Mathematics

Research

18 pages, 1326 KB

Open AccessArticle

Distributed Generalized Nash Equilibrium Seeking for Constrained Population Games via Consensus-Based Revision Protocols

by Jiajia Liu, Xuelei Fu and Ning Jiang

Mathematics 2026, 14(10), 1770; https://doi.org/10.3390/math14101770 - 21 May 2026

Viewed by 119

Abstract

This study addresses distributed decision-making in multi-agent systems under shared constraints. Existing methods often fail to guarantee strict constraint satisfaction or require sensitive parameter tuning. We propose a novel algorithm that integrates a revision protocol with a consensus mechanism. The key innovation is [...] Read more.

This study addresses distributed decision-making in multi-agent systems under shared constraints. Existing methods often fail to guarantee strict constraint satisfaction or require sensitive parameter tuning. We propose a novel algorithm that integrates a revision protocol with a consensus mechanism. The key innovation is a built-in, parameter-free constraint-checking function within the revision protocol, which automatically halts infeasible strategy updates. This approach enables agents using only local neighbor communication to seek a Generalized Nash Equilibrium (GNE). Theoretical analysis proves that the algorithm converges exponentially. Extensive simulations demonstrate its superiority: it achieves faster convergence and ensures strict per-iteration constraint satisfaction, significantly outperforming traditional gradient descent and penalty-based methods across various network topologies. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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35 pages, 1784 KB

Open AccessArticle

Hierarchical Multi-Population Cooperative Evolutionary Approach for Constrained Multi-Objective Optimization

by Xinyue Xiang, Xinyuan Gu, Jiaqi Li and Siyuan Zhao

Mathematics 2026, 14(5), 786; https://doi.org/10.3390/math14050786 - 26 Feb 2026

Viewed by 522

Abstract

To address the complexity of multi-constraint multi-objective optimization problems (mCMOPs), this paper proposes a novel multi-population evolutionary algorithm (MOEA). Multi-objective optimization problems (MOPs) are ubiquitous in scientific and engineering fields, while the introduction of multiple complex constraints significantly increases the difficulty of finding [...] Read more.

To address the complexity of multi-constraint multi-objective optimization problems (mCMOPs), this paper proposes a novel multi-population evolutionary algorithm (MOEA). Multi-objective optimization problems (MOPs) are ubiquitous in scientific and engineering fields, while the introduction of multiple complex constraints significantly increases the difficulty of finding solutions. To tackle this challenge, this work systematically analyzes the intrinsic relationships among the Single-Constraint Pareto Front (SCPF), Sub-Constraint Pareto Front (SSCPF), Unconstrained Pareto Front (UPF), and the Final Constrained Pareto Front (FCPF), and it investigates how these relationships can be leveraged to effectively enhance optimization performance. Based on this analysis, a Hierarchical Multi-Population Cooperative Evolutionary Approach (HMP-CE) is proposed. The approach constructs

C + 2

populations (where C represents the number of constraints) to search the UPF, SCPF, and SSCPF at appropriate stages, thereby driving the final solution approximation in a hierarchical manner. Meanwhile, HMP-CE introduces the following two key mechanisms: (1) the Population Activation–Dormancy Regulation (PADR) mechanism, which adaptively regulates the activation and dormancy of populations to reduce computational cost and accelerate convergence; (2) the Constraint Combination Timing Identification (CCTI) mechanism, which identifies suitable moments to jointly solve selected constraints in SSCPF, thereby enhancing cooperative efficiency among populations. Experimental results on 37 benchmark mCMOPs, and six real-world engineering problems demonstrate that the proposed algorithm exhibits superior performance in terms of convergence, feasibility, and solution diversity, providing a competitive approach for solving complex multi-constraint optimization problems. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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27 pages, 1280 KB

Open AccessFeature PaperArticle

Two-Stage Genetic-Based Optimization for Resource Provisioning and Scheduling of Multiple Workflows on the Cloud Under Resource Constraints

by Feng Li, Wen Jun Tan, Moongi Seok and Wentong Cai

Mathematics 2026, 14(2), 213; https://doi.org/10.3390/math14020213 - 6 Jan 2026

Viewed by 586

Abstract

Resource provisioning and scheduling are essential challenges in handling multiple workflow requests within cloud environments, particularly given the constraints imposed by limited resource availability. Although workflow scheduling has been extensively studied, few methods effectively integrate resource provisioning with scheduling, especially under cloud resource [...] Read more.

