Discrete Optimization Research Articles

2-SAT discrete Hopfield neural networks optimization via Crow search and fuzzy dynamical clustering approach

<abstract> <p>Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.</p> </abstract>

Modeling of 3 SAT discrete Hopfield neural network optimization using genetic algorithm optimized K-modes clustering

<p>The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.</p>

A Hybrid Discrete Grey Wolf Optimizer with Local Search for Multi-UAV Patrolling

This paper addresses the multi- UAV patrolling problem, a NP-hard optimization problem that is focused on minimizing idleness, which is defined as the time between consecutive visits to specific locations. We propose the Discrete Grey Wolf Optimizer (D-GWO), which is specifically developed to handle the discrete aspects of UAV patrolling routes. This new algorithm is enhanced with a 2-opt local search strategy, which integrates the global search capabilities of D-GWO with the precision of local optimization to effectively refine solutions. Comparative experimental results show that our algorithm outperforms established methods such as ant colony optimization and simulated annealing in terms of reducing global worst idleness and overall exploration time. Our findings suggest that the D-GWO algorithm is particularly effective for complex multi-UAV patrolling tasks, significantly enhancing efficiency in security and disaster response missions.

Multi-objective flexible job-shop scheduling in hospital using discrete particle swarm optimization algorithm with adaptive inertia weight (DPSO-AIW)

Multi-objective flexible job-shop scheduling in hospital using discrete particle swarm optimization algorithm with adaptive inertia weight (DPSO-AIW)

A Discrete Predator-Prey Brain Storm Optimization Technique for Optimal Allocation of Micro-PMUs in Distribution System State Estimation

This paper proposes a practical method for determining the optimal allocation of micro-Phasor Measurement Units (μPMUs) for distribution system state estimation (DSSE). The proposed method formulates the state estimation based on the nest structure of DistFlow, which is used for power flow calculation in distribution systems due to the redundancy of less than 1. The use of μPMUs can significantly improve the accuracy of estimates. However, Distribution System Operators (DSOs) need to consider optimization constraints before deciding on the optimal location of μPMUs. To tackle this issue, this paper proposes the use of DPPBSO (Discrete Predator-Prey Brain Storm Optimization) of Evolutionary Computation (EC) to optimize the location of μPMUs. PPBSO is an extension method that applies the Predator-Prey strategy to Brain Storm Optimization (BSO), and DPPSO is the discrete version of PPBSO used in solving combinatorial optimization problems. The Predator-Prey strategy is critical in improving the solution candidates by intensifying and diversifying solution searches in EC. Simulation results demonstrate the effectiveness of the proposed method in the IEEE 69-node distribution system.

Towards optimal space-time discretization for reachable sets of nonlinear control systems

Towards optimal space-time discretization for reachable sets of nonlinear control systems

NUMERICAL ANALYSIS OF BOREHOLE HEAT EXCHANGER PERFORMANCE IN SHALLOW GRAVEL AQUIFERS AND CLAY-DOMINATED SOIL

The performance of geothermal heat extraction in shallow aquifers depends on both Borehole Heat Exchanger (BHE) and soil or aquifer properties. In this work, an analysis of the thermal yield of a shallow geothermal reservoir was made numerically with the finite element method used to simulate heat and mass transfer in the three-dimensional reservoir. The main parameters for analysis which have been considered are the geometry and physical parameters of the BHE and grout, as well as aquifer matrix and groundwater fluid. Physical parameters are thermal conductivity, flow conductivity, expansion coefficient, porosity, volumetric heat capacity, anisotropy and dispersivity. The numerical tests have been performed in single BHE line source configuration representing numerically modelled thermal response test for the estimation of sustainable heat extraction. The domain size was a 100x100 meter rectangle with a depth of 200 meters. Three main lithological configurations have been modelled: gravel aquifer with low and high convection of groundwater fluid, as well as a shallow geothermal reservoir dominated by clay material without convection. For selected cases, the analysis for temporal and spatial discretization was also made. Three-dimensional transient modelling was made in FEFLOW® software with pre- and post-processing done in user-defined Python scripts. The results show the most influential parameters to be considered when setting up the real case simulation of geothermal heating and cooling, as well as optimal temporal and spatial discretization set-up with respect to expected thermal gradients in the reservoir.

