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Symmetry
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Symmetry in Mathematical Optimization Algorithm and Its Applications
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Symmetry in Mathematical Optimization Algorithm and Its Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 August 2026 | Viewed by 10301

Special Issue Editor

Dr. Zhe Sun

E-Mail Website
Guest Editor
School of Modern Posts, Nanjing University of Posts and Telecommunications, Nanjing, China
Interests: optimization algorithms; systems optimization; machine learning algorithm

Special Issue Information

Dear Colleagues,

The concept of symmetry is relevant in many real engineering fields and plays a crucial role in mathematical optimization problems, which are widely applied in system control, parameter identification and system modelling. Mathematical optimization algorithms also are an efficient tool for complex engineering problems that require a reasonable system configuration. Mathematical optimization algorithms refer to traditional optimization algorithms and intelligence optimization algorithms such as the machine learning algorithm, heuristic algorithm, swarm optimization algorithm, etc.

This Special Issue focuses on the theoretical and practical application of mathematical optimization algorithms. The scope of this Special Issue includes, but is not limited to, the enhanced application of traditional optimization algorithms, combinatorial optimization algorithms, the evolutionary algorithm and the swarm optimization algorithm. Contributions that explore the application of optimization algorithms in engineering, economics, traffic, logistics and other disciplines are also highly encouraged.

Dr. Zhe Sun
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • combinatorial optimization
  • machine learning algorithm
  • heuristic algorithm
  • algorithm design
  • systems optimization and control
  • NP hard Problem optimization
  • parameter identification

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Published Papers (10 papers)

