Discrete Problem Research Articles

Energy Loss in Frictional Hertzian Contact Subjected to Two-Dimensional Cyclic Loadings

We investigate the effect of three different harmonically varying loads as a function of the friction coefficient on energy loss in a three-dimensional discrete uncoupled frictional contact problem. Three loading cases include (1) a normal force is constant and a tangential force varies, (2) normal and tangential forces both vary, but the loading and unloading curves are identical, and (3) normal and tangential forces both vary, but the loading and unloading curves are different. For a higher coefficient of friction, three loading cases show different characteristics. If a normal force is constant and a tangential force varies, there is always some slip, but dissipation tends asymptotically to zero at large coefficient of friction. If normal and tangential forces both vary, but the loading and unloading curves are identical, there is no slip and no dissipation above a critical coefficient of friction. If the loading and unloading curves are different, dissipation occurs for all values of the coefficient of friction, and we expect that the dissipation is asymptotic to the relaxation damping value as the coefficient of friction approaches infinity. For lowering coefficient of friction, the three loading cases show similar behavior. Dissipation increases and reaches a maximum just before a state where gross slip is possible.

The velocity tracking problem for Navier–Stokes equations with pointwise-integral control constraints in time-space

In this work, we study the velocity tracking control problem of the evolutionary two-dimensional Navier–Stokes system. The regularizing Tikhonov term is not involved in the cost functional and pointwise-integral control constraints in time-space are considered. First- and second-order optimality conditions are established using a suitable extended cone for the sufficient ones. We also address the numerical approximation of the control problem, proving the convergence of the discrete problems and establishing error estimates between the optimal discrete and continuous states.

A Heterogeneous Multi-Satellite Dynamic Mission Planning Method Based on Metaheuristic Algorithms

The increasing demand for satellite communication necessitates efficient resource management, especially as next-generation high-throughput satellites (HTSs) face challenges in optimizing user connections. This paper presents a novel method that integrates a discretely improved Particle Swarm Optimization (PSO) algorithm with a dynamic task management framework to enhance satellite resource allocation efficiency. The PSO algorithm is adapted for discrete selection problems to maximize a fitness function based on user priority, user preference, and capacity satisfaction, thus ensuring accurate user–satellite matching. A main loop function iteratively updates user data and connection statuses, thus achieving continuous optimization of satellite connections. Through simulation studies, we validated the effectiveness of this method under dynamically changing user demands, demonstrating that the Discrete PSO algorithm significantly outperforms Simulated Annealing (SA) and the Genetic Algorithm (GA) in user satisfaction, maintaining levels above 0.996 even under high demand. Additionally, it effectively prioritizes high-demand users, ensuring their satisfaction remains above 0.95. Overall, our method enhances the management of daily communication tasks, significantly improving service quality and user satisfaction.

Application of the Gray Wolf Optimization Algorithm and Neural Networks for Solving Discrete Problems

Relevance. In recent decades, metaheuristic optimization methods have become popular for solving complex problems that require searching for global extrema. Algorithms such as genetic algorithm (GA), ant colony optimization (ACO), particle swarm optimization (PSO), as well as more modern approaches such as cat pack optimization (CSO) and gray wolf pack optimization (GWO) demonstrate high efficiency, but their application is often limited by the conditions of continuity and differentiability of the objective functions. This is a challenge when solving problems with discrete data, where such requirements are not met. In this context, the search for methods that allow adapting metaheuristic algorithms to work with discrete functions is of particular relevance.Aim. The study is aimed at testing the hypothesis about the possibility of using a neural network trained on a limited set of discrete data as an approximation of a function sufficient for the correct execution of the GWO algorithm when searching for a global minimum. The implementation of this hypothesis can significantly expand the scope of GWO, making it available for a wider range of problems where functions are defined on discrete sets.Methods. The study is based on the analysis of existing approaches and experimental verification of the hypothesis on two test functions: a linear function and a Booth function, which are widely used as standards for evaluating the performance of optimization algorithms. Numerical experiments were conducted using neural networks as an approximating model to obtain the results. Solution. During the experiments, an analysis of the applicability of neural networks for approximating discrete functions was carried out, which showed the success of this approach. It was found that neural networks can approximate discrete functions with high accuracy, creating conditions for a successful search for a global minimum using the GWO algorithm.Novelty. For the first time, a hypothesis was proposed and tested on the use of neural networks for approximating objective functions in metaheuristic optimization problems on discrete data. This direction has not previously received due coverage in the scientific literature, which adds significance to the obtained results and confirms the effectiveness of the proposed approach.Practical significance. The results of the study open up new prospects for the application of algorithms such as GWO in optimization problems based on discrete data, expanding the capabilities of metaheuristic methods and facilitating their implementation in a wider class of applied problems, including problems where the use of other methods is limited.