Comparing Variations of the Greedy Strategy to Find the Optimal Sugar Beet Processing Schedule

The article presents a mathematical model of sugar beet processing and finding the optimal schedule for this processing. Such task, with fully known data, reduces to a well-known assignment problem. However, in many applied problems of discrete optimization, it is not possible to use classical methods to find a solution in practice. This happens as a result of the fact that when setting the task at the initial moment of time, there is no complete information about some data. For example, in the assignment problem, information about the matrix on the basis of which the objective function is constructed may be incomplete. As a result, in practice it is necessary to search for heuristic algorithms for various possible situations. In particular, the greedy algorithm, as well as its various variations, claims to be such an algorithm. Computational experiment is being conducted that allows us to understand which of the variations of the greedy algorithm give an acceptable result when building an optimal schedule strategy for processing sugar beet batches.

Network programming - theory and application to manager building projects

The paper deals with network programming method and its application for solving multi-extreme and discrete optimization problems in Project Management. The purpose of the paper is to develop a new approach to solving optimization problems, called the network programming method and to solve a number of optimization problems of project management based on this method. This method is used to solve the problem of building a business-supporting project portfolio. The life cycle of a building project can be represented as a great number of interrelated and interacting actions leading to the achievement of the set goals in the presence of existing constraints (resources, time, cost, quality, reliability, safety, etc.). Such a representation is called a network representation. On the basis of the network programming method and its special case - the dichotomous programming method, new algorithms for solving a number of optimization problems in project management are proposed. The proposed method makes it possible to use it in solving the problems of managing the life cycle of a construction object. It can be used not only throughout the entire life cycle, but also independently during the planning, implementation and completion phases of project management.

Supported nondominated points as a representation of the nondominated set: An empirical analysis

AbstractThe nondominated set of a multiple objective discrete optimization problem is known to contain unsupported nondominated points, which outnumber the supported ones and are more difficult to obtain. We treat supported nondominated points as a representation and analyse their quality using different metrics beyond their sheer numbers. Under different data generation schemes on multiobjective knapsack and assignment problems, we observe that supported nondominated points almost always provide a good representation of the entire nondominated set.

Efficient and Stable Unsupervised Feature Selection Based on Novel Structured Graph and Data Discrepancy Learning.

Unsupervised feature selection is an important tool in data mining, machine learning, and pattern recognition. Although data labels are often missing, the number of data classes can be known and exploited in many scenarios. Therefore, a structured graph, whose number of connected components is identical to the number of data classes, has been proposed and is frequently applied in unsupervised feature selection. However, methods based on the structured graph learning face two problems. First, their structured graphs are not always guaranteed to maintain the same number of connected components as the data classes with existing optimization algorithms. Second, they usually lack strategies for choosing moderate hyperparameters. To solve these problems, an efficient and stable unsupervised feature selection method based on a novel structured graph and data discrepancy learning (ESUFS) is proposed. Specifically, the novel structured graph, consisting of a pairwise data similarity matrix and an indicator matrix, can be efficiently learned by solving a discrete optimization problem. Data discrepancy learning focuses on features that maximize the difference among data and helps in selecting discriminative features. Extensive experiments conducted on various datasets show that ESUFS outperforms state-of-the-art methods not only in accuracy (ACC) but also in stability and speed.