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Research

20 pages, 1642 KB  
Article
An Integrated Constrained Multi-Objective Evolutionary Algorithm with Feasibility-Driven Repair and Adaptive Parameter Control for Narrow-Band Optimization
by Hao Zhang, Junhua Ku and Jie Zhao
Symmetry 2026, 18(4), 641; https://doi.org/10.3390/sym18040641 - 10 Apr 2026
Viewed by 259
Abstract
Constrained multi-objective optimization (CMOP) is especially difficult when the feasible region is very narrow. In this study, we introduce Integrated-CMOEA, a clear and structured framework that uses structure-aware seeding, a projection-based repair operator, dual-population evolution, adaptive parameter control, and reference vector archiving. For [...] Read more.
Constrained multi-objective optimization (CMOP) is especially difficult when the feasible region is very narrow. In this study, we introduce Integrated-CMOEA, a clear and structured framework that uses structure-aware seeding, a projection-based repair operator, dual-population evolution, adaptive parameter control, and reference vector archiving. For the DC2-DTLZ1 problem, the repair step is handled as a continuous one-dimensional root-finding problem along a feasible search ray. This method provides clear rules for restoring feasibility when a valid bracket is found. Our results show that the method quickly finds and maintains strict feasibility and produces a well-distributed set of solutions near the constrained Pareto front. In tests with five independent runs, Integrated-CMOEA outperformed four other CMOEAs in both IGD and hypervolume. An ablation study shows that deterministic repair is the main reason for its strong performance on this narrow-band benchmark. Integrated-CMOEA is a reliable framework for analytically structured narrow-band CMOPs, though it has some limits when applied more broadly. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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35 pages, 22109 KB  
Article
MEBCMO: A Symmetry-Aware Multi-Strategy Enhanced Balancing Composite Motion Optimization Algorithm for Global Optimization and Feature Selection
by Gelin Zhang, Minghao Gao and Xianmeng Zhao
Symmetry 2026, 18(1), 40; https://doi.org/10.3390/sym18010040 - 24 Dec 2025
Viewed by 397
Abstract
To address the limitations of the traditional Balancing Composite Motion Optimization (BCMO) algorithm—namely weak directional global exploration, insufficient local exploitation accuracy, and a tendency to fall into local optima with reduced population diversity in feature selection tasks—this paper proposes a Multi-Strategy Enhanced Balancing [...] Read more.
To address the limitations of the traditional Balancing Composite Motion Optimization (BCMO) algorithm—namely weak directional global exploration, insufficient local exploitation accuracy, and a tendency to fall into local optima with reduced population diversity in feature selection tasks—this paper proposes a Multi-Strategy Enhanced Balancing Composite Motion Optimization algorithm (MEBCMO). From a symmetry perspective, MEBCMO exploits the symmetric and asymmetric relationships among candidate solutions in the search space to achieve a better balance between exploration and exploitation. The performance of MEBCMO is enhanced through three complementary strategies. First, an adaptive heat-conduction search mechanism is introduced to simulate thermal transmission behavior, where a Sigmoid function adjusts the heat-conduction coefficient α_T from 0.9 to 0.2 during iterations. By utilizing the symmetric fitness–distance relationship between the current solution and the global best, this mechanism improves the directionality and efficiency of global exploration. Second, a quadratic interpolation search strategy is designed. By constructing a quadratic model based on the current individual, a randomly selected individual, and the global best, the algorithm exploits local symmetric characteristics of the fitness landscape to strengthen local exploitation and alleviate performance degradation in high-dimensional spaces. Third, an elite population genetic strategy is incorporated, in which the top three individuals generate new candidates through symmetric linear combinations with non-elite individuals and Gaussian perturbations, preserving population diversity and preventing premature convergence. To evaluate MEBCMO, extensive global optimization experiments are conducted on the CEC2017 benchmark suite with dimensions of 30, 50, and 100, and comparisons are made with eight mainstream algorithms, including PSO, DE, and GWO. Experimental results demonstrate that MEBCMO achieves superior performance across unimodal, multimodal, hybrid, and composite functions. Furthermore, MEBCMO is combined with LightGBM to form the MEBCMO-LightGBM model for feature selection on 14 public datasets, yielding lower fitness values, higher classification accuracy, and fewer selected features. Statistical tests and convergence analyses confirm the effectiveness, stability, and rapid convergence of MEBCMO in symmetric and complex optimization landscapes. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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40 pages, 41737 KB  
Article
Multi-Threshold Image Segmentation Based on Reinforcement Learning–Thermal Conduction–Sine Cosine Algorithm (RLTCSCA): Symmetry-Driven Optimization for Image Processing
by Yijie Wang, Zuowen Bao, Qianqian Zhu and Xiang Lei
Symmetry 2025, 17(12), 2120; https://doi.org/10.3390/sym17122120 - 9 Dec 2025
Viewed by 525
Abstract
