Editorial for the Special Issue “Advances in Machine Learning and Mathematical Modeling for Optimization Problems”

Machine learning and deep learning have made tremendous progress over the last decade and have become the de facto standard across a wide range of image, video, text, and sound processing domains, from object recognition to image generation. Recently, deep learning and deep reinforcement learning have begun to develop end-to-end training to solve more complex operation research and combinatorial optimization problems, such as covering problems, vehicle routing problems, traveling salesman problems, scheduling problems, and other complex problems requiring general simulations. These methods also sometimes include classic search and optimization algorithms for machine learning, such as Monte Carlo Tree Search in AlphaGO.

Starting from the above considerations, this Special Issue aims to report the latest advances and trends concerning advanced machine learning and mathematical modeling for optimization problems. This Special Issue intends to provide a universally recognized international forum to present recent advances in mathematical modeling for optimization problems. We welcomed both theoretical contributions as well as papers describing interesting applications. Papers invited for this Special Issue considered aspects of this problem, including:

• Machine learning for optimization problems;
• Statistical learning;
• End-to-end machine learning;
• Graph neural networks;
• Combining classic optimization algorithms and machine learning;
• Mathematical models of problems for machine learning;
• Optimization method for machine learning;
• Evolutionary computation and optimization problems;
• Applications such as scheduling problems, smart cities, etc.

After reviewing submissions, we accepted a total of nine papers for publication.

The Internet of Things (IoT) encompasses many applications and service domains, from smart cities, autonomous vehicles, surveillance, and medical devices, to crop control. Most experts regard virtualization in wireless sensor networks (WSNs) as the most revolutionary technological technique in these areas. Due to node failure or communication latency and the regular identification of nodes in WSNs, virtualization in WSNs presents additional hurdles.

In the contribution by Othman et al. [1], “A Multi-Objective Crowding Optimization Solution for Efficient Sensing as a Service in Virtualized Wireless Sensor Networks”, the authors present a novel architecture for heterogeneous virtual networks on the Internet of Things. They propose to embed the architecture in WSN settings to improve fault tolerance and communication latency in service-oriented networking. Moreover, the authors utilize the Evolutionary Multi-Objective Crowding Algorithm (EMOCA) to maximize fault tolerance and minimize communication delay for virtual network embedding in WSN environments for service-oriented applications focusing on heterogeneous virtual networks in the IoT. Unlike the current wireless virtualization approach, which uses the Non-dominated Sorting Genetic Algorithm-II (NSGA-II), EMOCA uses both domination and diversity criteria in the evolving population for optimization problems. The analysis of the results demonstrates that the proposed framework successfully optimizes fault tolerance and communication delay for virtualization in WSNs.

Scholars have recently introduced various non-systematic satisfiability studies on Discrete Hopfield Neural Networks to address the lack of interpretation. Although a flexible structure was established to help generate a wide range of spatial solutions that converge on global minima, the fundamental issue is that the existing logic completely ignores the distribution and features of the probability dataset, as well as the literal status distribution.

In the study by Abdeen et al. [2], “S-Type Random k Satisfiability Logic in Discrete Hopfield Neural Network Using Probability Distribution: Performance Optimization and Analysis”, the authors consider a new type of non-systematic logic known as S-type Random k Satisfiability, which employs a novel layer of a Discrete Hopfield Neural Network and plays a significant role in identifying the predominant attribute likelihood of a binomial distribution dataset. Establishing the logical structure and assigning negative literals based on two specified statistical parameters is the objective of the probability logic phase. Abdeen et al. examined the performance of the proposed logic structure by comparing a proposed metric to current state-of-the-art logical rules. As a result, they discovered that the models have a high value in two parameters that efficiently introduce a logical structure in the probability logic phase. In addition, the study observed that implementing a Discrete Hopfield Neural Network reduced the cost function. The authors employed a novel statistical method of synaptic weight assessment to investigate the influence of the two proposed parameters on the logic structure. Overall, they revealed that regulating the two proposed parameters positively impacts synaptic weight management and the generation of global minimum solutions.

Traditional leak detection methods for gas pipelines necessitate task offloading decisions in the cloud, which has poor real-time performance. Edge computing provides a solution by allowing decisions to be made directly at the edge server, improving real-time performance; however, energy is the new bottleneck. In “Edge Computing Offloading Method Based on Deep Reinforcement Learning for Gas Pipeline Leak Detection”, Wei et al. [3] concentrate on the real-time detection of gas transmission pipeline leaks. As a result, the authors propose a novel detection algorithm that combines the benefits of both the heuristic algorithm and the advantage actor-critic (AAC) algorithm.