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

This paper is concerned with the numerical solution of the flow problem in a fractured porous medium where the fracture is treated as a lower dimensional object embedded in the rock matrix. We consider a space-time mixed variational formulation of such a reduced fracture model with mixed finite element approximations in space and discontinuous Galerkin discretization in time. Different spatial and temporal grids are used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. Analysis of the numerical scheme, including well-posedness of the discrete problem, stability and a priori error estimates, is presented. Using substructuring techniques, the coupled subdomain and fracture system is reduced to a space-time interface problem which is solved iteratively by GMRES. Each GMRES iteration involves solution of time-dependent problems in the subdomains using the method of lines with local spatial and temporal discretizations. The convergence of GMRES is proved by using the field-of-values analysis and the properties of the discrete space-time interface operator. Numerical experiments are carried out to illustrate the performance of the proposed iterative algorithm and the accuracy of the numerical solution.

An analysis of discontinuous Galerkin method for Electrical Impedance Tomography with partial data

In this work, we extend our previous work (Li and Wang, 2023) of the discontinuous Galerkin (DG) method for Electrical Impedance Tomography (EIT) with full data to partial data where the current and voltage measurements are taken only on part of the boundary. Additionally, we provide the convergence analysis of the DG approximation for EIT for partial data based on an iterative method with Tikhonov regularization. We prove that the minimizers of the discrete optimization problems converge to a minimizer of the continuous optimization problem as the mesh sizes of the discretization approach zero. Numerical results for the recovery of conductivities are tested with different types of partial data. The partial data results are demonstrated to be comparable to the full data results.

Ship Power System Network Reconfiguration Based on Swarm Exchange Particle Swarm Optimization Algorithm

As one of the important components of a ship, the ship’s integrated power system is an important safeguard for ships. In order to improve the service life of the ship’s power grid, the power system should be able to realize rapid reconstruction to ensure continuous power supply of important loads when the ship is attacked or fails suddenly. Therefore, it is of vital importance to study the reconfiguration technology of the ship’s integrated power system to ensure that it can quickly and stably cope with all kinds of emergencies in order to guarantee the safe and reliable navigation of the ship. This paper takes the ship’s ring power system as the research object and sets up the maximum recovery load and the minimum number of switching operations. The load is divided uniformly and the generator efficiency is balanced for the reconstruction of comprehensive function. It also sets up the system capacity, topology, and branch current limitations of the constraints to establish a mathematical model. The load branch correlation matrix method is used for branch capacity calculation and generator efficiency equalization calculation, and the load backup power supply path matrix is added on the basis of the matrix to judge the connectivity of some loads before reconfiguration. In this paper, for the network reconfiguration of the ship circular power system, which is a discrete nonlinear problem with multiple objectives, multiple time periods, and multiple constraints, we choose to use the particle swarm algorithm, which is suitable for global optimization, with a simple structure and fewer parameters; improve the particle swarm algorithm using the swarm exchange strategy by setting up two main and auxiliary swarms for global and local search; and exchange some of the particles with the golden ratio in order to keep the diversity of the populations. The simulation results of the network reconfiguration of the ship power system show that the improved algorithm can solve the power system network reconfiguration problem more effectively and provide a feasible reconfiguration scheme in a shorter time compared with the chaotic genetic algorithm under the same fault case test, and it also proves that the use of the swarm exchange particle swarm algorithm greatly improves the performance of reconfiguring the power grid of the ship.