To address the inherent limitations of the standard Sine Cosine Algorithm (SCA) in multi-threshold image segmentation, this paper proposes an enhanced algorithm named the Reinforcement Learning–Thermal Conduction–Sine Cosine Algorithm (RLTC-SCA), with symmetry as a core guiding principle. Symmetry, a fundamental property in nature [...] Read more.
To address the inherent limitations of the standard Sine Cosine Algorithm (SCA) in multi-threshold image segmentation, this paper proposes an enhanced algorithm named the Reinforcement Learning–Thermal Conduction–Sine Cosine Algorithm (RLTC-SCA), with symmetry as a core guiding principle. Symmetry, a fundamental property in nature and image processing, refers to the invariance or regularity of grayscale distributions, texture patterns, and structural features across image regions; this characteristic is widely exploited to improve segmentation accuracy by leveraging consistent spatial or intensity relationships. In multi-threshold segmentation, symmetry manifests in the balanced distribution of optimal thresholds within the grayscale space, as well as the symmetric response of segmentation metrics (e.g., PSNR, SSIM) to threshold adjustments. To evaluate the optimization performance of RLTC-SCA, comprehensive numerical experiments were conducted on the CEC2020 and CEC2022 benchmark test suites. The proposed algorithm was compared with several mainstream metaheuristic algorithms. An ablation study was designed to analyze the individual contribution and synergistic effects of the three enhancement strategies. The convergence behavior was characterized using indicators such as average fitness value, search trajectory, and convergence curve. Moreover, statistical stability was assessed using the Wilcoxon rank-sum test (at a significance level of p = 0.05) and the Friedman test. Experimental results demonstrate that RLTC-SCA outperforms all comparison algorithms in terms of average fitness, convergence speed, and stability, ranking first on both benchmark test suites. Furthermore, RLTC-SCA was applied to multi-threshold image segmentation tasks, where the Otsu method was adopted as the objective function. Segmentation performance was evaluated on multiple benchmark images under four threshold levels (2, 4, 6, and 8) using PSNR, FSIM, and SSIM as evaluation metrics. The results indicate that RLTC-SCA can efficiently obtain optimal segmentation thresholds, with PSNR, FSIM, and SSIM values consistently higher than those of competing algorithms—demonstrating superior segmentation accuracy and robustness. This study provides a reliable solution for improving the efficiency and precision of multi-threshold image segmentation and enriches the application of intelligent optimization algorithms in the field of image processing. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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26 pages, 5928 KB  
Article
A Chaos-Initiated and Adaptive Multi-Guide Control-Based Crayfish Optimization Algorithm for Image Analysis
by Ziyang Shen, Zhe Sun, Yunrui Bi and Zhixin Sun
Symmetry 2025, 17(11), 1940; https://doi.org/10.3390/sym17111940 - 12 Nov 2025
Cited by 3 | Viewed by 629
Abstract
Image clustering analysis faces the curse of dimensionality, distance concentration, multimodal landscapes, and rapid diversity loss that challenge meta-heuristics. Meanwhile, the standard Crayfish Optimization Algorithm (COA) has shown notable potential but often suffers from poor convergence speed and premature convergence. To address these [...] Read more.
Image clustering analysis faces the curse of dimensionality, distance concentration, multimodal landscapes, and rapid diversity loss that challenge meta-heuristics. Meanwhile, the standard Crayfish Optimization Algorithm (COA) has shown notable potential but often suffers from poor convergence speed and premature convergence. To address these issues, this paper introduces a Chaos-initiated and Adaptive Multi-guide Control-based COA (CMCOA). First, a chaotic initialization strategy is employed by explicitly exploiting the reflection symmetry of logistic-map chaotic sequences together with opposition-based learning, which enhances population diversity and facilitates early exploration of promising regions. Second, a fitness-feedback adaptive parameter control mechanism, motivated by the general idea of the MIT rule, is integrated to dynamically balance exploration and exploitation, thereby accelerating convergence while mitigating premature stagnation. Furthermore, a multi-guide stage-switching strategy is designed to avoid being trapped in local optima by promoting adaptive transitions between exploration phases and exploitation phases. CMCOA is benchmarked against competing algorithms on ten challenging test functions drawn from CEC2017, CEC2019, CEC2020, and CEC2022 suites. We also conducted multispectral clustering, where class differences often lie in reflectance magnitude; we adopt Euclidean distance for its efficiency and suitability in capturing such variations. Compared with other algorithms, CMCOA shows faster convergence, higher accuracy, and improved robustness, revealing its broader potential for image analysis tasks. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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22 pages, 3412 KB  
Article
Fault Identification Method for Photovoltaic Power Grids Based on an Improved GABP Neural Network and Fuzzy System
by Xiaofeng Dong, Houtao Sun, Zhongxiu Han, Yuanchen Xia, Hongjun Wang and Qingwen Mou
Symmetry 2025, 17(9), 1476; https://doi.org/10.3390/sym17091476 - 7 Sep 2025
Viewed by 819
Abstract
Fault detection and classification localization in photovoltaic power grids is a key challenge in photovoltaic power systems. Due to the greater fluctuation of power data in photovoltaic power grids, traditional grid fault detection methods suffer from inefficiency, low accuracy, and inaccurate fault localization [...] Read more.
