Discrete Optimization Problems Research Articles

A temperature-sensitive points selection method for machine tool based on rough set and multi-objective adaptive hybrid evolutionary algorithm

Reasonable deployment of temperature sensors is the key to accurately monitoring the temperature field of machine tools and improving the accuracy of thermal error prediction and compensation models. To determine the optimal deployment location of sensors, this paper proposes a temperature-sensitive points selection method tightly coupled with rough set and multi-objective optimization. Firstly, the importance of each temperature measurement point to the thermal error is calculated based on the rough set, and information entropy is introduced to amplify the importance difference among adjacent measurement points at the same heat source. Then, with the temperature measurement points groups as the variables, the number of temperature measurement points in the group, and the information importance of the group as the objectives, a multi-objective attribute reduction model is established, which transforms the temperature-sensitive points selection problem into a discrete multi-objective optimization problem. Finally, a multi-objective adaptive hybrid evolutionary algorithm is proposed, which designs a population initialization method based on mutual information and interval probability, and dynamic adaptive evolutionary parameters to achieve optimal temperature-sensitive points selection. Experiments on the high-speed dry hobbing machine verify the superiority and effectiveness of the proposed method.

A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints

A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints

A parallel social spider algorithm based on population mining

The social spider algorithm is a meta-heuristic algorithm inspired by the foraging behavior of biological social spiders, and is initially used to solve optimization problems with continuous functions. There exist many challenges when dealing with discrete combinatorial optimization problems. Therefore, a novel parallel social spider algorithm based on population mining is proposed. Its starting point is to mine dynamic heuristic information from the population of social spiders that cause vibrations to guide the spider’s next walk. It proposes the concept of vibration candidate sets and enhances the behavior of artificial spiders. Furthermore, it is implemented in parallel on the graphics processing unit. According to our investigation, it is the first parallel social spider algorithm. The classical traveling salesman problem is solved as an example. By solving 30 small-scale and medium-scale problems and comparing with other algorithms, it shows that its solution quality is significantly better. By solving 17 large-scale problems, compared with the parallel ant colony system and parallel iterative hill climbing algorithm on the GPU, it is found that both the solution quality and performance are also significantly better. Codes are available at https://github.com/BuptCIAGroup/P-SSA.

An improved nutcracker optimization algorithm for discrete and continuous optimization problems: Design, comprehensive analysis, and engineering applications

This study is presented to examine the performance of a newly proposed metaheuristic algorithm within discrete and continuous search spaces. Therefore, the multithresholding image segmentation problem and parameter estimation problem of both the proton exchange membrane fuel cell (PEMFC) and photovoltaic (PV) models, which have different search spaces, are used to test and verify this algorithm. The traditional techniques could not find approximate solutions for those problems in a reasonable amount of time, so researchers have used metaheuristic algorithms to overcome those shortcomings. However, the majority of metaheuristic algorithms still suffer from slow convergence speed and stagnation into local minima problems, which makes them unsuitable for tackling these optimization problems. Therefore, this study proposes an improved nutcracker optimization algorithm (INOA) for better solving those problems in an acceptable amount of time. INOA is based on improving the performance of the standard algorithm using a newly proposed convergence improvement strategy that aims to improve the convergence speed and prevent stagnation in local minima. This algorithm is first applied to estimating the unknown parameters of the single-diode, double-diode, and triple-diode models for a PV module and a solar cell. Second, four PEMFC modules are used to further observe INOA's performance for the continuous optimization challenge. Finally, the performance of INOA is investigated for solving the multi-thresholding image segmentation problem to test its effectiveness in a discrete search space. Several test images with different threshold levels were used to validate its effectiveness, stability, and scalability. Comparison to several rival optimizers using various performance indicators, such as convergence curve, standard deviation, average fitness value, and Wilcoxon rank-sum test, demonstrates that INOA is an effective alternative for solving both discrete and continuous optimization problems. Quantitively, INOA could solve those problems better than the other rival optimizers, with improvement rates for final results ranging between 0.8355 % and 3.34 % for discrete problems and 4.97 % and 99.9 % for continuous problems.

