System Of Linear Inequalities Research Articles

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.

Binary Extended Formulations and Sequential Convexification

A binarization of a bounded variable x is obtained via a system of linear inequalities that involve x together with additional variables [Formula: see text] in [Formula: see text] so that the integrality of x is implied by the integrality of [Formula: see text]. A binary extended formulation of a mixed-integer linear set is obtained by adding to its original description binarizations of its integer variables. Binary extended formulations are useful in mixed-integer programming as imposing integrality on 0/1 variables rather than on general integer variables has interesting convergence properties and has been studied from both the theoretical and practical points of view. We study the behavior of binary extended formulations with respect to sequential convexification. In particular, given a binary extended formulation and one of its variables x, we study a parameter that measures the progress made toward enforcing the integrality of x via application of sequential convexification. We formulate this parameter, which we call rank, as the solution of a set-covering problem and express it exactly for the classic binarizations from the literature. Funding: M. Aprile and M. Di Summa are supported by the University of Padova [SID 2019 Grant C94I20000280005].

A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems

This work is a short review of the state of the art aiming to contribute to the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. It shows that the algebraic structure of their solutions consists of the sum of a linear subspace, an acute cone, and a polytope, and that adequate software exists to obtain, in their simplest forms, these three components. The work describes, based on orthogonality and polarity, homogeneous and complete systems of inequalities, the associated compatibility problems, and their relations with convex polyhedra and polytopes, which are the only possible solution for bounded problems, the most common in real practice. The compatibility and the observability problems, including their symbolic forms, are analyzed and solved, identifying the subsets of unknowns with unique solutions and those unbounded, important items of information with practical relevance in artificial intelligence and automatic learning. Having infinitely many solutions of a given problem allows us to find solutions when some of the assumptions fail and unexpected constraints come into play, a common situation for engineers. The linear programming problem becomes trivial when the set of all solutions is available and all solutions are obtained, contrary to the case of standard programs that provide only one solution. Several examples of applications to several areas of knowledge are presented, illustrating the advantages of solving these systems of inequalities.

Criterion of solvability of the discriminant analysis problem underlying the medical organization management model

The article considers the task of building an effective and efficient organization of the management processes of a medical organization. A distinctive feature is noted - social orientation as the main function, and not just profit maximization with limited resources. This approach requires taking into account not only financial and resource components, but also the impact of many other external factors. However, the authors, realizing the difficulty of describing them, propose to take into account the effect of the "social orientation of the institution" in an implicit form when describing the financial condition of the organization. Taking into account implicit, including external, factors is possible when implementing models not only of statistical and financial analysis, as is usually accepted, but also their extensions to some image recognition methods, which often have a simple computer implementation. In particular, the formalization of the financial condition model of the organization is proposed, which reduces to the problem of mathematical programming with state vectors of both internal and external factors. The possibility of considering a decision-making model that would take into account the influence of possible external factors is also considered, while the "informality" of the problem in the sense of the absence of classical methods of solution is indicated. In this regard, it is proposed to reduce the model to the task of discriminant analysis, which is essentially a system of inequalities. Methods for solving systems of linear inequalities are known, but they are difficult to implement. For this reason, it is proposed to add an optimality criterion (objective function) to the discriminant analysis model, which allows to obtain a linear programming problem that already has simpler computer implementations of solutions. However, this possibility is present only in the case of linear constraints, which is a somewhat special case of the general problem. To solve the problem, in its broadest sense, it is proposed not only to reduce it to mathematical programming, and then to discriminant analysis, but also an effective recognition algorithm indicating the solvability of the problem. The process of constructive problem solving often requires a lot of time and effort, even with the powerful development of modern technologies. With the help of the given recognition algorithm, the discriminant analysis model in some formulation is output in a matrix form, which allows us to conclude when the model has the existence of a separating functional. The authors also propose the use of majority committee methods. The derived algorithm is quite simple to implement, but at the same time effective and efficient in practice, which makes it universal for this kind of tasks.

Prescribed-time robust ZNN models for solving equality and inequality systems

At present, there are few studies on solving time-variant linear equality and inequality systems (TVLEIS) under noise interference, and the numerical algorithm has limitations in solving the TVLEIS problems. Therefore, to determine the online solution of the TVLEIS in a complex environment, two prescribed-time robust zeroing neural network (PTRZNN) models are proposed, investigated, and verified in this paper. The PTRZNN models have a faster convergence rate and superior robustness compared with other zeroing neural network models activated by common activation functions. In addition, the detailed theoretical derivation of the prescribed-time convergence and robustness of the PTRZNN models is provided. The effectiveness and superiority of the PTRZNN models for determining the TVLEIS are further demonstrated by simulation results. It is worth mentioning that the design idea of the PTRZNN models is applied to the multi-agent system, which shows the practical value of the PTRZNN models.

