Multimodal Problems Research Articles

Mitigating local minima in extracting optimal parameters for photovoltaic models: An optimizer leveraging multiple initial populations (OLMIP)

The accurate identification of parameters in photovoltaic models is of paramount importance to accurately predict the electrical behavior of photovoltaic systems and improve their performance. Nowadays, this task poses a multimodal optimization problem, aiming to find the best combination of parameters that fit the model to real data. This article presents a proposal for an Optimizer Leveraging Multiple Initial Populations (OLMIP) that aims to achieve optimal solutions while effectively avoiding undesirable local optima. By utilizing a separate evolution strategy involving four distinct initial populations, followed by the construction of an elite population, the algorithm can explore multiple regions of the search space and escape local minima. Experiments were conducted with four models: the single diode, double diode, triple diode from RTC France cell, and Photowatt-PWP201 module. The mean squared errors obtained are 9.860219E-04, 9.824849E-04, 9.824849E-04, and 2.425075 E−03, respectively. These results indicate that the algorithm achieves superior or comparable accuracy to that of six competitors. Furthermore, the statistical analysis of the results, including the Wilcoxon and Friedman tests, confirms the robustness and effectiveness of the approach used for parameter estimation in photovoltaic systems. These findings open up new prospects for the improvement and optimization of photovoltaic systems.

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

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.

Multi-region hierarchical surrogate-assisted quantum-behaved particle swarm optimization for expensive optimization problems

Surrogate-assisted evolutionary algorithms (SAEAs) have been successfully applied to solve computationally expensive optimization problems. However, most SAEAs struggle to achieve good results in solving complex multimodal problems, especially high-dimensional ones. Moreover, for problems with complex landscapes, SAEAs typically require constructing complex global surrogates to model the landscape and performing many iterations to identify the surrogate’s optimum, thereby reducing the efficiency of SAEAs. To deal with these issues, this paper proposes a multi-region hierarchical surrogate-assisted quantum-behaved particle swarm optimization (MHS-QPSO) algorithm for expensive optimization problems. To better balance exploration and exploitation, a search behavior selection strategy is proposed, enabling MHS-QPSO to appropriately switch between global and local searches. For the global search, the search space is divided into multiple regions that can adaptively adjust the size of the areas. A surrogate is constructed in each region, requiring only a small number of QPSO iterations to find the optimum of each surrogate. Furthermore, a novel reliability-based criterion is proposed to screen candidate solutions in different regions for exact evaluations, which can save the number of exact function evaluations and can rapidly improve the fitting accuracy of the surrogates in regions with superior fitness. During local searches, a dynamic boundary adjustment strategy is introduced to guide the QPSO to faster approach the potential optimal region. Experimental results on seven benchmark functions with dimensions from 10 to 100, and on a complex real application, demonstrate that MHS-QPSO significantly outperforms several state-of-the-art algorithms within a limited computational budget. Code for MHS-QPSO is available at https://github.com/quanshuzhang/MHS-QPSO.git.

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.

Knowledge-Based Perturbation LaF-CMA-ES for Multimodal Optimization

Multimodal optimization presents a significant challenge in optimization problems due to the existence of multiple attraction basins. Balancing exploration and exploitation is essential for the efficiency of algorithms designed to solve these problems. In this paper, we propose the KbP-LaF-CMAES algorithm to address multimodal optimization problems based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) framework. The Leaders and Followers (LaF) and Knowledge-based Perturbation (KbP) strategies are the primary components of the KbP-LaF-CMAES algorithm. The LaF strategy is utilized to extensively explore the potential local spaces, where two cooperative populations evolve in synergy. The KbP strategy is employed to enhance exploration capabilities. Improved variants of CMA-ES are used to exploit specific domains containing local optima, thereby potentially identifying the global optimum. Simulation results on the test suite demonstrate that KbP-LaF-CMAES significantly outperforms other meta-heuristic algorithms.