Multi-area collision-free path planning and efficient task scheduling optimization for autonomous agricultural robots

Collision-free path planning and task scheduling optimization in multi-region operations of autonomous agricultural robots present a complex coupled problem. In addition to considering task access sequences and collision-free path planning, multiple factors such as task priorities, terrain complexity of farmland, and robot energy consumption must be comprehensively addressed. This study aims to explore a hierarchical decoupling approach to tackle the challenges of multi-region path planning. Firstly, we conduct path planning based on the A* algorithm to traverse paths for all tasks and obtain multi-region connected paths. Throughout this process, factors such as path length, turning points, and corner angles are thoroughly considered, and a cost matrix is constructed for subsequent optimization processes. Secondly, we reformulate the multi-region path planning problem into a discrete optimization problem and employ genetic algorithms to optimize the task sequence, thus identifying the optimal task execution order under energy constraints. We finally validate the feasibility of the multi-task planning algorithm proposed by conducting experiments in an open environment, a narrow environment and a large-scale environment. Experimental results demonstrate the method's capability to find feasible collision-free and cost-optimal task access paths in diverse and complex multi-region planning scenarios.

A memetic algorithm for improved joint route selection and split-level management in next-generation wireless communications

The complexity of next-generation wireless communications, especially Beyond 5G and 6G communication systems, will be handled by artificial intelligence-based management paradigms. The joint selection of routes and functional split levels involves critical decisions that network infrastructure providers need to make to support requests from virtual Mobile Network Operators (vMNOs). These decisions comprise the assignment and configuration of physical network resources, which must comply with the specific quality of service restrictions of each vMNO request. Recent work defined a detailed mathematical model for this complex challenge, its formulation as a constrained, discrete optimization problem, and the first algorithmic approaches. It was also found that an evolutionary algorithm delivers higher-quality solutions than an ad-hoc heuristic, and faster running times compared to a well-known commercial solver. This paper introduces a memetic algorithm that exploits the strengths of the former evolutionary method while incorporating several key innovations: a domain-specific recombination operator; a specialized repairing procedure; an enhanced fitness evaluation scheme; and a multiobjective archiving strategy that preserves promising solution trade-offs. We conduct a comprehensive evaluation of the performance and behavior of this proposal, as well as the contribution of each specific design component. The results highlight that our memetic algorithm consistently outperforms previous approaches from the literature, providing better trade-offs in terms of solution quality and the rate at which vMNO requests are successfully fulfilled.

Multi-objective discrete Cuckoo search algorithm for optimization of bag-of-tasks scheduling in fog computing environment

Internet of Things (IoT) applications are mostly time-sensitive and require reliable executions. The majority of IoT applications experience low quality of service (QoS) because of the intrinsic delay in utilizing the far-distant cloud infrastructures. To experience better QoS, fog computing as a complementary facility for cloud infrastructure is extended near the IoT devices. The bag of tasks (BoTs) IoT applications include a number of independent tasks each of which needs a certain amount of processing time so it is favorable to experience the lowest total execution time, makespan, for all applications. In addition, reliable execution guarantees the business continuity of IoT applications. Therefore, this paper presents a fog computing scheduler for the execution of BoT IoT applications in the fog platforms to minimize makespan and maximize the reliability execution at the same time. To this end, a new reliability model is presented. Then, the BoTs scheduling issue is formulated into a multi-objective discrete optimization problem with makespan and reliability optimality viewpoints. To solve this combinatorial problem, a multi-objective discrete cuckoo search algorithm (MoDCSA) is extended. The proposed MoDCSA optimization algorithm takes the benefit of several discrete operators each of which is smartly invoked. The quality of gained results from simulations of extensive scenarios verifies the effectiveness of the proposed MoDCSA scheduling algorithm which has 2.27%, 3.89%, 5.45%, 5.40%, and 4.55% improvement in terms of makespan and 1.99%, 5.11%, 6.39%, 7.95%, and 7.78% improvement in terms of reliability against MoBLA, MoGA, MoGWO, MoSA, and MoGSA comparative algorithms respectively.