Fault detection and classification localization in photovoltaic power grids is a key challenge in photovoltaic power systems. Due to the greater fluctuation of power data in photovoltaic power grids, traditional grid fault detection methods suffer from inefficiency, low accuracy, and inaccurate fault localization in photovoltaic scenarios. In this paper, a fuzzy control technique combined with an improved GABP neural network is used to identify potential fault nodes in the photovoltaic distribution network. The symmetric crossover operator of the genetic algorithm and the symmetry constraints of the neural network weight matrix are used to improve the model’s ability to capture the symmetric fluctuation characteristics of photovoltaic data, while a classification module consisting of three fuzzy controllers is used for fault identification. The simulation results show that the recognition method proposed in this paper has good performance and the fault classification accuracy reaches 92.75%, which provides a practical reference value for the management of photovoltaic distribution network. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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25 pages, 2525 KB  
Article
Symmetry-Enhanced Locally Adaptive COA-ELM for Short-Term Load Forecasting
by Shiyu Dai, Zhe Sun and Zhixin Sun
Symmetry 2025, 17(8), 1335; https://doi.org/10.3390/sym17081335 - 15 Aug 2025
Viewed by 731
Abstract
Reliable short-term electricity usage prediction is essential for preserving the stability of topologically symmetric power networks and their dynamic supply–demand equilibrium. To tackle this challenge, this paper proposes a novel approach derived from the standard Extreme Learning Machine (ELM) by integrating an enhanced [...] Read more.
Reliable short-term electricity usage prediction is essential for preserving the stability of topologically symmetric power networks and their dynamic supply–demand equilibrium. To tackle this challenge, this paper proposes a novel approach derived from the standard Extreme Learning Machine (ELM) by integrating an enhanced Crayfish Optimization Algorithm (DSYCOA). This algorithm combines Logistic chaotic mapping, local precise search, and dynamic parameter adjustment strategies designed to achieve a dynamic balance between exploration and exploitation, thereby optimizing the initial thresholds and weights of the ELM. Consequently, a new short-term power load forecasting model, namely the DSYCOA-ELM model, is developed. Experimental validation demonstrates that the improved DSYCOA exhibits fast convergence speed and high convergence accuracy, and successfully harmonizes global exploration and local exploitation capabilities while maintaining an empirical balance between exploration and exploitation. To additionally verify the effectiveness of DSYCOA in improving ELM, this paper conducts simulation comparison experiments among six models, including DSYCOA-ELM, ELM, and ELM improved by BWO (BWO-ELM). The findings demonstrate that the DSYCOA-ELM model outperforms the other five forecasting models in terms of Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and other indicators. Specifically, in terms of MAPE, DSYCOA-ELM reduces the error by 96.9% compared to ELM. This model demonstrates feasibility and effectiveness in solving the problem of short-term power load prediction, providing critical support for maintaining the stability of grid topological symmetry and supply–demand balance. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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22 pages, 8629 KB  
Article
3D UAV Route Optimization in Complex Environments Using an Enhanced Artificial Lemming Algorithm
by Yuxuan Xie, Zhe Sun, Kai Yuan and Zhixin Sun
Symmetry 2025, 17(6), 946; https://doi.org/10.3390/sym17060946 - 13 Jun 2025
Cited by 5 | Viewed by 1348
Abstract
The use of UAVs for logistics delivery has become a hot topic in current research, and how to plan a reasonable delivery route is the key to the problem. Therefore, this paper proposes a multi-environment logistics delivery route planning model that is based [...] Read more.
The use of UAVs for logistics delivery has become a hot topic in current research, and how to plan a reasonable delivery route is the key to the problem. Therefore, this paper proposes a multi-environment logistics delivery route planning model that is based on UAVs, is characterized by a 3D environment model, and aims at the shortest delivery route with minimum flight undulation. In order to find the optimal route in various environments, a multi-strategy improved artificial lemming algorithm, which integrates the Cubic chaotic map initialization, double adaptive t-distribution perturbation, and population dynamic optimization, is proposed. The symmetric nature of the t-distribution ensures that the lemmings conduct extensive searches in both directions within the solution space, thus improving the convergence speed and preventing them from falling into local optimal solutions. Through data experiments and simulation analysis, the improved algorithm can be successfully applied to the 3D route planning model, and the route quality is superior. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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22 pages, 482 KB  
Article
A Novel Symmetrical Inertial Alternating Direction Method of Multipliers with Proximal Term for Nonconvex Optimization with Applications
by Ji-Hong Li, Heng-You Lan and Si-Yuan Lin
Symmetry 2025, 17(6), 887; https://doi.org/10.3390/sym17060887 - 5 Jun 2025
Viewed by 836
Abstract
