Linear Inequalities Research Articles

Orbit-finite linear programming

An infinite set is orbit-finite if, up to permutations of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

Efficient Algorithm for Generating Optimal Inequality Candidates for MILP Modeling of Boolean Functions

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds. It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature. In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

Development Learning Design of Linear Inequality of Two Variables with GeoGebra Classroom Assisted Discovery Learning Model

Penelitian ini bertujuan untuk menghasilkan desain pembelajaran pertidaksamaan linier dua variabel dengan model pembelajaran kelas Discovery Learning GeoGebra yang valid, praktis dan efektif dengan menggunakan metode Educational Design Research (EDR). Tahapan yang dilakukan meliputi tahap analisis, tahap perancangan, tahap evaluasi, tahap pematangan intervensi dan tahap implementasi. Hasil validasi menunjukkan rata-rata nilai total validasi RPP sebesar 96,4%, rata-rata nilai total validasi Lembar Kinerja Siswa sebesar 93,2%, rata-rata nilai total validasi THB sebesar 95% dengan kategori sangat baik sehingga memenuhi valid kriteria. Hasil angket respon siswa >80% berada pada kategori baik, sehingga produk desain pembelajaran yang dikembangkan dapat dijelaskan secara praktis. Hasil belajar siswa yang menggunakan desain pembelajaran yang dikembangkan dijelaskan secara efektif karena lebih terlihat dibandingkan siswa yang tidak menggunakan desain pembelajaran yang dikembangkan, hal ini ditunjukkan dari hasil uji normalitas dan uji T dengan menggunakan SPSS. Oleh karena itu, disimpulkan bahwa desain pembelajaran yang dikembangkan dinyatakan valid, praktis dan efektif.

Positivity and semi-global polynomial stability of high-order Cohen–Grossberg BAM neural networks with multiple proportional delays

In this paper, we study positivity and semi-global polynomial stability (PS) of higher-order Cohen-Grossberg BAM neural networks with multiple proportional time delays. The proportional delays considered here are unbounded and time-varying, differing from constant, bounded, and distributional time delays. The system model cannot be represented using vector and matrices, making certain approaches within the vector-matrix framework unsuitable for applying. To address this limitation, a direct method based on the solution of the system is proposed to provide sufficient conditions guaranteeing the positivity and semi-global polynomial stability (PS) of the model under consideration. Furthermore, the direct method is applied to establish global PS conditions for BAM neural networks with multiple proportional delays. The obtained conditions contain only a few simple linear scalar inequalities that are easily solved. The applicability of the obtained PS conditions is verified by two numerical examples, and the solution of a linear programming problem is also obtained based on these theoretical results. Notably, this method can be applied to many system models with proportional delays after minor modifications.

Solving problems of volume scheduling for shipbuilding enterprises in conditions of limited resources

The shipbuilding industry development strategy for the period up to 2035 and beyond is aimed at creating a new competitive shipbuilding industry based on the development of scientific, technical and personnel potential, optimization of production capacities, their modernization and technical re-equipment [1]. The paper is devoted to solving problems of volume-calendar production planning for shipbuilding enterprises when coordinating the production capacity of the enterprise with the resource needs to fulfill obligations determined by the generated order portfolio. Shipbuilding and ship repair enterprises are typical representatives of enterprises with single and small-scale production and a long production cycle of the main products [2-4]. The problem is solved at the stage of forming a portfolio of orders, which is based on the limited volumes of resources available to the production system. It is required to determine the amount of missing resources by calendar periods that are necessary for the effective functioning of the production system. Non-stored resources such as labor resources, equipment operating time, are usually taken into account when solving the problems of volumetric or volume-scheduling planning. Such resources being unused in a certain planning cycle cannot be used in subsequent cycles. The procedure for finding missing volumes of resources is formulated as the problem of reducing an incompatible system of linear two-sided algebraic inequalities to a compatible one. To do this, it is proposed to “expand” the initial parameters of the system, taking into account the costs of expansion, so that the system becomes compatible. In the general case, formally, the problem under consideration is close to the problem of finding a Chebyshev point [5] for an incompatible system of linear algebraic equations, i.e. a point that is geometrically the least deviating from all hyperplanes formed by the system constraints. The paper proposes statements and solutions of volume-scheduling problems for the case when the structure of the problem constraints is modeled by a root-oriented weighted tree. This hierarchical structure is typical for shipbuilding enterprises [6].

