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Advances in Analysis and Application of Mathematical Optimization Algorithms

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 11720

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Dr. Man-Fai Leung

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School of Computing and Information Science, Faculty of Science and Engineering, Anglia Ruskin University, Cambridge, UK
Interests: computational intelligence; neural networks; optimization
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Dr. Wenming Cao

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School of Mathematics and Statistics, Chongqing Jiaotong University, 6 Xuefu Blvd, Nan'An, Chongqing, China 400074
Interests: Data Science and Machine Learning; Applied Mathematics for Machine Learning; Medical Image Analysis
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Dr. Hangjun Che

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Guest Editor

College of Electronic and Information Engineering, Southwest University, Chongqing, China
Interests: computational intelligence; neural networks; optimization
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Special Issue Information

Dear Colleagues,

In mathematics, computer science and operations research, mathematical optimization (or mathematical programming) refers to a collection of methods and techniques used for solving an optimization problem such as minimizing or maximizing an objective function. Over the last several decades, mathematical optimization has drawn a lot of attention due to its significance in many real-world applications, such as business, management, and engineering. Solving an optimization problem exactly may be very difficult or even impossible in practice, but by applying non-traditional algorithms being one very flexible and successful possibility, can often find some high-quality solutions. Besides, by combining population-based optimization algorithms with improvement techniques such as local search strategies and individual learning procedures, the capability of the algorithms could be enhanced for refined solutions. These algorithms exhibit good performance on various benchmark problems and real-world applications. As with problem-dependent improvement techniques, generating optimal solutions by the aforementioned approaches poses several unique issues such as the algorithm design and analysis. Besides, some scholars pointed out that the performance comparison via a large number of experiment tests cannot reveal the real strengths and weaknesses of the optimization algorithms. In particular, a few recent studies have shown that the good performance of some algorithms depends on the special characteristics of the test problems. This Special Issue will accept original research and review articles on novel mathematical optimization techniques and their applications. We also welcome analysis and design of optimization test problems, as well as performance evaluation indicators.

Potential topics include but are not limited to the following:

Theoretical analyses of optimization algorithms
Novel techniques and their applications
Thorough analysis and comparison of existing optimization algorithms
Analysis and design of optimization test problems and performance evaluation indicators
Optimization methods and techniques in machine learning
Application mathematical optimization in big data analytics
Optimization of machine learning and deep learning models
Robust optimization algorithms and their applications
Optimization techniques for IoT Applications
Single and multi- objective optimization algorithms.

Dr. Man-Fai Leung
Dr. Wenming Cao
Dr. Hangjun Che
Guest Editors

Manuscript Submission Information

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Keywords

optimization
neural networks
swarm intelligence
multi-objective Optimization
machine learning
applications

Published Papers (11 papers)

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Research

22 pages, 3934 KiB

Open AccessArticle

A Bibliometric Analysis of a Genetic Algorithm for Supply Chain Agility

by Weng Hoe Lam, Weng Siew Lam and Pei Fun Lee

Mathematics 2024, 12(8), 1199; https://doi.org/10.3390/math12081199 - 17 Apr 2024

Viewed by 603

Abstract

As a famous population-based metaheuristic algorithm, a genetic algorithm can be used to overcome optimization complexities. A genetic algorithm adopts probabilistic transition rules and is suitable for parallelism, which makes this algorithm attractive in many areas, including the logistics and supply chain sector. To obtain a comprehensive understanding of the development in this area, this paper presents a bibliometric analysis on the application of a genetic algorithm in logistics and supply chains using data from 1991 to 2024 from the Web of Science database. The authors found a growing trend in the number of publications and citations over the years. This paper serves as an important reference to researchers by highlighting important research areas, such as multi-objective optimization, metaheuristics, sustainability issues in logistics, and machine learning integration. This bibliometric analysis also underlines the importance of Non-Dominated Sorting Genetic Algorithm II (NSGA-II), sustainability, machine learning, and variable neighborhood search in the application of a genetic algorithm in logistics and supply chains in the near future. The integration of a genetic algorithm with machine learning is also a potential research gap to be filled to overcome the limitations of genetic algorithms, such as the long computational time, difficulties in obtaining optimal solutions, and convergence issues for application in logistics and supply chains. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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15 pages, 1445 KiB