Оптимизация мероприятий технической эксплуатации оросительных систем методами искусственного интеллекта

The purpose of the study is to develop a constructive solution for choosing priority objects of repair and restoration works using artificial intelligence methods in conditions of limited budget financing of operational measures of irrigation systems. The paper uses methods of system analysis and mathematical modeling, including binary variables for discrete optimization using evolutionary genetic programming methods. Based on the analysis of management decision support by means of mathematical support, the use of optimization methods for predicted impacts is justified, including the choice of discrete options to increase the functionality and effectiveness of planned technical operation measures. The allocation of limited financial resources is carried out on the model of multi-criteria optimization, including minimizing irrigation water losses, while increasing the area of irrigated land and increasing the financial indicators of the water management organization, which improves the quality of management impacts. The practical significance of research is determined by the development of innovative tools to solve the problem of allocating limited resources for carrying out repair and restoration work of the municipal water management complex using artificial intelligence methods. The approbation of the proposed solutions, carried out on the materials of the operation service of the State Budgetary Institution of the Republic of Crimea “Crimean Department of Water Management and Melioration”, proved the expediency of large-scale implementation of methods for quantitative assessment of management decisions. The technical and economic indicators of the planned activities correspond to the expected values and ensure the fulfillment of the following requirements: completeness, ensuring that they provide the necessary and sufficient information for decision-making; availability of reliable sources of reliable and accessible data for the information content of indicators and criteria.

A Novel Crossover based Discrete Artificial Algae Algorithm for Solving Traveling Salesman Problem

The Artificial Algae Algorithm (AAA) is a newly proposed metaheuristic algorithm that is inspired by microalgae behaviors. This algorithm has been proposed for solving continuous optimization problems and achieved good results for the continuous problems. In addition, binary versions of AAA are proposed in the literature. This paper presents a discrete version of AAA, which is named Discrete Artificial Algae Algorithm (DAAA). For discretization of AAA, Crossover operators (one-point and uniform) are used in the processes (helical movement, evolutionary process, and adaptation). In this study, in addition to crossover operators, transformation operators such as swapping, insertion, symmetry, and reversion are also used. DAAA’s performance was analyzed on a well-known discrete optimization problem called the Traveling Salesman Problem (TSP). DAAA was tested on thirty-two Benchmark instances of the TSP. These instances were small-sized, medium-sized, and large-sized. Firstly, the AAA processes (evolutionary process, adaptation, and helical movement) with the combination of nearest neighbor and transformation operators were tested for selected benchmark instances and this testing was called Process Analysis. After this process Analysis the best processes with which to continue were selected, and after this decision comparisons with other algorithms were started. The main comparison is between discrete Social Spider Algorithm (DSSA) and DAAA, and DAAA outperformed DSSA on most of the problems. Further, DAAA’s performance on some of the benchmark instances was compared with some of the well-known algorithms for TSP. In this comparison, DAAA has achieved better results than many other algorithms. Experimental results show that DAAA has the capability of solving discrete optimization problems and outperforming other algorithms.

ON THE OPTIMAL DISCRETIZATION OF THE SOLUTIONPOISSON’S EQUATION

The paper studies the problem of discretizing the solution of the Poisson equation with the right hand side f belonging to the multidimensional periodic Sobolev class. The research methodology is based on considering the problem of discretizing the solution of the Poisson equation as one of the concretizations of the general problem of optimal recovery of the operator Tf and using well known statements of approximation theory. Within the framework of this general optimal recovery problem, we first estimate from above the smallest discretization error N of the solution of the Poisson equation in the Hilbert metric using the discretization operator (l(N) , N) constructed from a finite set of Fourier coefficients of the function f. A lower estimate, coinciding in order with the upper estimate, for the smallest error N was obtained by involving all linear functionals defined on the multidimensional Sobolev class. It should be noted that the optimal discretization operator (l(N) , N) better approximates the solution under consideration in the Hilbert metric than any discretization operator constructed from values f at given points. Poisson’s equation is an elliptic partial differential equation and describes many physical phenomena such as electrostatic field, stationary temperature field, pressure field and velocity potential field in hydrodynamics. Therefore, the relevance of the research conducted here is beyond doubt.

