System Of Linear Inequalities Research Articles

Selectivity in probabilistic causality: Where psychology runs into quantum physics

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one determine, for each of the outputs, which of the inputs it is influenced by? The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of “what influences what” is equivalent to that of the existence of a joint distribution for a certain set of random variables. For inputs and outputs with finite sets of values this criterion translates into a test of consistency of a certain system of linear equations and inequalities (Linear Feasibility Test) which can be performed by means of linear programming. While new in the behavioral context, both this test and the Joint Distribution Criterion on which it is based have been previously proposed in quantum physics, in dealing with generalizations of Bell inequalities for the quantum entanglement problem. The parallels between this problem and that of selective influences in behavioral sciences are established by observing that noncommuting measurements in quantum physics are mutually exclusive and can therefore be treated as different levels of one and the same factor.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

Weakly linear systems of fuzzy relation inequalities and their applications: A brief survey

Weakly linear systems of fuzzy relation inequalities and equations have recently emerged from research in the theory of fuzzy automata. From the general aspect of the theory of fuzzy relation equations and inequalities homogeneous and heterogeneousweakly linear systems have been discussed in two recent papers. Here we give a brief overview of the main results from these two papers, as well as from a series of papers on applications of weakly linear systems in the state reduction of fuzzy automata, the study of simulation, bisimulation and equivalence of fuzzy automata, and in the social network analysis. Especially, we present algorithms for computing the greatest solutions to weakly linear systems.

Methods of finding dynamic flows in networks obeyed generalized Kirchhoff’s Law

Generalized Kirchhoff's law for flows in networks is modelled by a system of linear inequalities that has the structure of the corresponding graph. In the case when the graph has more than one cycle, certain difficulties arise in solving such a system of inequalities. This paper proposes a method for replacing a cycle in the graph by a star.

Solution Building for Arbitrary System of Linear Inequalities in an Explicit Form

The known Fourier-Chernikov algorithm of linear inequality system convolution is complemented with an original procedure of all dependent (redundant) inequalities deletion. The concept of “almost dependent” inequalities is defined and an algorithm for further reducing the system by deletion of these is considered. The concluding algorithm makes it possible to hold actual-time convolution of a general inequality system containing up to 50 variables with the rigorous method of dependent inequalities deletion and up to 100 variables with the approximate method of one. The main application of such an approach consists in solving linear inequality system in an explicit form. These results are illustrated with a series of computer experiments.

Investigation of Solutions of Linear Problems of Euclidean Combinatorial Optimization on Permutations with Additional Restrictions. Part III

The extension of the well-known method of solving the systems of linear inequalities for finding the general formula of the points of a permutation polyhedron with additional restrictions is studied.

Robust extraction of fuzzy rules with artificial neural network based on fuzzy inference system

The paper presents a method of parameters estimation for artificial neural network based on fuzzy inference system (ANNBFIS). It is based on deterministic annealing, e-insensitive learning by solving a system of linear inequalities and robust fuzzy c-means clustering. The proposed algorithm allows to improve the neuro-fuzzy modelling quality by increasing the generalisation ability and outliers robustness. To find the unknown number of fuzzy rules we proposed the procedure of robust clusters merging. The performance of the learning method is demonstrated through the benchmark sunspot prediction problem.

A very short algebraic proof of the Farkas Lemma

We present a very short algebraic proof of a generalisation of the Farkas Lemma: we set it in a vector space of finite or infinite dimension over a linearly ordered (possibly skew) field; the non-positivity of a finite homogeneous system of linear inequalities implies the non-positivity of a linear mapping whose image space is another linearly ordered vector space. In conclusion, we briefly discuss other algebraic proofs of the result, its special cases and related results.

Weakly linear systems of fuzzy relation inequalities: The heterogeneous case

New types of systems of fuzzy relation inequalities and equations, called weakly linear, have been recently introduced in [J. Ignjatović et al., On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets Syst. 161 (2010) 3081–3113]. The mentioned paper dealt with homogeneous weakly linear systems, composed of fuzzy relations on a single set, and a method for computing their greatest solutions has been provided. This method is based on the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations. Here we adapt this method for computing the greatest solutions of heterogeneous weakly linear systems, where the unknown fuzzy relation relates two possibly different sets. We also introduce and study quotient fuzzy relational systems and establish relationships between solutions to heterogeneous and homogeneous weakly linear systems. Besides, we point out to applications of the obtained results in the state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata, as well as in the positional analysis of fuzzy social networks.