Niching Global Optimisation: Systematic Literature Review

Niching in global optimisation refers to a set of techniques designed to identify multiple optimal solutions within a nonlinear, multimodal landscape. These algorithms enhance the exploratory capabilities of conventional metaheuristics by maintaining diversity and supporting coexisting subpopulations across a search space, thereby allowing a more deterministic approach to the true global optimum. Niching algorithms can be categorised into three primary subfamilies: sequential or temporal niching, parallel or spatial niching, and hybrid models which integrate various niching subparadigms. This research paper aims to explore the effectiveness and limitations of different niching algorithms by providing a systematic literature review of the theoretical frameworks within these subfamilies. Eleven major niching native subparadigms have been identified: fitness sharing, crowding, clearing, speciation, restricted tournament selection, clustering, multiobjectivisation, embedded hybrid methods, ensemble hybrid methods, and other hybrid approaches. This study offers a detailed examination of each paradigm’s theoretical foundation, including template algorithmic layouts, and delineates the unique elements of each approach. Research contributions from the inception of niching to 2024 have been aggregated from the SCOPUS database and systematically classified. Data aggregation included journal articles, conference papers, review papers, and research reports published in English only following the PRISMA framework. Application papers with novel theoretical ideas were also taken into account. In all, 203 research works were retained under the inclusion and exclusion criteria. This study concludes with overarching high-level recommendations for future research in modern niching optimisation, emphasising the development of space and time-scalable methods to enhance the adaptability and efficiency of optimisation algorithms in diverse, increasingly multivariable multimodal problems.

A Novel Reinforcement Learning-Based Particle Swarm Optimization Algorithm for Better Symmetry between Convergence Speed and Diversity

This paper introduces a novel Particle Swarm Optimization (RLPSO) algorithm based on reinforcement learning, embodying a fundamental symmetry between global and local search processes. This symmetry aims at addressing the trade-off issue between convergence speed and diversity in traditional algorithms. Traditional Particle Swarm Optimization (PSO) algorithms often struggle to maintain good convergence speed and particle diversity when solving multi-modal function problems. To tackle this challenge, we propose a new algorithm that incorporates the principles of reinforcement learning, enabling particles to intelligently learn and adjust their behavior for better convergence speed and richer exploration of the search space. This algorithm guides particle learning behavior through online updating of a Q-table, allowing particles to selectively learn effective information from other particles and dynamically adjust their strategies during the learning process, thus finding a better balance between convergence speed and diversity. The results demonstrate the superior performance of this algorithm on 16 benchmark functions of the CEC2005 test suite compared to three other algorithms. The RLPSO algorithm can find all global optimum solutions within a certain error range on all 16 benchmark functions, exhibiting outstanding performance and better robustness. Additionally, the algorithm’s performance was tested on 13 benchmark functions from CEC2017, where it outperformed six other algorithms by achieving the minimum value on 11 benchmark functions. Overall, the RLPSO algorithm shows significant improvements and advantages over traditional PSO algorithms in aspects such as local search strategy, parameter adaptive adjustment, convergence speed, and multi-modal problem handling, resulting in better performance and robustness in solving optimization problems. This study provides new insights and methods for the further development of Particle Swarm Optimization algorithms.

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.

CMA-ES with Learning Rate Adaptation

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving continuous black-box optimization problems. A practically useful aspect of CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This study comprehensively explores the impact of learning rate on CMA-ES performance and demonstrates the necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses the setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation mechanism for CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate the behavior of CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments and compare the results with those obtained for CMA-ES with a fixed learning rate and with population size adaptation. The results show that CMA-ES with the proposed learning rate adaptation works well for multimodal and/or noisy problems without extremely expensive learning rate tuning.

A New Approach to the Solution of Facility Layout Problems with Filled Function Method

Facility layout problems are generally solved by stochastic methods in the literature. The large number of iterations used in these methods is quite costly in terms of solution time. In this study, in order to get rid of this disadvantage, the facility layout problem was solved using the filled function method, which is known to be very successful in solving global optimization problems. In order to demonstrate the effectiveness of the filled function method, the classical linear facility layout problem was handled in a non-linear manner and the problem was deliberately made more difficult. In order to use the filled function method, the facility layout problem was transformed into an unconstrained and multimodal (including more than one local minima) global optimization problem by using the hyperbolic smoothing technique and the penalty function method. Thus, in this first study in the literature where a deterministic method is combined with the solution of the facility layout problem, it is shown that the non-convex facility layout problem can be solved with the filled function method with very few iterations and short solution times.