Resource provisioning and scheduling are essential challenges in handling multiple workflow requests within cloud environments, particularly given the constraints imposed by limited resource availability. Although workflow scheduling has been extensively studied, few methods effectively integrate resource provisioning with scheduling, especially under cloud resource limitations and the complexities of multiple workflows. To address this challenge, we propose an innovative two-stage genetic-based optimization approach. In the first stage, candidate cloud resources are selected for the resource pool under the given resource constraints. In the second stage, these resources are provisioned and task scheduling is optimized on the selected resources. A key advantage of our approach is that it reduces the search space in the first stage through a novel encoding scheme that enables a caching strategy, in which intermediate results are stored and reused to enhance optimization efficiency in the second stage. The proposed solution is evaluated through extensive simulation experiments, assessing both resource selection and task scheduling across a diverse range of workflows. The results demonstrate that the proposed approach outperforms existing algorithms, particularly for highly parallel workflows, highlighting its effectiveness in managing complex workflow scheduling under resource-constrained cloud environments. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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40 pages, 40982 KB

Open AccessArticle

Improved Enterprise Development Optimization with Historical Trend Updating for High-Precision Photovoltaic Model Parameter Estimation

by Zhiping Li, Yi Liao and Haoxiang Zhou

Mathematics 2026, 14(1), 121; https://doi.org/10.3390/math14010121 - 28 Dec 2025

Cited by 3 | Viewed by 606

Abstract

Accurate parameter estimation of photovoltaic (PV) models is fundamentally a challenging nonlinear optimization problem, characterized by strong nonlinearity, high dimensionality, and multiple local optima. These characteristics significantly hinder the convergence accuracy, stability, and efficiency of conventional metaheuristic algorithms when applied to PV parameter [...] Read more.

Accurate parameter estimation of photovoltaic (PV) models is fundamentally a challenging nonlinear optimization problem, characterized by strong nonlinearity, high dimensionality, and multiple local optima. These characteristics significantly hinder the convergence accuracy, stability, and efficiency of conventional metaheuristic algorithms when applied to PV parameter identification. Although the enterprise development (ED) optimization algorithm has shown promising performance in various optimization tasks, it still suffers from slow convergence, limited solution precision, and poor robustness in complex PV parameter estimation scenarios. To overcome these limitations, this paper proposes a multi-strategy enhanced enterprise development (MEED) optimization algorithm for high-precision PV model parameter estimation. In MEED, a hybrid initialization strategy combining chaotic mapping and adversarial learning is designed to enhance population diversity and improve the quality of initial solutions. Furthermore, a historical trend-guided position update mechanism is introduced to exploit accumulated search information and accelerate convergence toward the global optimum. In addition, a mirror-reflection boundary control strategy is employed to maintain population diversity and effectively prevent premature convergence. The proposed MEED algorithm is first evaluated on the IEEE CEC2017 benchmark suite, where it is compared with 11 state-of-the-art metaheuristic algorithms under 30-, 50-, and 100-dimensional settings. Quantitative experimental results demonstrate that MEED achieves superior solution accuracy, faster convergence speed, and stronger robustness, yielding lower mean fitness values and smaller standard deviations on the majority of test functions. Statistical analyses based on Wilcoxon rank-sum and Friedman tests further confirm the significant performance advantages of MEED. Moreover, MEED is applied to the parameter estimation of single-diode and double-diode PV models using real measurement data. The results show that MEED consistently attains lower root mean square error (RMSE) and integrated absolute error (IAE) than existing methods while exhibiting more stable convergence behavior. These findings demonstrate that MEED provides an efficient and reliable optimization framework for PV model parameter estimation and other complex engineering optimization problems. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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17 pages, 825 KB

Open AccessArticle

Research on Spare Parts Configuration Method for Marine Equipment Based on Spare Parts Utilization Rate

by Zhengxuan Gu, Yali Zhai and Songshi Shao

Mathematics 2026, 14(1), 35; https://doi.org/10.3390/math14010035 - 22 Dec 2025

Viewed by 420

Abstract

An optimization method for the spare parts configuration of marine equipment is investigated based on spare parts support probability and utilization rate indicators. First, a general expression for the spare parts utilization rate is presented, and analytical expressions for the spare parts support [...] Read more.