Mesh optimization for the virtual element method: How small can an agglomerated mesh become?

We present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). Once the local quality of the mesh elements is analyzed through a quality indicator specific to the VEM, groups of elements are agglomerated to optimize the global mesh quality. A user-set parameter regulates the percentage of mesh elements, and consequently of faces, edges, and vertices, to be removed. This significantly reduces the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, particularly for high-order formulations. We show how the VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones. We observe that the optimization has a regularization effect over low-quality meshes, removing the most pathological elements. In such cases, these “badly-shaped” elements yield a system matrix with very large condition number, which may cause the VEM to diverge, while the optimized meshes lead to convergence. We conclude by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.

Solving Elliptic Curve Discrete Logarithm Problem on Twisted Edwards Curves Using Quantum Annealing and Index Calculus Method

Abstract This paper presents an approach to solving the elliptic curve discrete logarithm problem on alternative curve models over prime fields using a quantum annealing and index calculus method. Part of the algorithm, relation searching, is transformed into the Quadratic Unconstrained Boolean Optimization (QUBO) problem and then is efficiently solved using the D-Wave computer by quantum annealing. As Faugère et al. showed, twisted Edwards curves, because of their symmetric shape, allow us to obtain solutions of relations searching step using Groebner basis faster than in the case of Weierstrass curves. Because of symmetries, a system of equations of relations searching step for twisted Edwards curves has many symmetric solutions. Using the Groebner basis and having many system solutions makes it easier to find any of them. The same is true using quantum annealing - it is easier to find any solution to the QUBO problem if many are correct. In this paper, we used this observation to find out that a properly constructed QUBO problem for the relations searching step for twisted Edwards curves allows us to find a solution faster for the same size of the base field than in the case of Weierstrass curves. Using the presented approach, we solved the discrete logarithm problem using quantum annealing and index calculus method for elliptic curve discrete logarithm problem defined on twisted Edwards curve over a field 𝔽1021 with order equal to 4 · 241. It is now the biggest field and size of the group, where the elliptic curve discrete logarithm problem was solved using quantum methods.

An Efficient Algorithm for Exact Segmentation of Large Compositional and Categorical Time Series

ABSTRACTChange‐point detection, also known as signal segmentation, is an essential preprocessing step in many applications, ranging from industrial monitoring to bioinformatics. In short, it consists in finding the temporal boundaries of homogeneous regimes in long and non‐stationary time series. While this area of research is active, most existing methods are designed for Euclidean data. However, in many practical scenarios, the collected time series are compositional, meaning that each observation belongs to the probability simplex (the set of non‐negative vectors whose components sum to one). In this work, we propose an algorithm detecting change‐points in large compositional signals with an underlying piecewise stationary model. We cast the change‐point detection task as a discrete optimization problem, whose solution is shown to converge to the true change‐points. We introduce a new and time‐efficient dynamic programming algorithm that solves exactly this problem. To limit the number of operations, we describe a novel pruning rule that allows us to reduce the set of candidate change‐point indices. Our method is tested on a thorough simulation study, which confirms its efficiency. Additionally, we apply our method to a human activity segmentation task, highlighting the necessity for such novel techniques compared to standard algorithms.

TrustCNAV: Certificateless aggregate authentication of civil navigation messages in GNSS

The Global Navigation Satellite System (GNSS) is capable of accurate positioning because it can provide high-precision data. These data are transmitted to the receiver in the form of navigation messages, called civil navigation messages (CNAV). As it is transmitted in an open, transparent environment without data integrity protection mechanisms and secure data transmission measures, the CNAV is suspected to spoofing attacks. In 2023, the OPSGROUP has received approximately 50 reports of GPS spoofing activity. A spoofed plane's navigation system will show it as being in a different place - a security risk if a jet is guided to fly into a hostile country's airspace. To prevent the forging of GNSS positioning data by spoofing attacks targeting CNAV, we propose a certificateless aggregation authentication for CNAV by using the elliptic curve discrete logarithm problem and the combination of the GNAV structural characteristics, called TrustCNAV. Security proof and performance analysis indicate that this authentication scheme can resist spoofing attacks and ensure data security of CNAV, also it avoids pairing operations with high computational complexity, thus meeting security requirements without causing too much time and communication consumption.