Center-enhanced video captioning model with multimodal semantic alignment

Video captioning aims at automatically generating descriptive sentences based on the given video, establishing an association between the visual contents and textual languages, has attracted great attention and plays a significant role in many practical applications. Previous researches focus more on the aspect of caption generation, ignoring the alignment of multimodal feature and just simply concatenating them. Besides, video feature extraction is usually done in an off-line manner, which leads to the fact that the extracted feature may not adapted to the subsequent caption generation task. To improve the applicability of extracted features for downstream caption generation and to address the issue of multimodal semantic alignment fusion, we propose an end-to-end center-enhanced video captioning model with multimodal semantic alignment, which integrates feature extraction and caption generation task into a unified framework. In order to enhance the completeness of semantic features, we design a center enhancement strategy where the visual–textual deep joint semantic feature can be captured via incremental clustering, then the cluster centers can serve as the guidance for better caption generation. Moreover, we propose to promote the visual–textual multimodal alignment fusion by learning the visual and textual representation in a shared latent semantic space, so as to alleviate the multimodal misalignment problem. Experimental results on two popular datasets MSVD and MSR-VTT demonstrate that the proposed model could outperform the state-of-the-art methods, obtaining higher-quality caption results.

Dynamic Niches-Based Hybrid Breeding Optimization Algorithm for Solving Multi-Modal Optimization Problem

Some problems exist in classical optimization algorithms to solve multi-modal optimization problems and other complex systems. A Dynamic Niches-based Improved Hybrid Breeding Optimization (DNIHBO) algorithm is proposed to address the multi-modal optimization problem in the paper. By dynamically adjusting the niche scale, it effectively addresses the issue of niche parameter sensitivity. The structure of the algorithm includes three distinct groups: maintainer, restorer, and sterile lines for updating operations. However, the maintainer individuals often stagnate, leading to the risk of the local optima. To overcome this, neighborhood search and elite mutation strategies are incorporated, enhancing the balance between exploration and exploitation. To further improve individual utilization within niches, a niche restart strategy is introduced, ensuring sustained population diversity. The efficacy of DNIHBO is validated through simulations on 16 multi-modal test functions, followed by comparative analyses with various multi-modal optimization algorithms. The results convincingly demonstrate that DNIHBO not only effectively locates multiple global optima but also consistently outperforms other algorithms on test functions. These findings underscore the superiority of DNIHBO as a high-performing solution for multi-modal optimization.

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.

Dynamic switched crowding-based multi-objective particle swarm optimization algorithm for solving multi-objective AC-DC optimal power flow problem

In this paper, the multi-objective AC-DC optimal power flow (MO/AC-DC OPF) problem in the presence of renewable energy sources (RESs), flexible AC transmission system (FACTS) devices and multi-terminal direct current (MTDC) systems is introduced for the first time. Conflicting objective functions and the high complexity of the objective and constraint spaces are the main challenges in finding optimal solutions for MO/AC-DC OPF. To overcome these challenges, twelve different versions of the dynamic switched crowding-based multi-objective particle swarm optimization (DSC-MOPSO) algorithm are introduced in this paper. Studies on multimodal optimization problems have shown that all DSC-MOPSO versions have better performance metrics than the MOPSO algorithm. Using the developed DSC-MOPSO and its strong competitors, the Pareto-optimal solution sets of the MO/AC-DC OPF problem are investigated. In these investigations, the performances of the algorithms are tested for the minimization of dual and triple objectives such as fuel cost, voltage level deviation, emission and power loss in a modified IEEE 30-bus power grid. According to the simulation results, the proposed DSC-MOPSO achieved an improvement in fuel cost between 0.02 % and 5.05 % and a reduction in active power loss between 0.44 % and 30.74 % compared to its competitors. The Hypervolume (HV) performance metric was used to evaluate the Pareto-front coverage performance of DSC-MOPSO and other optimizers. The results from nine case studies of the MO/AC-DC OPF were statistically analyzed by the Friedman test according to the 1/HV metric. According to the Friedman test results, the rankings of DSC-MOPSO and MOMA are 1.984 and 3.079, respectively, ranking first and second among all competitors. Finally, in this study, feasible solutions for MO/AC-DC OPF problem are identified for the first time and the stability of competitive algorithms in finding these solutions is analyzed for the first time. The success rates and search times of DSC-MOPSO and MOMA algorithms in finding feasible solutions for MO/AC-DC OPF are 91.01 % (30.641 s) and 82.01 % (46.038 s), respectively.

