System Of 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.

A low-complexity solution for LMIs of hardly constrained nonlinear MIMO tracking control systems

This research is dedicated to the discrete-time tracking control of nonlinear multi-input multi-output (MIMO) Newtonian mechanics systems with an equal number of inputs and outputs, in the presence of input saturation and high-order dynamics of the actuation system. Firstly, the continuous-time model of the plant and actuation system has been transformed into the discrete-time domain using Adams–Bashforth and Adams–Moulton discretization techniques. In the next step, considering saturation boundaries, the modified position and velocity references are derived using the discrete-time reaching law-based control approach. These modified references are obtained through an optimization algorithm that calculates the closest achievable position and velocity trajectories with respect to the desired reference trajectory. Modifying desired trajectories in MIMO systems results in the formation of a system of linear matrix inequalities (LMIs). Solving such systems of inequalities requires the use of complex methods, and this difficulty increases with the increase in the system’s degrees of freedom (DOFs). The proposed strategy offers a more straightforward approach to solving these inequalities. Once the modified references are calculated, the corresponding position and velocity control signals can be obtained using the mentioned reaching law strategies. The final step of the proposed algorithm is to combine the resulting position and velocity control signals using a Gaussian-weighted averaging method to form the main control signal. A desirable performance for the control system can be achieved by appropriately adjusting the standard deviations of the Gaussian weighing functions. To evaluate the performance characteristics of the proposed method, it has been applied in the tracking control procedure of a nonlinear 3-DOF mechanical system. The numerical simulation results demonstrate the effectiveness of the method.

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.

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%).

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.

Interval-based KKT framework for support vector machines and beyond

Our article proves inequalities for interval optimization and shows that feasible and descent directions do not intersect in constrained cases. Mainly, we establish some new interval inequalities for interval-valued functions by defining LC-partial order. We use LC-partial order to study Karush–Kuhn–Tucker (KKT) conditions and expands Gordan's theorems for interval linear inequality systems. By applying Gordan's theorem, we can determine the best outcomes for interval optimization problems (IOPs) that have constraints, such as Fritz John and KKT conditions. The optimality conditions are observed with inclusion relations rather than equality. We can use the KKT condition for binary classification with interval data and support vector machines(SVMs). We present some examples to illustrate our results.

Nutrient research of chopped semi-finished products enriched with a protein-carbohydrate composition

The difficulty in solving a multicomponent formulation problem is that five or more ingredients are currently used in product design. Solving a system of linear equations and inequalities with such a large number of variables manually presents significant difficulties, in which formulation errors are not excluded. In the work the methodology of computer modelling of multicomponent meat products is presented, which is realized on an example of technology of minced meat production. The aim of the research is to design the mathematical model of minced meat with the addition of new protein-carbohydrate compositions (PCC). The objects of research are minced meat, PCC, which includes chickpea flour, whey protein concentrate, ground soy and water for hydration. Implementation of the design method was carried out with the software system Microsoft Excel with the "Solution Finder" add-on. Work with the Excel spreadsheet processor is based on entering the data required for calculation, calculation formulas into the corresponding cells of the spreadsheet. The article presents the results of the study of the nutrient composition of minced meat with the addition of the new PCC. By means of mathematical modelling, the formulation of minced meat with PCC was optimized. Nutritional and energy value, vitamin, mineral and amino acid content of the obtained minced meat were determined. A targeted combination of ingredients made it possible to obtain food products with a given composition and properties.

Approximation of the Objective Function of Single-Machine Scheduling Problem

The problem of the approximation of the coefficients of the objective function of a scheduling problem for a single machine is considered. It is necessary to minimize the total weighted completion times of jobs with unknown weight coefficients when a set of problem instances with known optimal schedules is given. It is shown that the approximation problem can be reduced to finding a solution to a system of linear inequalities for weight coefficients. For the case of simultaneous job release times, a method for solving the corresponding system of inequalities has been developed. Based on it, a polynomial algorithm for finding values of weight coefficients that satisfy the given optimal schedules was constructed. The complexity of the algorithm is O(n2(N+n)) operations, where n is the number of jobs and N is the number of given instances with known optimal schedules. The accuracy of the algorithm is estimated by experimentally measuring the function ε(N,n)=1n∑j=1n∣wj−wj0∣wj0, which is an indicator of the average modulus of the relative deviation of the found values wj from the true values wj0. An analysis of the results shows a high correlation between the dependence ε(N,n) and a function of the form α(n)/N, where α(n) is a decreasing function of n.

Formally verifying decompositions of stochastic specifications

According to the principles of compositional verification, verifying that lower-level components satisfy their specification ensures that the whole system satisfies its top-level specification. The key step is to ensure that the lower-level specifications constitute a correct decomposition of the top-level specification. In a non-stochastic context, such decomposition can be analyzed using techniques of theorem proving. In industrial applications, especially in safety-critical systems, specifications are often of stochastic nature, for example, giving a bound on the probability that a system failure will occur before a given time. A decomposition of such a specification requires techniques beyond traditional theorem proving. The first contribution of the paper is a theoretical framework that allows the representation of, and reasoning about, stochastic and timed behavior of systems as well as specifications for such behavior. The framework is based on traces that describe the continuous-time evolution of a system, and specifications are formulated using timed automata combined with probabilistic acceptance conditions. The second contribution is a novel approach to verifying decompositions of such specifications by reducing the problem to checking emptiness of the solution space for a system of linear inequalities.