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

A Non- Linear Hyperbolic Variational Inequality

A Non- Linear Hyperbolic Variational Inequality

Physics-informed probabilistic slow feature analysis

This paper presents a novel approach called physics-informed probabilistic slow feature analysis. The probabilistic slow feature analysis method has been employed to extract slowly varying latent patterns from high-dimensional measured data. The extracted slow features have proven effective in industrial applications such as soft sensing and process monitoring. However, industrial processes come with various physical constraints that must be taken into account, such as energy requirements, equipment limitations, and safety considerations. The conventional black-box nature of the slow feature model often leads to physically inconsistent or unacceptable results. To address this issue, we propose integrating physics principles into the probabilistic slow feature model, ensuring that the extracted features adhere to physics laws. Our formulation incorporates two types of physical constraints: linear algebraic equality and inequality constraints. Through an industrial case study, we demonstrate the effectiveness of our methodology, showcasing the advantages of incorporating physics in feature extraction. These advantages include improved interpretability, reduced data dimensionality, and enhanced generalization performance.

Exponential qubit reduction in optimization for financial transaction settlement

We extend the qubit-efficient encoding presented in (Tan et al. in Quantum 5:454, 2021) and apply it to instances of the financial transaction settlement problem constructed from data provided by a regulated financial exchange. Our methods are directly applicable to any QUBO problem with linear inequality constraints. Our extension of previously proposed methods consists of a simplification in varying the number of qubits used to encode correlations as well as a new class of variational circuits which incorporate symmetries thereby reducing sampling overhead, improving numerical stability and recovering the expression of the cost objective as a Hermitian observable. We also propose optimality-preserving methods to reduce variance in real-world data and substitute continuous slack variables. We benchmark our methods against standard QAOA for problems consisting of 16 transactions and obtain competitive results. Our newly proposed variational ansatz performs best overall. We demonstrate tackling problems with 128 transactions on real quantum hardware, exceeding previous results bounded by NISQ hardware by almost two orders of magnitude.

ANALISIS KEMAMPUAN PEMECAHAN MASALAH MATEMATIKA PADA MATERI PERSAMAAN DAN PERTIDAKSAMAAN LINEAR SATU VARIABEL

This study was conducted with the primary aim of evaluating and deeply understanding students' abilities in solving mathematical problems, specifically focusing on Linear Equations and Inequalities with One Variable (SPLSV). A qualitative approach was chosen because it allows researchers to delve deeper into students' thought processes and how they approach given mathematical problems. Descriptive analysis was used to interpret data systematically and in detail, enabling researchers to identify specific patterns in test data and interviews. Adab Kesuma Bangsa Junior High School in Natar, South Lampung, was selected as the research site because it was considered representative of the student population being studied. A total of 32 students participated in this study, selected through Purposive Sampling method. The research instruments used included written tests and in-depth interviews. The main indicators focused on in this research were: (1) understanding of the problem, which measured how well students could identify and comprehend key elements of mathematical problems; (2) solution planning, which assessed students' ability to formulate effective strategies or steps to solve problems; (3) solution implementation, which observed to what extent students could apply their plans in practice and achieve correct solutions; and (4) re-evaluation, which evaluated students' ability to reflect on their processes and outcomes, identifying errors or areas for improvement. The research findings indicated that only a small proportion of students (21.5%) demonstrated high problem-solving abilities. Approximately 11.5% were at a moderate level, while the majority of students (67%) exhibited low problem-solving abilities. These findings provide valuable insights for educators and policymakers regarding the need for more effective educational interventions to enhance students' mathematical problem-solving abilities