Decoding The Playbook: Multi-Modal Characterization of Coordinated Influence Operations on Indian Social Media

Manipulation of online discourse through organized disinformation and propaganda campaigns is a threat to information integrity and democratic dialogue. Taking cues from on-ground reports and recent literature on online trend manipulation in the Indian political landscape, we argue that organizing and consequently detecting influence operations is a multi-modal problem, where coordination is organized around different modalities like tweet , retweet , image and temporal . In this paper, we examine three case studies of prominent hashtag campaigns on Indian Twitter. Building on prior coordination detection methods, we identify communities of coordinated users across each of the four modalities. An in-depth analysis of the coordinated communities offers unique insights into the playbook of coordination strategies employed in the Indian context. We find that tweet coordination is used for hashtag trending, while retweet coordination aids in amplifying messaging from influential right-wing accounts. Moreover, we find distinct roles of users across modalities, where users that disseminate content through tweet and image coordination ( disseminators ) are independent of users that amplify content through retweet coordination ( amplifiers ), suggesting the existence of distinct coordination campaigns and objectives within influence operations. We conclude by highlighting the multi-modal approach to coordination for comprehensively characterizing influence operations, the drawbacks of temporal filtering in coordination, and the transferability and implications of our findings.

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.

A clustering-assisted adaptive evolutionary algorithm based on decomposition for multimodal multiobjective optimization

A multimodal multiobjective optimization problem can have multiple equivalent Pareto Sets (PSs). Since the number of PSs may vary in different problems, if the population is restricted to a fixed size, the number of solutions found for each PS will inevitably fluctuate widely, which is undesirable for decision makers. To address the issue, this paper proposes a clustering-assisted adaptive evolutionary algorithm based on decomposition (CA-MMEA/D), whose search process can be roughly divided into two stages. In the first stage, an initial exploration of decision space is carried out, and then solutions with good convergence are used for clustering to estimate the number and location of multiple PSs. In the second stage, new search strategies are developed on the basis of clustering, which can take advantage of unimodal search methods. Experimental studies show that the proposed algorithm outperforms some state-of-the-art algorithms, and CA-MMEA/D can keep the number of solutions found for each PS at a relatively stable level for different problems, thus making it easier for decision makers to choose the desired solutions. The research in this paper provides new ideas for the design of decomposition-based multimodal multiobjective algorithms.

Multimodal urban mobility solutions for a smart campus using artificial neural networks for route determination and an algorithm for arrival time prediction

Artificial intelligence algorithms play a key role in solving multimodal urban mobility problems, this study outlines a scheme covering bus, boat, and pedestrian transport modes. This involves designing an Artificial Neural Network (ANN) model to classify the most suitable routes for passengers, together with an algorithm capable of estimating the arrival times of various transportation modes at designated stopping points or buildings within the smart campus. The information used to train the ANN is obtained from an Internet of Things (IoT) network and a database that includes the available routes within the campus (which is situated in the Brazilian Amazon), where there is a multimodal electric mobility service. In seeking to achieve the objectives, this ANN relied on the user's geographic location in the input layer, and route mapping data stored in the database in the output layer, as well as the backpropagation algorithm for computational processing and adjustments of synaptic weight. The algorithm for modal arrival time prediction used real-time geographic location data, together with the Haversine formula to calculate geographic distance, while the average speed was obtained from the IoT network. A range of different ANN parameters were included in the experiments. In the study case, the ANN results show an improvement of above 92% for determining the best routes and predicting arrival times. The findings show that ANNs can effectively find the best route within a real-world smart campus environment. Moreover, the results show the arrival time can be estimated by using real-time geographic location data.

Virtual Position Guided Strategy for Particle Swarm Optimization Algorithms on Multimodal Problems.

Premature convergence is a thorny problem for particle swarm optimization (PSO) algorithms, especially on multimodal problems, where maintaining swarm diversity is crucial. However, most enhancement strategies for PSO, including the existing diversity-guided strategies, have not fully addressed this issue. This paper proposes the virtual position guided (VPG) strategy for PSO algorithms. The VPG strategy calculates diversity values for two different populations and establishes a diversity baseline. It then dynamically guides the algorithm to conduct different search behaviors, through three phases - divergence, normal, and acceleration - in each iteration, based on the relationships among these diversity values and the baseline. Collectively, these phases orchestrate different schemes to balance exploration and exploitation, collaboratively steering the algorithm away from local optima and towards enhanced solution quality. The introduction of 'virtual position' caters to the strategy's adaptability across various PSO algorithms, ensuring the generality and effectiveness of the proposed VPG strategy. With a single hyperparameter and a recommended usual setup, VPG is easy to implement. The experimental results demonstrate that the VPG strategy is superior to several canonical and the state-of-the-art strategies for diversity guidance, and is effective in improving the search performance of most PSO algorithms on multimodal problems of various dimensionalities.