Open AccessArticle

A Patent Keyword Extraction Method Based on Corpus Classification

by Changjian Sun, Wentao Chen, Zhen Zhang and Tian Zhang

Mathematics 2024, 12(7), 1068; https://doi.org/10.3390/math12071068 - 2 Apr 2024

Viewed by 572

Abstract

The keyword extraction of patents is crucial for technicians to master the trends of technology. Traditional keyword extraction approaches only handle short text like title or claims, but ignore the comprehensive meaning of the description. This paper proposes a novel patent keyword extraction method based on corpus classification (PKECC), which simulates the patent understanding methods of human patent examiners. First of all, a corpus classification model based on multi-level attention mechanism adopts the Bert model and hierarchical attention mechanism to classify the sentences of patent description into four parts including technical field, technical problem, technical solution, and technical effect. Then, the proposed keyword extraction method based on the fusion of BiLSTM and CRF is incorporated to extract keywords from the four parts. The proposed PKECC simulates understanding style of patent examiner by extracting keywords from the description. Meanwhile, PKECC may reduce the complexity of extracting keywords from a long text and improve the accuracy of keyword extraction. The proposed PKECC is compared with 5 traditional or state-of-the-art models and achieves better accuracy, F1 score and recall rate; its recall rate is above 62%, its accuracy reaches over 84%, and the F1 score arrives at 69%. In addition, the experimental results shows the proposed PKECC has a better universality in keyword extraction. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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27 pages, 12022 KiB

Open AccessArticle

Optimizing the Three-Dimensional Multi-Objective of Feeder Bus Routes Considering the Timetable

by Xinhua Gao, Song Liu, Shan Jiang, Dennis Yu, Yong Peng, Xianting Ma and Wenting Lin

Mathematics 2024, 12(7), 930; https://doi.org/10.3390/math12070930 - 22 Mar 2024

Viewed by 589

Abstract

To optimize the evacuation process of rail transit passenger flows, the influence of the feeder bus network on bus demand is pivotal. This study first examines the transportation mode preferences of rail transit station passengers and addresses the feeder bus network’s optimization challenge within a three-dimensional framework, incorporating an elastic mechanism. Consequently, a strategic planning model is developed. Subsequently, a multi-objective optimization model is constructed to simultaneously increase passenger numbers and decrease both travel time costs and bus operational expenses. Due to the NP-hard nature of this optimization problem, we introduce an enhanced non-dominated sorting genetic algorithm, INSGA-II. This algorithm integrates innovative encoding and decoding rules, adaptive parameter adjustment strategies, and a combination of crowding distance and distribution entropy mechanisms alongside an external elite archive strategy to enhance population convergence and local search capabilities. The efficacy of the proposed model and algorithm is corroborated through simulations employing standard test functions and instances. The results demonstrate that the INSGA-II algorithm closely approximates the true Pareto front, attaining Pareto optimal solutions that are uniformly distributed. Additionally, an increase in the fleet size correlates with greater passenger volumes and higher operational costs, yet it substantially lowers the average travel cost per customer. An optimal fleet size of 11 vehicles is identified. Moreover, expanding feeder bus routes enhances passenger counts by 18.03%, raises operational costs by 32.33%, and cuts passenger travel time expenses by 21.23%. These findings necessitate revisions to the bus timetable. Therefore, for a bus network with elastic demand, it is essential to holistically optimize the actual passenger flow demand, fleet size, bus schedules, and departure frequencies. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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16 pages, 1978 KiB