PENGEMBANGAN INSTRUMEN TES MATEMATIKA BERBASIS HIGHER ORDER THINGKING SKILLS (HOTS) UNTUK SISWA SMP

The aim of this research is to develop a mathematical test instrument based on Higher Order Thingking Skills for high school students. The research methods used in this research are research and development methods or Research and Development. (R&D). The data-gathering instruments for this search are teacher lift, student perception lift, observation sheet, interview guide, and the definition test of Higher Order Thinking Skills. (HOTS). The study was tested on 25 students of the eighth grade of high school, the test was carried out on the material of linear equations and linear inequalities of one variable. The results of the research show that: 1) the test instrument Higher Order Thinking Skills developed is valid and reliable, 2) has an average difficulty level for each type of question, and 3) the test instruments Higher order thinking skills (HOTS) developed fall into the category both in different power tests. So, the tool developed by the researchers deserves to be used as an assessment to measure the Higher Order Thinking Skills (HOTS) test for high school students

Модификация модели прогнозирования финансово-экономического состояния для задачи управления развитием медицинских учреждений

The article analyzes the model of forecasting the financial and economic condition, considers the possibility of its transformation for managing the development of medical institutions, and provides an appropriate modification using methods of discriminant analysis and mathematical programming. The formulated modification of the model has a clear mathematical justification, which allows, on the one hand, to prove its validity, and on the other hand constructively shows the possibility of its use, taking into account the specifics of the medical industry. The purpose of the work. The purpose of this work is to study changes in the forecasting model of the financial and economic condition, taking into account the specifics of the industry, as well as to evaluate the application of methods of discriminant analysis, reducing them to a mathematical programming problem, the limitations of which are reduced to systems of linear inequalities obtained by building maximally joint subsystems. Materials and methods. It is proposed to use the method of discriminant analysis with extensible precedent sets. It is assumed that in order to improve the quality of solving this problem, expert judgments about objects that were not initially included in the training sample are involved. At the same time, the goal is to determine the best objects to involve them in the learning material in order to maximize the solution of the problem. To formalize the model based on the predicted values, we will encrypt all available data into a vector of states. The considered methods show their effectiveness. The results of the study. During the modification of the economic and financial condition model, a conclusion is made about the linearization of the model, which is also associated with the differentiation operation. The possibility of using neural networks is also demonstrated for the case of differentiation of complex functions, and in the case of convex programming, for marginal estimates obtained using the directional differentiation operation. Conclusion. The results obtained allow us to assert that the use of strict mathematical tools, a modified model of the economic and financial condition for the medical industry is easy to use for development management, since it can be implemented as some component of a computer program (neural network) into which the administrator enters data, and at the output receives a processed result.

Student Errors in Solving System of Linear Inequalities Story Questions Based on Newman’s Error Analysis (NEA)

This paper aims to describe the types of error and the causes of student error in solve math story questions on the subject matter of systems of linear inequalities based on Newman's Error Analysis (NEA). The linear inequalities system material was chosen for this study because the linear inequalities system has components that students must understand so they can easily determine the error made by student. This study is a qualitative descriptive. The subjects of this study were taken using a purposive sampling technique and obtained 10 students in the 5th semester of math education study program, Muhammadiyah University of Tangerang. This research was divided into three stage, namely the preparation of research instrument, implementation and colection of data study and data analysis. The instrument used in this research is a description of 3 questions. Based on data analysis, the factor that led to student error in solve math story question on the subject matter of the system of linear inequalities based on NEA were inaccurate, unable to read questions, unable to understand the question, and unable to carry out procedure or steps that would used to problem solving. This study contributes to providing information on errors that are commonly made by student in solve math story questions on the subject matter system linear inequalities based on NEA.