A time optimal control problem for the collision‐free robot motion planning

AbstractA time optimal control problem for the computation of the fastest collision‐free trajectory of a robot surrounded by obstacles is presented. The dynamics of the robot is governed by ordinary differential equations. To build the collision avoidance criterion, the robot and the obstacles are represented as finite unions of convex polyhedra. Each polyhedron is described by a system of linear inequalities. This description combined with linear programming arguments allows us to include the collision avoidance as linear state constraints in the optimal control problem. The resulting system is then solved by a sequential quadratic programming method. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Correction of inconsistent systems of linear inequalities with block matrices by minimax criterion

The problem of correcting inconsistent systems of linear inequalities (SLI) with block matrices by the minimax criterion on weighted Euclidean norm of the blocks of correction matrices.

Lagrangian-Penalization Algorithm for Constrained Optimization and Variational Inequalities

Let X, Y be real Hilbert spaces. Consider a bounded linear operator A : X → Y and a nonempty closed convex set \(\mathcal{C}\subset Y\). In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems $$(\mathcal{P})\qquad\qquad\qquad \inf \{f(x)\ :\ Ax\in \mathcal{C}\}, $$ where f is a proper lower-semicontinuous convex function on X; and variational inequalities $$(\mathcal{VI})\qquad\qquad\qquad 0\in\mathcal{M} x+A^*N_{\mathcal{C}}(Ax), $$ where \(\mathcal{M}:X\rightrightarrows X\) is a maximal monotone operator and \(N_{\mathcal{C}}\) denotes the normal cone to the set \(\mathcal{C}\). Our method combines a penalization procedure involving a bounded sequence of parameters, with the predictor corrector proximal multiplier method of Chen and Teboulle (Math Program 64(1, Ser A):81–101, 1994). Under suitable assumptions the sequences generated by our algorithm are proved to converge weakly to solutions of \((\mathcal{P})\) and \((\mathcal{VI})\). As applications, we describe how the algorithm can be used to find sparse solutions of linear inequality systems and solve partial differential equations by domain decomposition.

Relaxation methods for solving linear inequality systems: converging results

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. In González-Gutiérrez and Todorov (Optim. Lett. doi:10.1007/s11590-010-0244-4, 2011), an algorithm, called extended relaxation method, for solving the feasibility problem has been proposed by the authors. Convergence of the algorithm has been proven. In this paper, we consider a class of extended relaxation methods depending on a parameter and prove their convergence. Numerical experiments have been provided, as well.

Stability and stabilization of the family of pseudolinear differential systems

New methods for the analysis of the robust stability of equilibrium states are developed for some classes of nonlinear differential systems. We formulate sufficient conditions for the stability of the trivial solutions of the families of pseudolinear controlled systems with undetermined coefficient matrices and feedback from the measured output. A method is developed for the analysis of stability according to the first approximation to the family of nonlinear systems. The application of the obtained results is reduced to the solution of systems of linear differential matrix inequalities. We present an example of a system of stabilization of a double inverted pendulum.

Decomposition method in correction problems for inconsistent systems of linear inequalities with partitioned matrices

Correction problems for inconsistent systems of linear inequalities with partitioned matrices are examined. The criteria used are based on the minimum of the sum of squares and the minimax weighted Euclidean norm of a block in the correction matrix.

Viability criteria for differential inclusions

A method of verifying the viability criterion at a given point for a region with nonsmooth boundary, which is expressed by a quasidifferentiabl function, under a differential inclusion which is a convex hull of finitely many functions, is proposed. By this method, determining the viability is transformed into solving a number of systems of linear inequalities, or equivalently solving a number of linear programming problems. For the other differential inclusion, called the generalized convex process, it is shown that viability condition holds for a polytope if and only if it holds at all of its vertices. This result is an extension of corresponding one for a linear control system.

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l ∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

The inertia sets of symmetric tridiagonal sign-patterns

We give a complete description of the possible inertias of real symmetric matrices with a given (arbitrarily prescribed) tridiagonal symmetric sign pattern. The inertia set of such a pattern is a subset of Z2 given by a system of linear inequalities, whose parameters are determined from the sign sequence of the pattern’s diagonal entries.

Data envelopment analysis classification machine

This paper establishes the equivalent relationship between the data classification machine and the data envelopment analysis (DEA) model, and thus build up a DEA based classification machine. A data is characterized by a set of values. Without loss of the generality, it is assumed that the data with a set of smaller values is preferred. The classification is to label if a particular data belongs to a specified group according to a set of predetermined characteristic or attribute values. We treat such a data as a decision making unit ( DMU) with these given attribute values as input and an artificial output of identical value 1. Then classifying a data is equivalent to testing if the DMU is in the production possibility set, called acceptance domain, constructed by a sample training data set. The proposed DEA classification machine consists of an acceptance domain and a classification function. The acceptance domain is given by an explicit system of linear inequalities. This makes the classification process computationally convenient. We then discuss the preference cone restricted classification process. The method can be applied to classifying large amount of data. Furthermore, the research finds that DEA classification machines based on different DEA models have the same format. Input-oriented and output-oriented DEA classification machines have similar properties. The method developed has great potential in practice with its computational efficiency.

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).