Open AccessFeature PaperArticle

An Enhanced FCM Clustering Method Based on Multi-Strategy Tuna Swarm Optimization

by Changkang Sun, Qinglong Shao, Ziqi Zhou and Junxiao Zhang

Mathematics 2024, 12(3), 453; https://doi.org/10.3390/math12030453 - 31 Jan 2024

Viewed by 607

Abstract

To overcome the shortcoming of the Fuzzy C-means algorithm (FCM)—that it is easy to fall into local optima due to the dependence of sub-spatial clustering on initialization—a Multi-Strategy Tuna Swarm Optimization-Fuzzy C-means (MSTSO-FCM) algorithm is proposed. Firstly, a chaotic local search strategy and an offset distribution estimation strategy algorithm are proposed to improve the performance, enhance the population diversity of the Tuna Swarm Optimization (TSO) algorithm, and avoid falling into local optima. Secondly, the search and development characteristics of the MSTSO algorithm are introduced into the fuzzy matrix of Fuzzy C-means (FCM), which overcomes the defects of poor global searchability and sensitive initialization. Not only has the searchability of the Multi-Strategy Tuna Swarm Optimization algorithm been employed, but the fuzzy mathematical ideas of FCM have been retained, to improve the clustering accuracy, stability, and accuracy of the FCM algorithm. Finally, two sets of artificial datasets and multiple sets of the University of California Irvine (UCI) datasets are used to do the testing, and four indicators are introduced for evaluation. The results show that the MSTSO-FCM algorithm has better convergence speed than the Tuna Swarm Optimization Fuzzy C-means (TSO-FCM) algorithm, and its accuracies in the heart, liver, and iris datasets are 89.46%, 63.58%, 98.67%, respectively, which is an outstanding improvement. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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33 pages, 6128 KiB

Open AccessArticle

Mathematical Models for the Design of GRID Systems to Solve Resource-Intensive Problems

by Valeriya V. Tynchenko, Vadim S. Tynchenko, Vladimir A. Nelyub, Vladimir V. Bukhtoyarov, Aleksey S. Borodulin, Sergei O. Kurashkin, Andrei P. Gantimurov and Vladislav V. Kukartsev

Mathematics 2024, 12(2), 276; https://doi.org/10.3390/math12020276 - 15 Jan 2024

Cited by 3 | Viewed by 773

Abstract

Artificial neural networks are successfully used to solve a wide variety of scientific and technical problems. The purpose of the study is to increase the efficiency of distributed solutions for problems involving structural-parametric synthesis of neural network models of complex systems based on GRID (geographically disperse computing resources) technology through the integrated application of the apparatus of evolutionary optimization and queuing theory. During the course of the research, the following was obtained: (i) New mathematical models for assessing the performance and reliability of GRID systems; (ii) A new multi-criteria optimization model for designing GRID systems to solve high-resource computing problems; and (iii) A new decision support system for the design of GRID systems using a multi-criteria genetic algorithm. Fonseca and Fleming’s genetic algorithm with a dynamic penalty function was used as a method for solving the stated multi-constrained optimization problem. The developed program system was used to solve the problem of choosing an effective structure of a centralized GRID system that was configured to solve the problem of structural-parametric synthesis of neural network models. To test the proposed approach, a Pareto-optimal configuration of the GRID system was built with the following characteristics: average performance–103.483 GFLOPS, cost–500 rubles per day, availability rate–99.92%, and minimum performance–51 GFLOPS. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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15 pages, 18490 KiB

Open AccessArticle

GA-KELM: Genetic-Algorithm-Improved Kernel Extreme Learning Machine for Traffic Flow Forecasting

by Wenguang Chai, Yuexin Zheng, Lin Tian, Jing Qin and Teng Zhou

Mathematics 2023, 11(16), 3574; https://doi.org/10.3390/math11163574 - 18 Aug 2023