On the geometry and refined rate of primal–dual hybrid gradient for linear programming

AbstractWe study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

Development of Electronic Teaching Materials Based on Contextual Teaching and Learning (CTL) Approach to Improve High School Students' Mathematical Problem Solving Skills

The aim of this research is to develop approach-based electronic teaching materials Contextual Teaching And Learning (CTL) on systems of equations and linear inequalities to improve high school students' valid and practical mathematical problem solving abilities. This research method uses research and development with a 4D model. Data was collected using questionnaires, observations and interviews. The data analysis technique used is quantitative descriptive analysis. Teaching materials are said to be valid and practical if the score obtained is more than 70%. The results of this research show that electronic teaching materials are based The CTL approach was developed to increase mathematical problem solving abilities for high school students get score mean 89% with very valid category. The results of student responses in the limited trial obtained a score of 90.28% in the very practical category. The student response results in the field trial were 90.35% with very practical criteria. The teacher response results were 94.14% with the very practical category and the observation sheet 90.6% with the very practical criteria. From the results of validity and practicality, it can be seen that the teaching materials developed are suitable for use as learning resources.

A note on the generalized Hessian of the least squares associated with systems of linear inequalities

The goal of this note is to point out an erroneous formula for the generalized Hessian of the least squares associated with a system of linear inequalities, that was given in the paper ‘A finite Newton method for classification’ by Mangasarian (Optim. Methods Softw. 17 (2002), pp. 913–929) and reproduced multiple times in other publications. We also provide sufficient contiditions for the validity of Mangasarian's formula and show that Slater's condition guarantees that some particular elements from the set defined by Mangasarian belong to the generalized Hessian of the corresponding function.

Analisis Hambatan Belajar Pada Materi Sistem Persamaan Dan Pertidaksamaan Linear Satu Variabel Di SMP Negeri 2 Tatapaan

This research was conducted to find out what factors hinder the learning of class VII students of SMP Negeri 2 Tatapaan for the 2023/2024 academic year on the material Systems of Linear Equations and Inequalities in One Variable. The population and sample of this research is 40. This research is a quantitative descriptive research that uses survey methods and data collection using research instruments in the form of questionnaires and interviews. The questionnaire created had 24 statement items. The results of the research show that the internal aspects of learning barriers in the One Variable System of Linear Equations and Inequalities material fall into the Moderately Obstructed category (65.74%), with the lowest indicator being student readiness which falls into the Quite Impeded category (48%). Furthermore, the external aspect of barriers to student learning in the material Systems of Linear Equations and Inequalities with One Variable was included in the Moderately Obstructed category (68.6%), with the lowest indicator being Family, which was included in the Moderately Obstructed category (55.5%).

Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data

We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.

Quantitative stability for stochastic tensor variational inequalities

ABSTRACT The goal of this article is to investigate a class of stochastic tensor variational inequalities (denoted by STVI), which is a generalization of the stochastic linear variational inequalities (denoted by SLVI) and a subclass of stochastic variational inequalities (denoted by SVI). The quantitative stability of the expected residual minimization (ERM) problem for the STVI is analysed. Firstly, the existence of solutions of the ERM problem and its perturbed problem are considered. Then, the quantitative stability of the ERM problem is derived under suitable probability metrics. Finally, under some suitable assumptions, the rates of convergence of the sample average approximation (SAA) method for the optimal solution sets of the perturbed problems are discussed.

Using Constraint Propagation to Bound Linear Programs

We present an approach to compute bounds on the optimal value of linear programs based on constraint propagation. Given a feasible dual solution, we apply constraint propagation to the complementary slackness conditions and, if propagation succeeds to prove these conditions infeasible, the infeasibility certificate (in the sense of Farkas’ lemma) is reconstructed from the propagation history. This certificate is a dual-improving direction, which allows us to improve the bound. As constraint propagation need not always detect infeasibility of a linear inequality system, the method is not guaranteed to converge to a global solution of the linear program but only to an upper bound, whose tightness depends on the structure of the program and the used propagation method. The approach is suited for large sparse linear programs (such as LP relaxations of combinatorial optimization problems), for which the classical LP algorithms may be infeasible, if only for their super-linear space complexity. The approach can be seen as a generalization of the Virtual Arc Consistency (VAC) algorithm to bound the LP relaxation of the Weighted CSP (WCSP). We newly apply it to the LP relaxation of the Weighted Max-SAT problem, experimentally showing that the obtained bounds are often not far from optima of the relaxation and proving that they are exact for known tractable subclasses of Weighted Max-SAT.