Expansion planning of hybrid electrical and thermal systems using reconfiguration and adaptive bat algorithm

_ This study introduces a comprehensive model for the concurrent expansion planning of various energy systems and their associated equipment. The need to reduce network costs, emissions, losses, and feeder loading, as well as to enhance network reliability and voltage profile, mandates the utilization of proper multi-objective planning models that respect all network constraints. The introduced framework includes units for generating both electrical and thermal energies. The model leverages conventional expansion alternatives such as the installation of new lines, network reconfiguration, rewiring, and the addition of new thermal and electrical generating units to the network. Expansion planning involves determining the optimal time, location, and type of new installations to meet future energy demands while minimizing costs and emissions. Reconfiguration refers to altering the network topology to improve reliability and reduce losses. The proposed expansion planning is formulated as a discrete, nonlinear, and non-convex optimization problem, which is solved using the Self Adaptive Learning Bat Algorithm (SALBA). This algorithm improves convergence speed and increases the diversity of the search population, enhancing the likelihood of finding the global optimum. Numerical simulations of the proposed methodology on two modified standard IEEE test systems corroborate the efficacy and feasibility of the suggested approach. Key innovations include the comprehensive modeling for concurrent expansion planning, the use of an advanced optimization algorithm, and a focus on reducing costs, emissions, losses, and feeder loading while enhancing network reliability and voltage profile.

Gate-Based Variational Quantum Algorithm for Truss Structure Size Optimization Problem

Quantum computing has become a pivotal innovation in computational science, offering novel avenues for tackling the increasingly complex and high-dimensional optimization challenges inherent in engineering design. This paradigm shift is particularly pertinent in the domain of structural optimization, where the intricate interplay of design variables and constraints necessitates advanced computational strategies. In this vein, the gate-based variational quantum algorithm utilizes quantum superposition and entanglement to improve search efficiency in large solution spaces. This paper delves into the gate-based variational quantum algorithm for the discrete variable truss structure size optimization problem. By reformulating this optimization challenge into a quadratic, unconstrained binary optimization framework, we bridge the gap between the discrete nature of engineering optimization tasks and the quantum computational paradigm. A detailed algorithm is outlined, encompassing the translation of the truss optimization problem into the quantum problem, the initialization and iterative evolution of a quantum circuit tailored to this problem, and the integration of classical optimization techniques for parameter tuning. The proposed approach demonstrates the feasibility and potential of quantum computing to transform engineering design and optimization, with numerical experiments validating the effectiveness of the method and paving the way for future explorations in quantum-assisted engineering optimizations.

Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness

We present a variational approach aimed at enhancing the training of physics-informed neural networks (PINNs) and more general surrogate models for learning partial differential equations (PDE). In particular, we extend our formerly introduced notion of Sobolev cubatures to negative orders, enabling the approximation of negative order Sobolev norms. We mathematically prove the effect of negative order Sobolev cubatures in improving the condition number of discrete PDE learning problems, providing balancing scalars that mitigate numerical stiffness issues caused by loss imbalances. Additionally, we consider polynomial surrogate models (PSMs), which maintain the flexibility of PINN formulations while preserving the convexity structure of the PDE operators. The combination of negative order Sobolev cubatures and PSMs delivers well-conditioned discrete optimization problems, solvable via an exponentially fast convergent gradient descent for λ-convex losses. Our theoretical contributions are supported by numerical experiments, addressing linear and non-linear, forward and inverse PDE problems. These experiments show that the Sobolev cubature-based PSMs emerge as the superior state-of-the-art PINN technique.