The proposed detection algorithm seeks to optimize and ensure real-time pipeline mapping analysis tasks and maximize the survival time of portable gas leak detectors. Because the computing power of portable detection devices is limited due to their battery power, the main problem posed in this study is how to account for node energy overhead while ensuring system performance requirements.

Wei et al. establish the optimization model by introducing the concept of edge computing and using the mapping relationship between resource occupation and energy consumption as a starting point to optimize the total system cost (TSC). This is constituted of the transmission energy consumption of the node, the local computing energy consumption, and the residual electricity weight.

To reduce TSC, the algorithm employs the AAC network to make task scheduling decisions and determine whether tasks should be offloaded. Furthermore, it uses heuristic strategies and the Cauchy–Buniakowsky–Schwarz inequality to allocate communication resources.

Their experiments show that their proposed algorithm can meet the detector’s real-time requirements while consuming less energy. Compared to the Deep Q Network (DQN) algorithm, their proposed algorithm saves approximately 56% of the system energy. It saves 21%, 38%, 30%, 31%, and 44% of energy consumption compared to the artificial gorilla troops Optimizer (GTO), the black widow optimization algorithm (BWOA), the exploration-enhanced grey wolf optimizer (EEGWO), the African vulture optimization algorithm (AVOA), and the driving training-based optimization (DTBO). Moreover, it saves 50% and 30% compared to entirely local computing and fully offloading algorithms, respectively. Meanwhile, this algorithm’s task completion rate is 96.3%, the best real-time performance among these algorithms.

The pickup and delivery problems are pertinent problems in our interconnected world. Efficiently moving goods and people can decrease costs, emissions, and time. In the contribution by Little et al. [4], “Comparison of Genetic Operators for the Multi-Objective Pickup and Delivery Problem”, the authors develop a genetic algorithm to solve the multi-objective capacitated pickup-and-delivery problem by adapting standard benchmarks.

They aim to reduce the total distance traveled and the number of vehicles employed. Based on NSGA-II, the authors investigate the effects of inter-route and intra-route mutations on the final solution. Little et al. introduce six inter-route operations and sixteen intra-route operations. Then, they calculate the hypervolume to compare their impact directly. In addition, the authors present two unique crossover operators tailored to this problem.

Their methodology identified optimal results in 23% of the instances in the first benchmark. In most other models, it generated a Pareto front within 1 vehicle and 20% of the best-known distance. Users can select the routes that best suit their requirements due to the presence of multiple solutions.

In a disaster, the road network is often compromised in capacity and usability conditions. This is a challenge for humanitarian operations delivering critical medical supplies. In the contribution by Anuar et al. [5], “A Multi-Depot Dynamic Vehicle Routing Problem with Stochastic Road Capacity: An MDP Model and Dynamic Policy for Post-Decision State Rollout Algorithm in Reinforcement Learning”, the authors optimize vehicle routing for a Multi-Depot Dynamic Vehicle-Routing Problem with Stochastic Road Capacity (MDDVRPSRC) using the Markov Decision Processes (MDP) model. They use the Post-Decision State Rollout Algorithm (PDS-RA) as a look-ahead approach in an Approximate Dynamic Programming (ADP) solution method. The authors execute a PDS-RA for all assigned vehicles to effectively solve the problem. The agent then decides at the end.

For the PDS-RA, Anuar et al. propose five types of constructive base heuristics. Firstly, they propose the Teach Base Insertion Heuristic (TBIH-1) to investigate the partial random construction approach for non-obvious decisions. The paper presents TBIH-2 and TBIH-3 as extensions to the TBIH-1 to demonstrate how experts could execute the Sequential Insertion Heuristic (I1) and Clarke and Wright (CW) in a dynamic setting, respectively. Additionally, the authors propose TBIH-4 and TBIH-5 (TBIH-1 with the addition of Dynamic Look-ahead SIH (DLASIH) and Dynamic Look-ahead CW (DLACW)). The goal is to improve the on-the-fly constructed decision rule (dynamic policy on the fly) in look-ahead simulations.

COVID-19 has shaken the world economy and affected millions of people in a brief period. COVID-19 has countless overlapping symptoms with other upper respiratory conditions, making it challenging for diagnosticians to diagnose correctly. Several mathematical models have been presented for their diagnosis and treatment. In “An Optimized Decision Support Model for COVID-19 Diagnostics Based on Complex Fuzzy Hypersoft Mapping”, Saeed et al. [6] propose a mathematical framework based on a novel agile fuzzy-like arrangement, the complex fuzzy hypersoft (CFHS) set, a combination of the complex fuzzy (CF) and the hypersoft sets (an extension of the soft set).