Обобщение неэлементарных линейных регрессий

<p>Earlier, the author developed a non-elementary linear regression consisting of a linear part and all possible combinations of min and max binary operations. This article is devoted to its generalization. For the first time a non-elementary linear regression with a linear part and all possible combinations of binary, ternary, ..., l-ary operations min and max has been introduced. The proposed model generalizes both linear regression and the Leontief function, and can be effectively used both for predicting and for interpreting the study object functioning. An estimation algorithm was developed using the method of least squares for non-elementary linear regressions without a linear part and with an l-ary operation min (max), i.e. regressions with specification in the form of a Leontief function. The essence of the algorithm is to form a set of possible values of slope coefficients, from which a point is selected with the minimum value of the residual sum of squares. A system of linear inequalities is identified that makes it possible to form such a set. Using the algorithm, a model of the gross regional product of the Irkutsk region was construct and its interpretation was given.</p>

Activity propagation in systems of linear inequalities and its relation to block-coordinate descent in linear programs

Activity propagation in systems of linear inequalities and its relation to block-coordinate descent in linear programs

Development and Validation of Learning Activity Sheets in Rational Expressions and Linear Equations and Inequalities

The purpose of this research study is to develop Learning Activity Sheets (LAS) in Rational Expressions and Linear Equations and Inequalities for the students. Data were collected via survey questionnaire from three sets of evaluators. The LAS was also administered from 100 students of New Society National High School to further scrutinize its validity. The results were interpreted and analysed using the rating scale and statistical tools including on-way ANOVA and mean.
 Based from the results of the study, the researcher concluded that: there were fourteen competencies included in the developed LAS which are the not mastered and least mastered competencies of the students. The competency that received the highest mean percentage on the competency test was LAS 17 (graphs and illustrates a linear function and its (a) domain; (b) range; (c) table of values; (d) intercepts; and (e) slope). On the other hand, the competency that received the lowest mean percentage on the competency test was LAS 12 (solves problems involving systems of linear inequalities in two variables). The evaluators found the developed LAS to be effective and useful in terms of objectives, concepts, skills, usability, appropriateness, and adequacy with an interpreted mean rating of very high extent. The researcher also concluded that there is no significant difference in the standard response of the evaluators which is composed of Grade 8 math teachers, master teachers, and LRMDS-trained teachers. Based on the mean percentage results of the administered LAS, there was also a considerable improvement in the students' learning.

Design of continuous-time recurrent neural networks with piecewise-linear activation function for generation of prescribed sequences of bipolar vectors

A recurrent neural network (RNN) can generate a sequence of patterns as the temporal evolution of the output vector. This paper focuses on a continuous-time RNN model with a piecewise-linear activation function that has neither external inputs nor hidden neurons, and studies the problem of finding the parameters of the model so that it generates a given sequence of bipolar vectors. First, a sufficient condition for the model to generate the desired sequence is derived, which is expressed as a system of linear inequalities in the parameters. Next, three approaches to finding solutions of the system of linear inequalities are proposed: One is formulated as a convex quadratic programming problem and others are linear programming problems. Then, two types of sequences of bipolar vectors that can be generated by the model are presented. Finally, the case where the model generates a periodic sequence of bipolar vectors is considered, and a sufficient condition for the trajectory of the state vector to converge to a limit cycle is provided.

Development of a two-variable linear inequality system mathematics learning design with a problem-based learning model

Teachers are designers, implementers, and evaluators of learning. The teacher's ability to design learning that favors students will greatly affect the quality of the learning process and the learning outcomes that will then be obtained by students. This study aims to develop a problem-based learning design for the more interesting topic of systems of two-variable linear inequalities. This development research uses the Dick Carey development model. The Dick Carey model can be an alternative for teachers in designing pro-student learning because, by design, this model starts with student and context analysis as well as instructional analysis, which will provide sufficient information and evidence about the characteristics and needs of students as well as instructional objectives that are appropriate to the level of their education and stage of development.

On Cm solutions to systems of linear inequalities

Recent work of C. Fefferman and the first author [8] has demonstrated that the linear system of equations∑j=1MAij(x)Fj(x)=fi(x)(i=1,…,N), has a Cm solution F=(F1,…,FM) if and only if f1,…,fN satisfy a certain finite collection of partial differential equations. Here, the Aij are fixed semialgebraic functions.In this paper, we consider the analogous problem for systems of linear inequalities:∑j=1MAij(x)Fj(x)≤fi(x)(i=1,…,N). Our main result is a negative one, demonstrated by counterexample: the existence of a Cm solution F may not, in general, be determined via an analogous finite set of partial differential inequalities in f1,…,fN.