Roulette wheel-based level learning evolutionary algorithm for feature selection of high-dimensional data

Feature selection in high-dimensional data is a large-scale sparse and discrete optimization problem. Most evolutionary algorithms are designed to tackle continuous optimization problems. However, when dealing with high-dimensional feature selection tasks, they often suffer from poor population diversity and are computationally expensive. To address these challenges, this work introduces a roulette wheel-based level learning evolutionary algorithm (RWLLEA). RWLLEA integrates two key components. Firstly, it employs a leveled population mode. Individuals from higher levels provide guidance to those at lower levels during the evolutionary process, thereby exploring the potential combinatorial effects among features. Secondly, recognizing the characteristics associated with the high-dimensional feature selection task, a roulette wheel-based update method is devised to dynamically reduce search space and harmonize the algorithm's exploitation and exploration capacities across different stages. The performance of the proposed method was evaluated by comparing it with six other feature selection techniques across a range of fifteen diverse datasets. The experimental findings demonstrate that the proposed method can achieve a reduced feature set with a shorter runtime and exhibit superior classification accuracy.

Metaheuristics for variable-size mixed optimization problems: A unified taxonomy and survey

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle VMVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy or comprehensive survey for handling this important family of optimization problems. This paper presents an unified taxonomy for metaheuristic solutions for solving VMVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of VMVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

The turnpike property for high‐dimensional interacting agent systems in discrete time

AbstractWe investigate the interior turnpike phenomenon for discrete‐time multi‐agent optimal control problems. While for continuous systems the turnpike property has been established, we focus here on first‐order discretizations of such systems. It is shown that the resulting time‐discrete system inherits the turnpike property with estimates of the same type as in the continuous case. In particular, we prove that the discrete time optimal control problem is strictly dissipative and the cheap control assumption holds.

Pairwise Distances and the Problem of Multiple Optima.

Discrete optimization problems arise in many biological contexts and, in many cases, we seek to make inferences from the optimal solutions. However, the number of optimal solutions is frequently very large and making inferences from any single solution may result in conclusions that are not supported by other optimal solutions. We describe a general approach for efficiently (polynomial time) and exactly (without sampling) computing statistics on the space of optimal solutions. These statistics provide insights into the space of optimal solutions that can be used to support the use of a single optimum (e.g., when the optimal solutions are similar) or justify the need for selecting multiple optima (e.g., when the solution space is large and diverse) from which to make inferences. We demonstrate this approach on two well-known problems and identify the properties of these problems that make them amenable to this method.

The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems

Many solution algorithms for multiobjective optimization problems are based on scalarization methods that transform the multiobjective problem into a scalar-valued optimization problem. In this article, we study the theory of weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarizations. These methods minimize the distance induced by a weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm between the image of a feasible solution and a given reference point. We provide a comprehensive theory of the set of eligible weights and, in particular, analyze the topological structure of the normalized weight set. This set is composed of connected subsets, called weight set components which are in a one-to-one relation with the set of optimal images of the corresponding weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarization. Our work generalizes and complements existing results for the weighted sum and the weighted Tchebycheff scalarization and provides new insights into the structure of the set of all Pareto optimal solutions.

Research on the Human–Robot Collaborative Disassembly Line Balancing of Spent Lithium Batteries with a Human Factor Load