An optimization method for the spare parts configuration of marine equipment is investigated based on spare parts support probability and utilization rate indicators. First, a general expression for the spare parts utilization rate is presented, and analytical expressions for the spare parts support probability and utilization rate are derived for Gamma-type spare parts with non-exponential life distributions, which are commonly found in marine spare parts. The relationship between two calculation methods for the utilization rate of exponential-type spare parts, as a special case of Gamma-type spare parts, is discussed. Second, as the analytical expression for the Gamma-type spare parts utilization rate is relatively complex, an approximate calculation method for the spare parts utilization rate is provided. Under the condition of given support probability requirements, the spare parts utilization rate is calculated through three approaches: theoretical calculation, approximate calculation, and simulation experiment. The calculation results demonstrate the validity of the analytical expression for the spare parts utilization rate and the applicability of the approximate algorithm. Furthermore, an approximate algorithm for the Weibull-type spare parts utilization rate is presented and verified through simulation calculations. Subsequently, considering both spare parts support probability requirements and utilization rate requirements, a spare parts configuration optimization model is established with the objective of maximizing the cost ratio, and the calculation procedures are provided. Finally, the feasibility of the spare parts configuration optimization model is illustrated through case analysis. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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17 pages, 1742 KB

Open AccessFeature PaperArticle

Hessian-Enhanced Likelihood Optimization for Gravitational Wave Parameter Estimation: A Second-Order Approach to Machine Learning-Based Inference

by Zhuopeng Peng and Fan Zhang

Mathematics 2025, 13(24), 4014; https://doi.org/10.3390/math13244014 - 17 Dec 2025

Viewed by 805

Abstract

We introduce a new method for estimating gravitational wave parameters. This approach uses a second-order likelihood optimization framework built into a machine learning system (JimGW). Current methods often rely on first-order approximations, which can miss important details, while our method incorporates the full [...] Read more.

We introduce a new method for estimating gravitational wave parameters. This approach uses a second-order likelihood optimization framework built into a machine learning system (JimGW). Current methods often rely on first-order approximations, which can miss important details, while our method incorporates the full Hessian matrix of the likelihood function. This allows us to better capture the shape of the parameter space for gravitational waves. Our theoretical framework demonstrates that the trace of the Hessian matrix, when properly normalized, provides a coordinate-invariant measure of the local likelihood geometry that significantly enhances parameter recovery accuracy for gravitational wave sources. We test our second-order method using data from the three gravitational wave events. Take GW150914 as an example; the results show large gains in precision for parameter estimation, with accuracy gains exceeding 93% across all inferred parameters compared to standard first-order implementations. We use Jensen–Shannon divergence to compare the resulting posterior distributions. The JSD values range from 0.366 to 0.948, which correlate directly with improved parameter recovery as validated through injection studies. The method remains computationally efficient with only a 20% increase in runtime. At the same time, it produces seven times more effective samples. Our results show that machine learning methods using only first-order information can lead to systematic errors in gravitational wave parameter estimation. The incorporation of second-order corrections emerges not as an optional refinement but as a necessary component for achieving theoretically optimal inference. It also matters for ongoing gravitational wave analyses, future detector networks, and the broader application of machine learning methods in precision scientific measurement. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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21 pages, 2695 KB

Open AccessArticle

Ship Spare Parts Optimization Model Based on Ideal Point Method and Improved Ant Colony Algorithm

by Tianyu Ma, Huiling Sun, Rui Qi and Xiangjun Li

Mathematics 2025, 13(23), 3862; https://doi.org/10.3390/math13233862 - 2 Dec 2025

Cited by 3 | Viewed by 554

Abstract

This study proposes an optimization approach for ship spare parts allocation by integrating the ideal point method (IPM) with an improved ant colony algorithm. The traditional R/C (Reliability–Cost Ratio) model is constrained by single-objective formulations that fail to reconcile cost efficiency with system [...] Read more.

