System Of Linear Inequalities Research Articles

Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas’ lemma for a parametric linear inequality system with affinely adjustable variables. Our strong duality theorem not only shows that the primal and dual program values are equal, but also allows one to find the value of a two-stage robust linear program by solving a semidefinite linear program. In the case of an ellipsoidal uncertainty set, it yields a corresponding strong duality result with a second-order cone program as its dual. To illustrate the efficacy of our results, we show how optimal storage cost of an adjustable two-stage lot-sizing problem under a ball uncertainty set can be found by solving its dual semidefinite program, using a commonly available software.

Linear Programming from Fibonacci to Farkas

ABSTRACT At the beginning of the 13th century Fibonacci described the rules for making mixtures of all kinds, using the Hindu-Arabic system of arithmetic. His work was repeated in the early printed books of arithmetic, many of which contained chapters on ‘alligation', as the subject became known. But the rules were expressed in words, so the subject often appeared difficult, and occasionally mysterious. Some clarity began to appear when Thomas Harriot introduced a modern form of algebraic notation around 1600, and he was almost certainly the first to express the basic rule of alligation in algebraic terms. Thus a link was forged with the work on Diophantine problems that occupied mathematicians like John Pell and John Kersey in the 17th century. Joseph Fourier's work on mechanics led him to suggest a procedure for handling linear inequalities based on a combination of logic and algebra; he also introduced the idea of describing the set of feasible solutions geometrically. In 1898, inspired by Fourier’s work, Gyula Farkas proved a fundamental theorem about systems of linear inequalities. This topic eventually found many applications, and it became known as Linear Programming. The theorem of Farkas also plays a significant role in Game Theory.

Scalable Parallel Algorithm for Solving Non-stationary Systems of Linear Inequalities

In this paper, a scalable iterative projection-type algorithm for solving non-stationary systems of linear inequalities is considered. A non-stationary system is understood as a large-scale system of inequalities in which coefficients and constant terms can change during the calculation process. The proposed parallel algorithm uses the concept of pseudo-projection which generalizes the notion of orthogonal projection. The parallel pseudo-projection algorithm is implemented using the parallel BSF-skeleton. An analytical estimation of the algorithm scalability boundary is obtained on the base of the BSF cost metric. The large-scale computational experiments were performed on a cluster computing system. The obtained results confirm the efficiency of the proposed approach.

Gowers norms control diophantine inequalities

A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the M\"{o}bius function over certain sequences. The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues.

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in {mathbb {R}}^{n}. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of {mathbb {R}} ^{n+1}. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques.

Dual Diophantine Systems of Linear Inequalities

The paper suggests a modified version of the ℒ-algorithm for constructing an infinite sequence of integral solutions of dual systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ of linear inequalities in d + 1 variables consisting of k⊥ and k*⊥ inequalities, respectively, where k⊥ + k*⊥ = d + 1. Solutions are obtained from two recurrence relations of order d + 1. Approximation in the inequality systems $$ \mathcal{S} $$ and $$ {\mathcal{S}}^{\ast } $$ is effected with the Diophantine exponents $$ \frac{d+1-{k}^{\perp }}{k^{\perp }}-\upvarrho $$ and $$ \frac{d+1-{k}^{\ast \perp }}{k^{\ast \perp }}-\upvarrho $$ , respectively, where the deviation ϱ > 0 can be made arbitrarily small by appropriately choosing the recurrence relations. The ℒ-algorithm is based on a method for localizing units in algebraic number fields.

Lipschitz lower semicontinuity moduli for linear inequality systems

The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings.

On the extended set of weights of the OWA functions

ABSTRACTThe paper offers an extension and generalization of the definitions of the weighted arithmetic mean WAM, the ordered weighted averaging function OWA and the discrete Choquet integral using not only positive but also negative weights. Negative weights are important in various contexts, including statistics and robust aggregation, and they directly affect the monotonicity of the mentioned functions. The paper offers insights into the extended WAM, OWA and Choquet integral-based averaging functions with respect to their directional monotonicity, and establish the systems of linear inequalities which define the cone of monotonicity of these functions.