The disassembly of spent lithium batteries is a prerequisite for efficient product recycling, the first link in remanufacturing, and its operational form has gradually changed from traditional manual disassembly to robot-assisted human–robot cooperative disassembly. Robots exhibit robust load-bearing capacity and perform stable repetitive tasks, while humans possess subjective experiences and tacit knowledge. It makes the disassembly activity more adaptable and ergonomic. However, existing human–robot collaborative disassembly studies have neglected to account for time-varying human conditions, such as safety, cognitive behavior, workload, and human pose shifts. Firstly, in order to overcome the limitations of existing research, we propose a model for balancing human–robot collaborative disassembly lines that take into consideration the load factor related to human involvement. This entails the development of a multi-objective mathematical model aimed at minimizing both the cycle time of the disassembly line and its associated costs while also aiming to reduce the integrated smoothing exponent. Secondly, we propose a modified multi-objective fruit fly optimization algorithm. The proposed algorithm combines chaos theory and the global cooperation mechanism to improve the performance of the algorithm. We add Gaussian mutation and crowding distance to efficiently solve the discrete optimization problem. Finally, we demonstrate the effectiveness and sensitivity of the improved multi-objective fruit fly optimization algorithm by solving and analyzing an example of Mercedes battery pack disassembly.

Stochastic Gradient Descent-like relaxation is equivalent to Metropolis dynamics in discrete optimization and inference problems

Is Stochastic Gradient Descent (SGD) substantially different from Metropolis Monte Carlo dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g. SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.

Advancing feature ranking with hybrid feature ranking weighted majority model: a weighted majority voting strategy enhanced by the Harris hawks optimizer

Abstract Feature selection (FS) is vital in improving the performance of machine learning (ML) algorithms. Despite its importance, identifying the most important features remains challenging, highlighting the need for advanced optimization techniques. In this study, we propose a novel hybrid feature ranking technique called the Hybrid Feature Ranking Weighted Majority Model (HFRWM2). HFRWM2 combines ML models with the Harris Hawks Optimizer (HHO) metaheuristic. HHO is known for its versatility in addressing various optimization challenges, thanks to its ability to handle continuous, discrete, and combinatorial optimization problems. It achieves a balance between exploration and exploitation by mimicking the cooperative hunting behavior of Harris’s hawks, thus thoroughly exploring the search space and converging toward optimal solutions. Our approach operates in two phases. First, an odd number of ML models, in conjunction with HHO, generate feature encodings along with performance metrics. These encodings are then weighted based on their metrics and vertically aggregated. This process produces feature rankings, facilitating the extraction of the top-K features. The motivation behind our research is 2-fold: to enhance the precision of ML algorithms through optimized FS and to improve the overall efficiency of predictive models. To evaluate the effectiveness of HFRWM2, we conducted rigorous tests on two datasets: “Australian” and “Fertility.” Our findings demonstrate the effectiveness of HFRWM2 in navigating the search space and identifying optimal solutions. We compared HFRWM2 with 12 other feature ranking techniques and found it to outperform them. This superiority was particularly evident in the graphical comparison of the “Australian” dataset, where HFRWM2 showed significant advancements in feature ranking.

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

Phasor measurement units (PMUs) are deployed at power grid nodes around the transmission grid, determining precise power system monitoring conditions. In real life, it is not realistic to place a PMU at every power grid node; thus, the lowest PMU number is optimally selected for the full observation of the entire network. In this study, the PMU placement model is reconsidered, taking into account single- and multi-capacity placement models rather than the well-studied PMU placement model with an unrestricted number of channels. A restricted number of channels per monitoring device is used, instead of supposing that a PMU is able to observe all incident buses through the transmission connectivity lines. The optimization models are declared closely to the power dominating set and minimum edge cover problem in graph theory. These discrete optimization problems are directly related with the minimum set covering problem. Initially, the allocation model is declared as a constrained mixed-integer linear program implemented by mathematical and stochastic algorithms. Then, the 0/1 integer linear problem is reformulated into a non-convex constraint program to find optimality. The mathematical models are solved either in binary form or in the continuous domain using specialized optimization libraries, and are all implemented in YALMIP software in conjunction with MATLAB. Mixed-integer linear solvers, nonlinear programming solvers, and heuristic algorithms are utilized in the aforementioned software packages to locate the global solution for each instance solved in this application, which considers the transformation of the existing power grids to smart grids.

Learning to sample initial solution for solving 0–1 discrete optimization problem by local search

Learning to sample initial solution for solving 0–1 discrete optimization problem by local search