In this paper, we propose a novel alternating direction method of multipliers based on acceleration technique involving two symmetrical inertial terms for a class of nonconvex optimization problems with a two-block structure. To address the nonconvex subproblem, we introduce a proximal term to [...] Read more.
In this paper, we propose a novel alternating direction method of multipliers based on acceleration technique involving two symmetrical inertial terms for a class of nonconvex optimization problems with a two-block structure. To address the nonconvex subproblem, we introduce a proximal term to reduce the difficulty of solving this subproblem. For the smooth subproblem, we employ a gradient descent method on the augmented Lagrangian function, which significantly reduces the computational complexity. Under appropriate assumptions, we prove subsequential convergence of the algorithm. Moreover, when the generated sequence is bounded and the auxiliary function satisfies Kurdyka–Łojasiewicz property, we establish global convergence of the algorithm. Finally, effectiveness and superior performance of the proposed algorithm are validated through numerical experiments in signal processing and smoothly clipped absolute deviation penalty problems. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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34 pages, 17954 KB  
Article
Unmanned Aerial Vehicle Path Planning Method Based on Improved Dung Beetle Optimization Algorithm
by Fengjun Lv, Yongbo Jian, Kai Yuan and Yubin Lu
Symmetry 2025, 17(3), 367; https://doi.org/10.3390/sym17030367 - 28 Feb 2025
Cited by 7 | Viewed by 2062
Abstract
To address the problem of UAV path planning in complex mountainous terrains, this paper comprehensively considers constraints such as natural mountain and obstacle collision threats, the shortest path, and flight altitude. We propose a more practical UAV path planning model that better reflects [...] Read more.
To address the problem of UAV path planning in complex mountainous terrains, this paper comprehensively considers constraints such as natural mountain and obstacle collision threats, the shortest path, and flight altitude. We propose a more practical UAV path planning model that better reflects the actual UAV path planning situation in complex mountainous areas. In order to solve this model, this paper improves the traditional dung beetle optimization (DBO) algorithm and proposes an improved dung beetle optimization (IDBO) algorithm. The IDBO algorithm optimizes the population initialization method based on the concept of symmetry, ensuring that the population is more evenly distributed within the solution space. Additionally, the algorithm introduces a sine–cosine function-based movement strategy, inspired by the symmetry principle, to enhance the search efficiency of individual population members. Furthermore, a population evolution strategy is incorporated to prevent the algorithm from getting stuck in local optima. To demonstrate the algorithm’s performance, tests were conducted using 23 commonly used benchmark functions provided by the CEC 2005 competition and six commonly used engineering problem models provided by the CEC 2020 competition. The results indicate that IDBO significantly outperforms DBO in terms of convergence performance, effectively solving various engineering optimization problems. Finally, experimental tests under three different threat scenarios show that the proposed IDBO algorithm has scientific validity when applied to UAV path planning. This solution method effectively reduces UAV flight energy consumption costs and obstacle collision threats while improving the efficiency and accuracy of UAV path planning. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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22 pages, 2930 KB  
Article
Type-2 Backstepping T-S Fuzzy Bionic Control Based on Niche Symmetry Function
by Yunli Hao, Maohua Wang, Jian Tang, Ziyue Zhang and Jiangling Xiong
Symmetry 2025, 17(1), 121; https://doi.org/10.3390/sym17010121 - 14 Jan 2025
Cited by 1 | Viewed by 1264
Abstract
Niche can reflect the changes in the quality of the ecological environment and the balance of ecological state. The more advanced the ecosystem, the more complex and higher-order nonlinearities and uncertainties that are presented. For such an uncertain parameter system with complex nonlinearity, [...] Read more.
Niche can reflect the changes in the quality of the ecological environment and the balance of ecological state. The more advanced the ecosystem, the more complex and higher-order nonlinearities and uncertainties that are presented. For such an uncertain parameter system with complex nonlinearity, backstepping fuzzy control is a good control method. When the backstepping control method is introduced into the Type-2 fuzzy T-S control principle, the equality index symmetry function composed of ecological factors is used as the backstepping control consequence, and the Lyapunov function is constructed to analyze the stability and find out the adaptive law of the ecological factors in the equality index symmetry function of the control consequence. This reflects that the individual organisms always develop in their own favorable direction, highlighting the bionic intelligent control of the method. Through simulation analysis, the Type-2 Backstepping control method is effective in stability and parameter tracking, which reflects the self-development ability and self-coordination ability of individual organisms, highlighting the physical background and symmetry of the bionic intelligent control of this method. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Optimization Algorithm and Its Applications)
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