First, the authors develop the CFHS elementary theory, which considers the amplitude term (A-term) and phase term (P-term) of complex numbers simultaneously to address uncertainty, ambivalence, and mediocrity of data. This new fuzzy-like hybrid theory is versatile in two parts.

First, it provides access to a wide range of membership function values by broadening them to the unit circle on an Argand plane and incorporating an additional term, the P-term, to account for the periodic nature of the data. Second, it divides the distinct attributes into corresponding sub-valued sets for easier comprehension. The CFHS set and CFHS mapping, with its inverse mapping (INM), can manage such issues. They validate their proposed framework by connecting COVID-19 symptoms to medications. This work also includes a generalized CFHS mapping [6], which can assist a specialist in extracting the patient’s health record and predicting how long it will take to overcome the infection.

With the fourth industrial revolution developing, the way factories operate will no longer be the same. Factory automation can save labor and avoid equipment failures with online fault-detection systems. In recent years, various signal-processing methods have received much attention in the problem of fault-detection systems. In the article by Lee et al. [7], “Application of ANN in Induction-Motor Fault-Detection System Established with MRA and CFFS”, the authors propose a fault-detection system for faulty induction motors (bearing faults, inter-turn shorts, and broken rotor bars) based on a multiresolution analysis (MRA), correlation and fitness values-based feature selection (CFFS), and artificial neural network (ANN).

For induction–motor–current signature analysis, Lee et al. compare two feature-extraction methods: the MRA and the Hilbert Huang transform (HHT). This work compares feature-selection methods to reduce the number of features while maintaining the best detection system accuracy to reduce operating costs. In addition, the proposed detection system is tested with additive white Gaussian noise, and the best signal-processing and feature-selection methods are chosen to create the best detection system. According to their results, features extracted from MRA outperform HHT using CFFS and ANN. The authors also confirm that the CFFS significantly reduces operation costs (95% of the features) while maintaining 93% accuracy using ANN in their proposed detection system.

Detection and recognition of scene text, such as automatic license plate recognition, is a technology with various applications. Although numerous studies have been conducted to increase detection performance, accuracy decreases when low-resolution and low-quality legacy license plate images are input into a recognition module.

In “HIFA-LPR: High-Frequency Augmented License Plate Recognition in Low-Quality Legacy Conditions via Gradual End-to-End Learning”, Lee, S.-J. et al. [8] propose a model for high-frequency augmented license plate recognition. They integrate and collaboratively train the super-resolution and the license plate recognition modules using a proposed gradual end-to-end learning-based optimization. To train their model optimally, the authors propose a holistic feature extraction method that effectively precludes the generation of grid patterns from the super-resolved image during training.

Moreover, to exploit high-frequency information that affects license plate recognition performance, the authors propose a high-frequency augmentation-based license plate recognition module. In addition, they present a three-step, gradual, and end-to-end learning process based on weight immobilization. Their three-step methodological approach optimizes each module for robust performance in recognition. The experimental outcomes demonstrate that their model outperforms extant methods in low-quality legacy conditions for the UFPR and Greek vehicle datasets.

In machine learning, the convex minimization problem in the sum of two convex functions is fundamental. Many authors have analyzed this problem due to its applications in various fields, such as data science, computer science, statistics, engineering, physics, and medical science. These applications include signal processing, compressed sensing, medical image reconstruction, digital image processing, and data prediction and classification. In the contribution by Chumpungam et al. [9], “An Accelerated Convex Optimization Algorithm with Line Search and Applications in Machine Learning”, the authors introduce a new line search technique and use it to build a novel accelerated forward–backward algorithm for solving convex minimization problems in the sum of two convex functions, one of which is smooth in a real Hilbert space.

The authors demonstrate a weak convergence to a solution of the proposed algorithm in the absence of the Lipschitz assumption on the gradient of the objective function. Furthermore, they evaluate its performance by applying the proposed algorithm to classification problems on various data sets and comparing it to other line search algorithms. The authors’ experiments show that their proposed algorithm outperforms other line search algorithms.

The articles presented in this Special Issue provide insights into fields related to “Advances in Machine Learning and Mathematical Modeling for Optimization Problems”, including models, performance evaluation and improvements, and application developments. We wish that readers can benefit from the insights of these papers and contribute to these rapidly growing areas. We also hope that this Special Issue sheds light on major developments in the area of machine learning and mathematical modeling for optimization problems and attracts the attention of the scientific community to pursue further investigations leading to the rapid implementation of these techniques.