This study proposes an optimization approach for ship spare parts allocation by integrating the ideal point method (IPM) with an improved ant colony algorithm. The traditional R/C (Reliability–Cost Ratio) model is constrained by single-objective formulations that fail to reconcile cost efficiency with system reliability, often producing the paradoxical result that fewer spare parts correspond to higher reliability. To address this limitation, a multi-objective model was constructed for reliability–cost optimization, while the enhanced ant colony algorithm identifies optimal spare part configurations that achieve balanced trade-offs. Nine representative scenarios were analyzed, with simulation outcomes compared between the IPM and R/C model approaches. Sensitivity analyses of critical parameters were conducted, and the effectiveness of both approaches was evaluated. The results demonstrate that the IPM consistently achieves higher reliability, particularly under stringent reliability requirements and tighter spare parts constraints. The findings provide a robust analytical foundation for evidence-based decision-making in ship equipment support. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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19 pages, 498 KB

Open AccessArticle

Optimization of Direct Convolution Algorithms on ARM Processors for Deep Learning Inference

by Shang Li, Fei Yu, Shankou Zhang, Huige Yin and Hairong Lin

Mathematics 2025, 13(5), 787; https://doi.org/10.3390/math13050787 - 27 Feb 2025

Cited by 2 | Viewed by 3296

Abstract

In deep learning, convolutional layers typically bear the majority of the computational workload and are often the primary contributors to performance bottlenecks. The widely used convolution algorithm is based on the IM2COL transform to take advantage of the highly optimized GEMM (General Matrix [...] Read more.

In deep learning, convolutional layers typically bear the majority of the computational workload and are often the primary contributors to performance bottlenecks. The widely used convolution algorithm is based on the IM2COL transform to take advantage of the highly optimized GEMM (General Matrix Multiplication) kernel acceleration, using the highly optimized BLAS (Basic Linear Algebra Subroutine) library, which tends to incur additional memory overhead. Recent studies have indicated that direct convolution approaches can outperform traditional convolution implementations without additional memory overhead. In this paper, we propose a high-performance implementation of the direct convolution algorithm for inference that preserves the channel-first data layout of the convolutional layer inputs/outputs. We evaluate the performance of our proposed algorithm on a multi-core ARM CPU platform and compare it with state-of-the-art convolution optimization techniques. Experimental results demonstrate that our new algorithm performs better across the evaluated scenarios and platforms. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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46 pages, 9513 KB

Open AccessArticle

Multi-Strategy Improved Binary Secretarial Bird Optimization Algorithm for Feature Selection

by Fuqiang Chen, Shitong Ye, Jianfeng Wang and Jia Luo

Mathematics 2025, 13(4), 668; https://doi.org/10.3390/math13040668 - 18 Feb 2025

Cited by 5 | Viewed by 1675

Abstract

With the rapid development of large model technology, data storage as well as collection is very important to improve the accuracy of model training, and Feature Selection (FS) methods can greatly eliminate redundant features in the data warehouse and improve the interpretability of [...] Read more.

With the rapid development of large model technology, data storage as well as collection is very important to improve the accuracy of model training, and Feature Selection (FS) methods can greatly eliminate redundant features in the data warehouse and improve the interpretability of the model, which makes it particularly important in the field of large model training. In order to better reduce redundant features in data warehouses, this paper proposes an enhanced Secretarial Bird Optimization Algorithm (SBOA), called BSFSBOA, by combining three learning strategies. First, for the problem of insufficient algorithmic population diversity in SBOA, the best-rand exploration strategy is proposed, which utilizes the randomness and optimality of random individuals as well as optimal individuals to effectively improve the population diversity of the algorithm. Second, to address the imbalance in the exploration/exploitation phase of SBOA, the segmented balance strategy is proposed to improve the balance by segmenting the individuals in the population, targeting individuals of different natures with different degrees of exploration and exploitation performance, and improving the quality of the FS subset when the algorithm is solved. Finally, for the problem of insufficient exploitation performance of SBOA, a four-role exploitation strategy is proposed, which strengthens the effective exploitation ability of the algorithm and enhances the classification accuracy of the FS subset by different degrees of guidance through the four natures of individuals in the population. Subsequently, the proposed BSFSBOA-based FS method is applied to solve 36 FS problems involving low, medium, and high dimensions, and the experimental results show that, compared to SBOA, BSFSBOA improves the performance of classification accuracy by more than 60%, also ranks first in feature subset size, obtains the least runtime, and confirms that the BSFSBOA-based FS method is a robust FS method with efficient solution performance, high stability, and high practicality. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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22 pages, 1918 KB

Open AccessArticle

Research on Multi-Center Path Optimization for Emergency Events Based on an Improved Particle Swarm Optimization Algorithm

by Zeyu Zou, Hui Zeng, Xiaodong Zheng and Junming Chen

Mathematics 2025, 13(4), 654; https://doi.org/10.3390/math13040654 - 17 Feb 2025

Cited by 4 | Viewed by 1655

Abstract

Emergency events pose critical challenges to national and social stability, requiring efficient and timely responses to mitigate their impact. In the initial stages of an emergency, decision-makers face the dual challenge of minimizing transportation costs while adhering to stringent rescue time constraints. To [...] Read more.