An Innovative Enhanced JAYA Algorithm for the Optimization of Continuous and Discrete Problems

Metaheuristic algorithms have gained popularity in the past decade due to their remarkable ability to address various optimization challenges. Among these, the JAYA algorithm has emerged as a recent contender that demonstrates strong performance across different optimization problems, largely attributed to its simplicity. However, real-world problems have become increasingly complex in today’s era, creating a demand for more robust and effective solutions to tackle these intricate challenges and achieve outstanding results. This article proposes an enhanced JAYA (EJAYA) method that addresses its inherent shortcomings, resulting in improved convergence and search capabilities when dealing with diverse problems. The current study evaluates the performance of the proposed optimization methods on both continuous and discontinuous problems. Initially, EJAYA is applied to solve 20 prominent test functions and is validated by comparison with other contemporary algorithms in the literature, including moth–flame optimization, particle swarm optimization, the dragonfly algorithm, and the sine–cosine algorithm. The effectiveness of the proposed approach in discrete scenarios is tested using feature selection and compared to existing optimization strategies. Evaluations across various scenarios demonstrate that the proposed enhancements significantly improve the JAYA algorithm’s performance, facilitating escape from local minima, achieving faster convergence, and expanding the search capabilities.

Urban freight distribution with electric vehicles: comparing some solution procedures

The Vehicle Routing Problem (VRP) is a well-known discrete optimization problem that has an impact on theoretical and practical applications. In this paper, a freight distribution model that includes a charging system located at the depot, making it feasible for real world-implementation, is proposed. Two different solution methods are proposed and compared: a genetic algorithm (GA) and a population-based simulated annealing (PBSA) with the number of moves increasing during the iterations. Among the variety of algorithm used to solve the VRP, population-based search methods are the most useful, due to the ability to update the memory at each iteration. To demonstrate the practical aspects of the proposed solution a case study is solved using travel time on a real network to evaluate the potentiality for a real-world application.

Tighter bound for generalized multiple discrete logarithm problem via MDS matrix method

Discrete logarithm problem (DLP) is one of the fundamental hard problems used in cryptography. For 1≤k≤n, solving the k-out-of-n DLP instances is an important problem emerging in certain scenarios in public-key cryptography. Ying and Kunihiro (ACNS 2017) pioneered in studying k-out-of-n instance solutions of DLP, which is a generalized version of multiple DLP. By reducing the multiple DLP to the generalized version, they established lower bounds on the computational complexity of k-out-of-n DLP for different parameter values of k.In this paper, we further reduce the reduction complexity presented in Ying and Kunihiro's work and increase the range of k and n for the tight lower bound of k-out-of-n DLP in the generic group model, which has applications in related cryptographic schemes. To achieve the goal, the key technique is to utilize a variant of fast multipoint evaluation. We divide the discussion into two cases. In the special case when n divides p−1, by leveraging Number Theory Transform (NTT) technique, we expand k and n to a larger range. In the general case, by using a variant of fast multipoint evaluation, we increase k and n to a moderately larger range.

An Enhanced Algorithm for Hybrid Flow Shop Scheduling Using Discrete Sparrow Optimization and Rough Set Theory

Aiming at the shortcomings of the sparrow search algorithm, such as it is easy to fall into local optimum and unable to solve discrete optimization problems, an improved discrete sparrow search algorithm is proposed. Firstly, the position update formula of the original sparrow search algorithm is abstracted, a new discrete heuristic position update strategy is designed according to the different identities of individuals, and the encoding and decoding methods are designed for the hybrid flow shop scheduling problem; Secondly, the rough data-deduction theory is introduced, and the feasibility and rationality of the above theory are explained by mathematical proofs, which provides theoretical support for the algorithm and improves the interpretability; Then, the nature of upper approximation is adopted to expand the search space, improve the population diversity, avoid prematurity of the algorithm, combine division and rough data-deduction to propose three strategies to promote information sharing among populations, regulate the exploitation ability and exploration ability of populations, and reduce the probability of the algorithm falling into local optimum; Finally, the improved discrete sparrow search algorithm is used to solve the hybrid flow shop scheduling problem. Simulation experiments are carried out on three small-scale practical examples and Liao's classic test set to verify the feasibility of the improved discrete sparrow search algorithm to solve the hybrid flow shop scheduling problem, and to prove the superiority of the proposed algorithm and the effectiveness of the improved strategy through comparison experiments with other algorithms.