Multi-material structural discrete variable topology optimization with minimum length scale control under mass constraint

Topology optimization of multi-material structure has a larger design space compared with single material optimization, and it also requires efficient material selection methods to provide references for designers. The multi-material additive manufacturing ensures the manufacturability of topology design of multi-material structures, but also puts forward new manufacturability requirements, such as the minimum size of multi-material and the connection interface. This paper focuses on the development of a topology optimization algorithm for multi-material structures under the total mass constraint based on discrete variables. Recursive multiphase materials interpolation (RMMI) scheme is utilized to establish the relationship between discrete design variables and element elastic modulus and density, and sequential approximate integer programming (SAIP) and canonical relaxation algorithms are constructed to update the discrete design variables. It is shown that this method can obtain multi-material topology design with clear interface display, and has the ability of free material coexistence and degradation, which indicates that the discrete variables have more advantages than the continuous variables in the RMMI scheme. Based on this, the material selection under mass constraints is discussed in depth, and it is pointed out that the traditional determination of material coexistence or degradation in design based on specific stiffness has limitations. Thus, the present work expands the necessary conditions of material coexistence. Then, a minimum size control method of multi-material design based on post-processing is proposed to solve the minimum size geometry constraint problem by SAIP. Numerical examples show that this method can not only precisely control the minimum size of each material phase (including void phase), but also resolve the problem of interface and boundary diffusion of the multi-material phase, and significantly improve the manufacturability of multi-material design. Finally, it is applied to optimization problems with complex design domain and irregular meshes which are closer to engineering design.

Optimal discretization of geothermal boreholes for the calculation of g-functions

The effect of the discretization of geothermal boreholes on the accuracy of g-function evaluations is studied. A data set of 557,056 bore field configurations covering a large range of geometrical parameters is generated using pygfunction. A nonuniform discretization of borehole segments geometrically expanding in length toward the middle of the borehole is proposed. A nonuniform discretization is shown to achieve better accuracy than a uniform discretization. The nonuniform discretization is optimized to minimize the maximum absolute percentage error over the entire data set. The discretization is optimized for each bore field configuration, and an artificial neural network (ANN) is trained to predict the optimal discretization given only geometrical and thermal parameters of the boreholes, excluding the borehole positions. Thermal parameters that quantify the bore field temperature distribution are introduced as inputs to the ANN. The maximum absolute percentage error using a uniform discretization is 99.0% in the worst studied case of a dense rectangular field of Nb = 1116 boreholes with lengths of 418.8 m and spacings of 3.14 m and 3.18 m along rows and columns, while only 1% of the cases feature an error above 26.7%. The error is reduced to 3.6% using the global optimal discretization and 3.3% using the ANN.

A Modified Horse Herd Optimization Algorithm and Its Application in the Program Source Code Clustering

Maintenance is one of the costliest phases in the software development process. If architectural design models are accessible, software maintenance can be made more straightforward. When the software’s source code is the only available resource, comprehending the program profoundly impacts the costs associated with software maintenance. The primary objective of comprehending the source code is extracting information used during the software maintenance phase. Generating a structural model based on the program source code is an effective way of reducing overall software maintenance costs. Software module clustering is considered a tremendous reverse engineering technique for constructing structural design models from the program source code. The main objectives of clustering modules are to reduce the quantity of connections between clusters, increase connections within clusters, and improve the quality of clustering. Finding the perfect clustering model is considered an NP-complete problem, and many previous approaches had significant issues in addressing this problem, such as low success rates, instability, and poor modularization quality. This paper applied the horse herd optimization algorithm, a distinctive population-based and discrete metaheuristic technique, in clustering software modules. The proposed method’s effectiveness in addressing the module clustering problem was examined by ten real-world standard software test benchmarks. Based on the experimental data, the quality of the clustered models produced is approximately 3.219, with a standard deviation of 0.0718 across the ten benchmarks. The proposed method surpasses former methods in convergence, modularization quality, and result stability. Furthermore, the experimental results demonstrate the versatility of this approach in effectively addressing various real-world discrete optimization challenges.

Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates

In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and C. elegans.In this work, we developed a computationally efficient 3D Finite Element (FE) unified framework for the locomotion of limbless organisms on frictional substrates. Muscle activity is simulated with a multiplicative decomposition of the deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D contractile elastic solids.We propose a two-field FE formulation based on positions and velocities. Governing partial differential equations are transformed into equivalent time-continuous differential–algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure is discussed, and Symplectic Euler time integration of optimality conditions are employed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backward Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal strategy out of distinct locomotion patterns adopted by limbless organisms.

Improved Algorithm of Partial Transmit Sequence Based on Discrete Particle Swarm Optimization

Orthogonal frequency division multiplexing (OFDM) in 5G has many advantages; however, one of the disadvantages is that the superposition of a large number of subcarriers leads to a high peak-to-average power ratio (PAPR) of the transmit signal. A high PAPR results in high-power amplifier distortion and performance degradation. The partial transmit sequence (PTS) algorithm is commonly used for PAPR reduction. It enumerates all combinations of phase factors, weighs the signal using each phase factor combination, and finds the set of phase factors that minimizes the PAPR value of the OFDM signal. The advantage of the PTS is that it determines the optimal solution through enumeration; however, its major drawback is the higher complexity caused by the use of enumeration. Some studies have introduced the discrete particle swarm optimization (DPSO) algorithm instead of enumeration to determine the optimal solution of the PTS algorithm. As an excellent optimization method, the DPSO algorithm represents each individual as a solution during the optimization. Through iterative updates of the initial population, individuals in the population continuously move closer to the optimal solution. This approach significantly reduces complexity compared with the exhaustive enumeration used in the traditional PTS algorithm. However, the disadvantage of the general DPSO algorithm is that it can result in premature and early convergence, which leads to degradation of the PAPR reduction performance. In this study, we propose an improved method based on the general DPSO-based PTS algorithm, and the improved algorithm MDPSO-PTS adopts dynamic time-varying learning factors, which can find the optimal combination of phase factors more efficiently. The MDPSO-PTS algorithm expands the search space when seeking the optimal combination of phase factors. This avoids the drawback of premature convergence commonly observed in general DPSO-PTS algorithms, preventing early consideration of local optima as global optima. A comparative simulation of the improved MDPSO-PTS algorithm with the general DPSO-PTS algorithm shows that the improved algorithm has stronger PAPR reduction, whereas the complexity remains basically unchanged. A comparative simulation with the traditional PTS algorithm shows a significant reduction in complexity, with only a slight, acceptable loss of reduction performance.

Decoding Nature with Nature's Tools: Heterotic Line Bundle Models of Particle Physics with Genetic Algorithms and Quantum Annealing

AbstractThe string theory landscape may include a multitude of ultraviolet embeddings of the Standard Model, but identifying these has proven difficult due to the enormous number of available string compactifications. Genetic Algorithms (GAs) represent a powerful class of discrete optimisation techniques that can efficiently deal with the immensity of the string landscape, especially when enhanced with input from quantum annealers. In this letter, we focus on geometric compactifications of the heterotic string theory compactified on smooth Calabi‐Yau threefolds with Abelian bundles. We make use of analytic formulae for bundle‐valued cohomology to impose the entire range of spectrum requirements, something that has not been possible so far. For manifolds with a relatively low number of Kähler parameters, we compare the GA search results with results from previous systematic scans, showing that GAs can find nearly all the viable solutions while visiting only a tiny fraction of the solution space. Moreover, we carry out GA searches on manifolds with a larger numbers of Kähler parameters where systematic searches are not feasible.