Cited by 12 | Viewed by 1202

Abstract

A prompt and precise estimation of traffic conditions on the scale of a few minutes by analyzing past data is crucial for establishing an effective intelligent traffic management system. Nevertheless, because of the irregularity and nonlinear features of traffic flow data, developing a prediction model with excellent robustness poses a significant obstacle. Therefore, we propose genetic-search-algorithm-improved kernel extreme learning machine, termed GA-KELM, to unleash the potential of improved prediction accuracy and generalization performance. By substituting the inner product with a kernel function, the accuracy of short-term traffic flow forecasting using extreme learning machines is enhanced. The genetic algorithm evades manual traversal of all possible parameters in searching for the optimal solution. The prediction performance of GA-KELM is evaluated on eleven benchmark datasets and compared with several state-of-the-art models. There are four benchmark datasets from the A1, A2, A4, and A8 highways near the ring road of Amsterdam, and the others are D1, D2, D3, D4, D5, D6, and P, close to Heathrow airport on the M25 expressway. On A1, A2, A4, and A8, the RMSEs of the GA-KELM model are 284.67 vehs/h, 193.83 vehs/h, 220.89 vehs/h, and 163.02 vehs/h, respectively, while the MAPEs of the GA-KELM model are 11.67%, 9.83%, 11.31%, and 12.59%, respectively. The results illustrate that the GA-KELM model is obviously superior to state-of-the-art models. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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18 pages, 20871 KiB

Open AccessArticle

Robust Low-Rank Graph Multi-View Clustering via Cauchy Norm Minimization

by Xinyu Pu, Baicheng Pan and Hangjun Che

Mathematics 2023, 11(13), 2940; https://doi.org/10.3390/math11132940 - 30 Jun 2023

Cited by 2 | Viewed by 940

Abstract

Graph-based multi-view clustering methods aim to explore the partition patterns by utilizing a similarity graph. However, many existing methods construct a consensus similarity graph based on the original multi-view space, which may result in the lack of information on the underlying low-dimensional space. Additionally, these methods often fail to effectively handle the noise present in the graph. To address these issues, a novel graph-based multi-view clustering method which combines spectral embedding, non-convex low-rank approximation and noise processing into a unit framework is proposed. In detail, the proposed method constructs a tensor by stacking the inner product of normalized spectral embedding matrices obtained from each similarity matrix. Then, the obtained tensor is decomposed into a low-rank tensor and a noise tensor. The low-rank tensor is constrained via nonconvex low-rank tensor approximation and a novel Cauchy norm with an upper bound is proposed to handle the noise. Finally, we derive the consensus similarity graph from the denoised low-rank tensor. The experiments on five datasets demonstrate that the proposed method outperforms other state-of-the-art methods on five datasets. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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17 pages, 4080 KiB

Open AccessArticle

Active Learning: Encoder-Decoder-Outlayer and Vector Space Diversification Sampling

by Hongyi Zeng and Fanyi Kong

Mathematics 2023, 11(13), 2819; https://doi.org/10.3390/math11132819 - 23 Jun 2023

Viewed by 1844

Abstract

This study introduces a training pipeline comprising two components: the Encoder-Decoder-Outlayer framework and the Vector Space Diversification Sampling method. This framework efficiently separates the pre-training and fine-tuning stages, while the sampling method employs pivot nodes to divide the subvector space and selectively choose unlabeled data, thereby reducing the reliance on human labeling. The pipeline offers numerous advantages, including rapid training, parallelization, buffer capability, flexibility, low GPU memory usage, and a sample method with nearly linear time complexity. Experimental results demonstrate that models trained with the proposed sampling algorithm generally outperform those trained with random sampling on small datasets. These characteristics make it a highly efficient and effective training approach for machine learning models. Further details can be found in the project repository on GitHub. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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19 pages, 1104 KiB

Open AccessArticle

Swarm Robots Search for Multiple Targets Based on Historical Optimal Weighting Grey Wolf Optimization

by Qian Zhu, Yongqing Li and Zhen Zhang

Mathematics 2023, 11(12), 2630; https://doi.org/10.3390/math11122630 - 8 Jun 2023