Discrete Logarithm Factory

The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields (except for the weak small characteristic case). The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. A precomputation, solely dependent on an approximate finite field size and an extension degree, allows to efficiently compute discrete logarithms in a constant proportion of the finite fields of the given approximate size and extension degree. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination improves the asymptotic complexity. We also notice that combining our approach with the MNFS variant would be an unnecessary complication as all the potential gain of MNFS is subsumed by our Factory variant anyway. Furthermore, we demonstrate how Chebotarev's density theorem allows to compute the density of finite fields that can be solved with a given precomputation. Finally, we provide experimental data in order to assess the practical reach of our approach.

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

Solving discrete network design problem using disjunctive constraints

AbstractThis paper introduces a deterministic algorithm to solve the discrete network design problem (DNDP) efficiently. This non‐convex bilevel optimization problem is well‐known as an non deterministic polynomial (NP)‐hard problem in strategic transportation planning. The proposed algorithm optimizes budget allocation for large‐scale network improvements deterministically and with computational efficiency. It integrates disjunctive programming with an improved partial linearized subgradient method to enhance performance without significantly affecting solution quality. We evaluated our algorithm on the mid‐scale Sioux Falls and large‐scale Chicago networks. We assess the proposed algorithm's accuracy by examining the objective function's value, specifically the total travel time within the network. When tested on the mid‐scale Sioux Falls network, the algorithm achieved an average 46% improvement in computational efficiency, compared to the best‐performing method discussed in this paper, albeit with a 4.17% higher total travel time than the most accurate one, as the value of the objective function. In the application to the large‐scale Chicago network, the efficiency improved by an average of 99.48% while the total travel time experienced a 4.34% increase. These findings indicate that the deterministic algorithm proposed in this research improves the computational speed while presenting a limited trade‐off with solution precision. This deterministic approach offers a structured, predictable, and repeatable method for solving DNDP, which can advance transportation planning, particularly for large‐scale network applications where computational efficiency is paramount.

General

For this General review, I have selected five exciting books dealing with religion. I am very happy to report that we now find selected papers of Robert Parker, one of the luminaries in the field of Greek religion, in a handsomely produced and affordable volume published in the Kernos Suppléments series, whose many virtues I have often extolled on these pages.1 The collection contains twenty articles published over the span of thirty-five years, and, in a way, provides the ‘best of’ of Parker's opera minora. But these are minora in name only: all articles gathered in this volume have been, and remain, highly influential and represent question-defining studies that shaped the way we think about discrete problems in Greek religion.

Bayesian network modeling applied to food risks: Data from General Administration of Customs of China as an example

We introduce a multidimensional Bayesian Network (BN) modeling approach as an alternative to the classical multivariate regression approach commonly used for risk factor analysis. BN modeling is a rich and flexible analytical technique capable of elucidating complex food data. It is a data-driven graphical modeling technique and exploratory tool that visually presents dependencies and causal associations while retaining statistical rigor in holistic level inference. We used data on monthly unauthorized food imports to China for demonstrating a full Additive BN model in the open-source R language with all code necessary to reproduce our analyses. Compared to the classical regression model, the Additive BN model of this study described more relationships between variables, and overcome the problems of classical discretization of general BN. The analysis showed that food category and import weight were two actionable drivers directly related to risk and can be controlled to manage risk. The risk was also related to where the food is produced, but not where it is likely to be sold. The BN model also is able to systematically elaborate uncertainty and obtain information through probabilistic inference, which might be useful for giving additional new insights potentially not captured by classical methods. Finally, we discuss the potential and evolution of BN, such as dynamic BN.