Traveling wavefronts for a Lotka–Volterra competition model with partially nonlocal interactions

The purpose of this work is to investigate the existence and stability of monostable traveling wavefronts for a Lotka–Volterra competition model with partially nonlocal interactions. We first establish an innovative lemma for the existence of positive solutions to a system of linear inequalities. By this lemma, we can construct a pair of sub-super-solutions and derive the existence result by applying the technique of monotone iteration method. It is found that if the ratio of the diffusive rate of the species without nonlocal interactions to that of the other species is not greater than a specific value, then the minimal wave speed of the wavefronts is linearly determined. Moreover, by the spectral analysis of the linearized operators, we show that the traveling wavefronts are essentially unstable in the space of uniformly continuous functions. However, if the initial perturbations of the traveling wavefronts belong to certain exponential weighted spaces, then we prove that the traveling wavefronts with noncritical wave speed are asymptotically stable in the exponential weighted spaces.

Formal Chronicle Analyses and Comparisons: How to Deal with Negative Behaviors

The overall context of this paper is the event-based behavior analysis and focuses on modeling and analyzing behaviors of interest involving time information. Any behavior of interest from any time event system is concisely defined as a set of time constrained events that must occur (positive behavior) and a set of time constrained events that must not occur (negative behavior). This article proposes a formal extension of the chronicle formalism that allows for the concise description of positive and negative behaviors. Based on this new formalism, several criteria are introduced, they formally characterize and compare a set of chronicles. A fully proved implementation of the proposed criteria is then described; it relies on the use of polyhedron techniques to solve systems of linear inequalities.

New Discrete-Time Models of Zeroing Neural Network Solving Systems of Time-Variant Linear and Nonlinear Inequalities

In this paper, a new one-step-ahead numerical differentiation rule termed 5-instant discretization formula is proposed for the first-order derivative approximation with higher computational precision. Then, by exploiting the proposed formula to discretize the continuous-time zeroing neural network [or termed, continuous-time Zhang neural network (ZNN)] models, two new discrete-time zeroing neural network [or termed, discrete-time ZNN (DTZNN)] models are proposed, analyzed and investigated for solving systems of discrete time-variant inequalities, including the system of discrete time-variant linear inequalities and the system of discrete time-variant nonlinear inequalities. For comparative purposes, the recently developed Taylor-type DTZNN models and the widely used Euler-type DTZNN models are also presented. Theoretical analyses show that the proposed DTZNN models are convergent, and their steady-state residual errors have an O(g <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) pattern with g denoting the sampling gap. Comparative numerical experimental results further substantiate the efficacy and superiority of the proposed DTZNN models for solving the systems of discrete time-variant inequalities.

Nonlinear modelling and optimal control via Takagi-Sugeno fuzzy techniques: A quadrotor stabilization

AbstractUsing the principles of Takagi-Sugeno fuzzy modelling allows the integration of flexible fuzzy approaches and rigorous mathematical tools of linear system theory into one common framework. The rule-based T-S fuzzy model splits a nonlinear system into several linear subsystems. Parallel Distributed Compensation (PDC) controller synthesis uses these T-S fuzzy model rules. The resulting fuzzy controller is nonlinear, based on fuzzy aggregation of state controllers of individual linear subsystems. The system is optimized by the linear quadratic control (LQC) method, its stability is analysed using the Lyapunov method. Stability conditions are guaranteed by a system of linear matrix inequalities (LMIs) formulated and solved for the closed loop system with the proposed PDC controller. The additional GA optimization procedure is introduced, and a new type of its fitness function is proposed to improve the closed-loop system performance.

Locally Polynomial Method for Solving Systems of Linear Inequalities

A numerical method combining a gradient technique with the projection onto a linear manifold is proposed for solving systems of linear inequalities. It is shown that the method converges in a finite number of iterations and its running time is estimated as a polynomial in the space dimension and the number of inequalities in the system.

Accelerating Fourier–Motzkin elimination using bit pattern trees

The paper concerns the elimination of a set of variables from a system of linear inequalities. We employ the widely used Fourier–Motzkin elimination method extended with the Chernikov rules. A straightforward implementation of the algorithm results in extensive enumeration during the most computationally demanding stage. We propose a new way of checking Chernikov rules using bit pattern trees as an accelerating data structure to avoid extensive enumeration. The bit pattern tree is a data structure based on k-d tree used to accelerate the double description method. First we describe an adaptation of that approach to check the second Chernikov rule in Fourier–Motzkin elimination. We also propose a new algorithm that employs bit pattern trees to accelerate both Chernikov rules. Presented results of computational evaluation prove competitiveness of the proposed algorithms.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

Robust stability and evaluation of the quality functional for linear control systems with matrix uncertainty

New methods of robust stability analysis for equilibrium states and optimization of linear dynamic systems are developed. Sufficient stability conditions of the zero state are formulated for a linear control systems with uncertain coefficient matrices and measurable output feedback. In addition, a general quadratic Lyapunov function and ellipsoidal set of stabilizing matrices for the feedback amplification coefficients are given. Application of the results is reduced to solving the systems of linear matrix inequalities.