Switched max-plus linear-dual inequalities: cycle time analysis and applications

AbstractP-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the primal and dual operations of the max-plus algebra. We define a new class of models called switched LDIs (SLDIs), which allow to switch between different modes of operation, each corresponding to a set of LDIs, according to a sequence of modes called schedule. In this paper, we focus on the analysis of SLDIs when the considered schedule is fixed and either periodic or intermittently periodic. We show that SLDIs can model a wide range of applications including single-robot multi-product processing networks, in which every product has different processing requirements and corresponds to a specific mode of operation. Based on the analysis of SLDIs, we propose algorithms to compute: i. minimum and maximum cycle times for these processes, improving the time complexity of other existing approaches; ii. a complete trajectory of the robot including start-up and shut-down transients.

The Difficulty of Students’ Reflective Thinking in Problems Solving of Linear Program

The identification in this study with the aim is to describe how difficult it is for students to think reflectively when solving math problems, especially in linear programming material. Based on the purpose of this study, the type of research is qualitative with a descriptive exploratory approach. Data collection techniques used are: (1) test instruments; (2) interview instruments, and 3) documentation. Analysis of research data namely: (1) research data reduction, (2) data exposure, (3) data triangulation, and (4) drawing conclusions. The subjects in this study were 24 high school students. Then 2 students were selected as subjects for each category (high, medium, and low). The results of the study show that students with high mathematical abilities have difficulty in reflecting, namely, 1) difficulty connecting new information with previous understanding, so they are not careful when identifying stories in the form of mathematical models, 2) difficulties in aspects of finding relationships and formulating solutions, students mistake the sign of linear inequality two variables, 3) difficulty in evaluating aspects of the completion process, students find it difficult to recall the function graph material to solve problems using the graphical method. Students with moderate mathematical abilities, namely: 1) difficulties in the aspect of connecting new knowledge with previous understanding, students need to be careful in solving contextual problems, 2) difficulties in aspects of finding relationships and formulating solutions, students have difficulty recalling function graph material, difficult to shade the area of settlement, 3) difficulties when students evaluate the completion process. Students find it difficult to prove whether the answer is correct or not by using the graphical method. Students with low mathematical ability, 1) difficulties in the aspect of connecting new knowledge with previous understanding, students find it difficult to translate story problems into mathematical models, it is difficult to recall the material of a two-variable linear inequality system, 2) difficulties in the aspect of finding relationships and formulating solutions, students have difficulty finding coordinates, drawing graphs, finding intersection points, substituting corner points into the objective function. 3) difficulties in evaluating aspects of the completion process, students find it difficult to prove the correctness of the answers obtained by the graphical method. The difficulties experienced by students in reflective thinking were caused by students not remembering previous material related to linear programming, as well as students’ difficulties in the dimensions of fact, concept and procedural knowledge.

АЛГЕБРАЇЧНИЙ МЕТОД СИНТЕЗУ БЕЗПОМИЛКОВОЇ БІНАРНОЇ НЕЙРОННОЇ МЕРЕЖІ

A mathematical model of the problem of calculating the weighting coefficients of a binary neural network is given. It is proved that in the case of step functions of neuron activation, this model is a system of linear inequalities, which is incompatible for most practical problems. A method of analyzing the system of inequalities is proposed, which allows calculating the values of the weighting coefficients and synthesizing the structure of the neural network, which ensures the absolute accuracy of the output signals. The algorithm and an implementation example are given. Keywords: neural network, mathematical model, analysis, synthesis, error.

Admissible Control Laws for Constrained Linear Power Flow: The General Case

Linearized power flow with line flow and voltage constraints can be modeled as a system of linear inequalities depending on the power injections. When some injections are controlled by the grid operator while others are determined exogenously, robust control aims at determining grid operator's actions under imperfect system observability such that the grid state is feasible for all possible realizations of the exogenous actions. It was shown how to design and analyze such admissible control policies efficiently for the subclass of affine control laws, but this paper shows that there are important cases that require more general control policies. We show that, for the constrained linear power flow setting, general admissible control policies can always be chosen as piecewise affine (PWA) mappings. The PWA mapping can be explicitly characterized offline or control actions can be computed online by solving an optimization problem given a current observation. For the latter setup, we provide an algorithm that verifies offline that the online control scheme is admissible. This verification step is a crucial precondition to apply the online control scheme in real, safety-critical power grids. The proposed framework is demonstrated with applications to voltage regulation in active distribution grids subject to uncertain prosumers and low observability.

On the Construction of the Graph of Discrete States of a Switched Affine System

The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.