Emergency events pose critical challenges to national and social stability, requiring efficient and timely responses to mitigate their impact. In the initial stages of an emergency, decision-makers face the dual challenge of minimizing transportation costs while adhering to stringent rescue time constraints. To address these issues, this study proposes a two-stage optimization model aimed at ensuring the equitable distribution of disaster relief materials across multiple distribution centers. The model seeks to minimize the overall cost, encompassing vehicle dispatch expenses, fuel consumption, and time window penalty costs, thereby achieving a balance between efficiency and fairness. To solve this complex optimization problem, a hybrid algorithm combining genetic algorithms and particle swarm optimization was designed. This hybrid approach leverages the global exploration capability of genetic algorithms and the fast convergence of particle swarm optimization to achieve superior performance in solving real-world logistics challenges. Case studies were conducted to evaluate the feasibility and effectiveness of both the proposed model and the algorithm. Results indicate that the model accurately reflects the dynamics of emergency logistics operations, while the hybrid algorithm exhibits strong local optimization capabilities and robust performance in handling diverse and complex scenarios. Experimental findings underscore the potential of the proposed approach in optimizing emergency response logistics. The hybrid algorithm consistently achieves significant reductions in total cost while maintaining fairness in material distribution. These results demonstrate the algorithm’s applicability to a wide range of disaster scenarios, offering a reliable and efficient tool for emergency planners. This study not only contributes to the body of knowledge in emergency logistics optimization but also provides practical insights for policymakers and practitioners striving to improve disaster response strategies. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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37 pages, 9637 KB

Open AccessFeature PaperArticle

An Optimized Method for Solving the Green Permutation Flow Shop Scheduling Problem Using a Combination of Deep Reinforcement Learning and Improved Genetic Algorithm

by Yongxin Lu, Yiping Yuan, Jiarula Yasenjiang, Adilanmu Sitahong, Yongsheng Chao and Yunxuan Wang

Mathematics 2025, 13(4), 545; https://doi.org/10.3390/math13040545 - 7 Feb 2025

Cited by 8 | Viewed by 3197

Abstract

This paper tackles the green permutation flow shop scheduling problem (GPFSP) with the goal of minimizing both the maximum completion time and energy consumption. It introduces a novel hybrid approach that combines end-to-end deep reinforcement learning with an improved genetic algorithm. Firstly, the [...] Read more.

This paper tackles the green permutation flow shop scheduling problem (GPFSP) with the goal of minimizing both the maximum completion time and energy consumption. It introduces a novel hybrid approach that combines end-to-end deep reinforcement learning with an improved genetic algorithm. Firstly, the PFSP is modeled using an end-to-end deep reinforcement learning (DRL) approach, named PFSP_NET, which is designed based on the characteristics of the PFSP, with the actor–critic algorithm employed to train the model. Once trained, this model can quickly and directly produce relatively high-quality solutions. Secondly, to further enhance the quality of the solutions, the outputs from PFSP_NET are used as the initial population for the improved genetic algorithm (IGA). Building upon the traditional genetic algorithm, the IGA utilizes three crossover operators, four mutation operators, and incorporates hamming distance, effectively preventing the algorithm from prematurely converging to local optimal solutions. Then, to optimize energy consumption, an energy-saving strategy is proposed that reasonably adjusts the job scheduling order by shifting jobs backward without increasing the maximum completion time. Finally, extensive experimental validation is conducted on the 120 test instances of the Taillard standard dataset. By comparing the proposed method with algorithms such as the standard genetic algorithm (SGA), elite genetic algorithm (EGA), hybrid genetic algorithm (HGA), discrete self-organizing migrating algorithm (DSOMA), discrete water wave optimization algorithm (DWWO), and hybrid monkey search algorithm (HMSA), the results demonstrate the effectiveness of the proposed method. Optimal solutions are achieved in 28 test instances, and the latest solutions are updated in instances Ta005 and Ta068 with values of 1235 and 5101, respectively. Additionally, experiments on 30 instances, including Taillard 20-10, Taillard 50-10, and Taillard 100-10, indicate that the proposed energy strategy can effectively reduce energy consumption. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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