PENGARUH MODEL TREFFINGER TERHADAP HASIL BELAJAR KOGNITIF SISWA PADA MATERI SISTEM PERTIDAKSAMAAN LINEAR DUA VARIABEL

Students’ knowledge about apperception material is important in supporting the success of a learning. One of the learning innovations to maximize apperception material can be done with treffinger learning model. This research aims to find out how the effect of the learning model treffinger on students’ cognitive learning outcomes for the subject of two variables linear inequality system. This research was conducted at SMA Negeri 1 Gunungsari, West Lombok. The research design used is quasi experimental design. Sampling in this study uses simple random sampling technique. X MIA 3 class as an experimental class and X MIA 1 class as a control class. The research instrument was a test sheet and observation sheet for the implementation of the learning model treffinger. The results of the research were obtained by using t-test, the average score of cognitive results of the experimental class students was 68.88 and the control class was 64.10, obtained tcount&gt;ttable (2.557 &gt; 2.042). So it can be conclude that H0 is rejected and Ha is accepted, this means that there are significant treffinger learning model towards students cognitive learning outcomes in materials twi variable linear inequality system at class X grade of SMA Negeri 1 Gunungsari academic year 2018/2019.

Newton-Type Method for Solving Systems of Linear Equations and Inequalities

A Newton-type method is proposed for numerical minimization of convex piecewise quadratic functions, and its convergence is analyzed. Previously, a similar method was successfully applied to optimization problems arising in mesh generation. It is shown that the method is applicable to computing the projection of a given point onto the set of nonnegative solutions of a system of linear equations and to determining the distance between two convex polyhedra. The performance of the method is tested on a set of problems from the NETLIB repository.

Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements

This paper proposes a novel Takagi-Sugeno fuzzy predictor observer to tackle the problem of the constant and known delay in the measurements. The proposed observer is developed for a trajectory-tracking problem of a wheeled mobile robot where a parallel-distributed compensation control is used to control the robot. The L2-stability of the proposed observer is also proven in the paper. Both, the control and the observer gains are obtained by solving the proposed system of linear matrix inequalities. To illustrate the efficiency of the proposed approach, an experimental comparison with another predictor observer was done.

Linear Matrix Inequalities in Stability Problems: Retrospective and Theoretical Aspects

Some aspects of the development of the theory of linear matrix inequalities are considered. A number of results obtained at the initial stage of the development of this theory, both in the development of numerical methods and in obtaining analytical conditions for their solvability, are highlighted. The main attention is focused on the system of linear matrix inequalities arising in solving the absolute stabi lity problem. E. S. Pyatnitskiy and his followers showed that the solvability of this system is a criterion for the existence of a quadratic Lyapunov function and a sufficient condition for absolute stability. The prerequisites leading to this result are considered here. The use of the considered system of inequalities for studying the stability of hybrid systems described by differential inclusions and switching systems is shown. An analysis is given of citing some works of Pyatnitskiy’s school on the theory of stability and the theory of systems of linear matrix inequalities, from which the relevance of the results of these works at the present time follows.In developing numerical methods, it was first shown in the work of Pyatnitskiy and Skorodinskiy that the solvability problem for a system of linear matrix inequalities reduces to a convex programming problem. An interesting gradient algorithm for finding solutions to such a system is also presented. In analyzing analytical conditions of solvability, an unsolvability criterion for the system of our interest obtained by Kamenetskiy and Pyatnitskiy is noted. In modern terms, this result can be considered as a description of an admissible set in the dual semidefinite programming problem. A similar result is given in the famous book by S. Boyd et al. The paper shows that the result of Boyd et al. is a simple corollary of the unsolvability criterion. Here the unsolvability criterion is generalized and refined.