Viewed by 789

Abstract

This study investigates the problem of swarm robots searching for multiple targets in an unknown environment. We propose the Historical Optimal Weighting Grey Wolf Optimization (HOWGWO) algorithm based on an improved grouping strategy. In the HOWGWO algorithm, we gather and update every individual grey wolf’s historical optimal position and rank grey wolves based on the merit of their historical optimal position. The position of the prey is dynamically estimated by the leader wolf, and all grey wolves move towards the prey’s estimated position. To solve the multi-target problem of swarm robots search, we integrate the HOWGWO algorithm with an improved grouping strategy and divide the algorithm into two stages: the random walk stage and the dynamic grouping stage. During the random walk stage, grey wolves move randomly and update their historical optimal positions. During the dynamic grouping stage, the HOWGWO algorithm generates search auxiliary points (SAPs) by adopting an improved grouping strategy based on individual grey wolves’ historical optimal positions. These SAPs are then utilized for grouping grey wolves to search for different prey. The SAPs are re-generated using the optimum historical positions of every single grey wolf after positions have been updated, rather than just those belonging to a specific group. The effectiveness of the proposed HOWGWO algorithm is extensively assessed in 30 dimensions using the CEC 2017 test suite, which simulates unimodal, multimodal, hybrid, and composition problems. Then, the obtained results are compared with competitors, including GWO, PSO and EGWO, and the results are statistically analyzed through Friedman’s test. Ultimately, simulations are performed to simulate the problem of searching multiple targets by swarm robots in a real environment. The experimental results and statistical analysis confirm that the proposed HOWGWO algorithm has a fast convergence speed and solution quality for solving global optimization problems and swarm robots searching multiple targets problems. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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21 pages, 805 KiB

Open AccessArticle

Dispatch for a Continuous-Time Microgrid Based on a Modified Differential Evolution Algorithm

by Lei Zhang and Rui Tang

Mathematics 2023, 11(2), 271; https://doi.org/10.3390/math11020271 - 4 Jan 2023

Cited by 1 | Viewed by 980

Abstract

The carbon trading mechanism is proposed to remit global warming and it can be considered in a microgrid. There is a lack of continuous-time methods in a microgrid, so a continuous-time model is proposed and solved by differential evolution (DE) in this work. This research aims to create effective methods to obtain some useful results in a microgrid. Batteries, microturbines, and the exchange with the main grid are considered. Considering carbon trading, the objective function is the sum of a quadratic function and an absolute value function. In addition, a composite electricity price model has been put forward to conclude the common kinds of electricity prices. DE is utilized to solve the constrained optimization problems (COPs) proposed in this paper. A modified DE is raised in this work, which uses multiple mutation and selection strategies. In the case study, the proposed algorithm is compared with the other seven algorithms and the outperforming one is selected to compare two different types of electricity prices. The results show the proposed algorithm performs best. The Wilcoxon Signed Rank Test is also used to verify its significant superiority. The other result is that time-of-use pricing (ToUP) is economic in the off-peak period while inclining block rates (IBRs) are economic in the peak and shoulder periods. The composite electricity price model can be applied in social production and life. In addition, the proposed algorithm puts forward a new variety of DE and enriches the theory of DE. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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26 pages, 2073 KiB

Open AccessArticle

Recurrent Neural Network Models Based on Optimization Methods

by Predrag S. Stanimirović, Spyridon D. Mourtas, Vasilios N. Katsikis, Lev A. Kazakovtsev and Vladimir N. Krutikov

Mathematics 2022, 10(22), 4292; https://doi.org/10.3390/math10224292 - 16 Nov 2022

Cited by 3 | Viewed by 1472

Abstract

Many researchers have addressed problems involving time-varying (TV) general linear matrix equations (GLMEs) because of their importance in science and engineering. This research discusses and solves the topic of solving TV GLME using the zeroing neural network (ZNN) design. Five new ZNN models based on novel error functions arising from gradient-descent and Newton optimization methods are presented and compared to each other and to the standard ZNN design. Pseudoinversion is involved in four proposed ZNN models, while three of them are related to Newton’s optimization method. Heterogeneous numerical examples show that all models successfully solve TV GLMEs, although their effectiveness varies and depends on the input matrix. Full article

(This article belongs to the Special Issue Advances in Analysis and Application of Mathematical Optimization Algorithms)

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