algorithm, we comprehensively conduct experiments on the multimodal optimization of TSP benchmark and the well-known TSPLIB library. The experimental results reveal that the proposed method outperforms other methods, especially for large-scale cases.

Differential evolution with nearest better clustering for multimodal multiobjective optimization

The main aim of multimodal multiobjective optimization algorithms is to find multiple optimal solutions for multiple objectives. The problem with multiple objectives is to maintain the diversity and convergence balance in both decision and objective space. Better performance in one space can lead to deterioration of performance in another space. To effectively perform exploration and exploitation in the decision space, we have proposed a niching technique in our proposed algorithm based on differential evolution to form groups of the population. The niching technique also incorporates a balancing strategy to average the size of niches and a generation strategy to add new elements by finding promising areas. We have also proposed an archive strategy to store the possible optimal solutions that help to explore the decision space and ensure diversity as well as the convergence of the solutions. To show the efficiency of our proposed algorithm, we have performed exhaustive experimentation on the benchmark test suite and compared the results with state-of-the-art algorithms. The experimental results obtained using the proposed algorithm highlight the efficiency and effectiveness of our proposed algorithm in both decision and objective space.

Fault Reconfiguration in Distribution Networks Based on Improved Discrete Multimodal Multi-Objective Particle Swarm Optimization Algorithm.

Distribution network reconfiguration involves altering the topology structure of distribution networks by adjusting the switch states, which plays an important role in the smart grid since it can effectively isolate faults, reduce the power loss, and improve the system stability. However, the fault reconfiguration of the distribution network is often regarded as a single-objective or multi-objective optimization problem, and its multimodality is often ignored in existing studies. Therefore, the obtained solutions may be unsuitable or infeasible when the environment changes. To improve the availability and robustness of the solutions, an improved discrete multimodal multi-objective particle swarm optimization (IDMMPSO) algorithm is proposed to solve the fault reconfiguration problem of the distribution network. To demonstrate the performance of the proposed IDMMPSO algorithm, the IEEE33-bus distribution system is used in the experiment. Moreover, the proposed algorithm is compared with other competitors. Experimental results show that the proposed algorithm can provide different equivalent solutions for decision-makers in solving the fault reconfiguration problem of the distribution network.

A ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) involve multiple equivalent Pareto sets (PSs) with identical Pareto front (PF). Popular multimodal multi-objective evolutionary algorithms (MMEAs) are capable of finding multiple equivalent PSs. However, most of MMEAs lead to imbalanced or local PSs are dominated and lost when tackling several MMOPs with the imbalance between convergence and diversity (MMOP-ICD) or MMOP with local Pareto solutions (MMOPL). To tackle this issue, we propose a ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization. A ring-based niche technique is used based on the Pareto-based ranking hierarchy. Each hierarchy and its upper and lower neighbors hierarchy form a ring-hierarchy topology structure. Subsequently, a local convergence quality that considers the dominance relationship and objective values between all individuals is involved in the ring-hierarchy-based evolutionary strategy. It updates individuals and improves the population convergence quality. Moreover, a distance-based dominance selection that considers the distance between the neighbors and the dominance relationship is also developed. In this case, some individuals that approach imbalanced PS and local PS are maintained in the population instead of being dominated. Meanwhile, a dual-crowding distance is also involved in distance-based dominance selection to select diverse individuals. The proposed algorithm and several state-of-the-art MMEAs are tested on several MMOPs benchmarks. The experimental results demonstrate that the proposed algorithm is competitive and is capable of locating imbalanced PSs and local PSs.

A Zoning Search-Based Multimodal Multi-Objective Brain Storm Optimization Algorithm for Multimodal Multi-Objective Optimization

For multimodal multi-objective optimization problems (MMOPs), there are multiple equivalent Pareto optimal solutions in the decision space that are corresponding to the same objective value. Therefore, the main tasks of multimodal multi-objective optimization (MMO) are to find a high-quality PF approximation in the objective space and maintain the population diversity in the decision space. To achieve the above objectives, this article proposes a zoning search-based multimodal multi-objective brain storm optimization algorithm (ZS-MMBSO). At first, the search space segmentation method is employed to divide the search space into some sub-regions. Moreover, a novel individual generation strategy is incorporated into the multimodal multi-objective brain storm optimization algorithm, which can improve the search performance of the search engineering. The proposed algorithm is compared with five famous multimodal multi-objective evolutionary algorithms (MMOEAs) on IEEE CEC2019 MMOPs benchmark test suite. Experimental results indicate that the overall performance of the ZS-MMBSO is the best among all competitors.