System Of Linear Inequalities Research Articles

An algorithm for constructing the skeleton graph of degenerate systems of linear inequalities.

Derive the quantitative predictions of constraint-based models require of conversion algorithms to enumerate and construct the skeleton graph conformed by the extreme points of the feasible region, where all constraints in the model are fulfilled. The conversion is problematic when the system of linear constraints is degenerate. This paper describes a conversion algorithm that combines the best of two methods: the incremental slicing of cones that defeats degeneracy and pivoting for a swift traversal of the set of extreme points. An extensive computational practice uncovers two complementary classes of conversion problems. The two classes are distinguished by a practical measure of complexity that involves the input and output sizes. Detailed characterizations of the complexity classes and the corresponding performances of the algorithm are presented. For the benefit of implementors, a simple example illustrates the stages of the exposition.

Attraction domain estimate for single-input affine systems with constrained control

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

An algorithm for solving two-sided interval system of max-plus linear equations

We study six types of solutions (weak, strong, tolerance, control, Left-localized, and Right-localized solutions) to a two-sided interval system of max-plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions. These conditions are in the form of two-sided systems of max-plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems. Therefore, we could work on the interval inclusion linear systems, instead of directly solving these six types of solutions themselves. An optimization problem with the two-sided interval system of max-plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems.

A MATLAB Toolbox for Estimating the Second Moments of Earthquake Ruptures

ABSTRACT I present a set of routines in MATLAB for estimating the second‐degree moments of an earthquake’s rupture from far‐field body waves. The second moments describe the length, width, duration, and directivity of a rupture. The second‐moments approach is particularly useful when a seismic dataset is dense enough to resolve the primary finite‐source properties but the geodetic data needed for a well‐resolved finite‐fault inversion are not available. In particular for M w 3–6 earthquakes, this approach can be a useful way to estimate rupture area without the assumptions of typical corner‐frequency approaches. The provided software utilizes empirical Green’s function deconvolution to isolate the apparent source time function (ASTF) for each station and phase. The spatial variations in the duration of the ASTF are quantified and inverted for the second moments. The inverse problem is solved using MATLAB’s convex optimization routines for systems of linear matrix inequalities. An error analysis using the jackknife and bootstrap methods is included. An example M w 4.7 earthquake from the San Jacinto fault is used to demonstrate the method.

Methods of search for stable solutions of some optimization problems in the theory of recognition by precedents

In constructing some models of recognition algorithms, there arise a number of optimization problems. The search for the optimal consistent subsystem of a given system of linear inequalities plays an important role in the process of data analysis in the theory of recognition by precedent. The optimality of a required subsystem is defined by a number of the conditions imposed on it. Earlier the author developed several algorithms for solving the problems of this type. These algorithms are based on exhaustive search for nodal subsystems of a given system of linear inequalities. In the search for optimal consistent subsystem, these algorithms find boundary decisions. However, in practical application often it is necessary to find a stable solution. So, when looking for logical regularities of a special type, it is required to find a set of non-degenerate polyhedra describing a certain class of objects in a space of features. Therefore, linear inequalities systems corresponding to these polyhedra must be stable. In this paper, we propose a method for modifying the previously developed algorithms to select the stable consistent subsystem of highest possible power and find its stable solution.

Output Stabilization and Weighted Suppression of Disturbances in Discrete-Time Control Systems

Output stabilizability conditions for some classes of discrete-time linear and nonlinear control systems are formulated by using dynamic controllers. Methods for constructing static and dynamic controllers that provide a specified evaluation of the weighted damping level of external and initial disturbances are proposed for systems with controllable and observable outputs. The implementation of these methods with the use of static state feedback or complete order dynamic controllers is based on the solution of systems of linear matrix inequalities. The example of a controller synthesis for a two-mass mechanical system is given.

(Sub-)Gradient Formulae for Probability Functions of Random Inequality Systems under Gaussian Distribution

We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in the case of linear random inequality systems. In the case of a constant coefficient matrix, an upper estimate for even the smaller Mordukhovich subdifferential is proven.

Dynamic Pricing to Minimize Maximum Regret

We consider a dynamic pricing problem that involves selling a given inventory of a single product over a short, two‐period selling season. There is insufficient time to replenish inventory during this season, hence sales are made entirely from inventory. The demand for the product is a stochastic, nonincreasing function of price. We assume interval uncertainty for demand, that is, knowledge of upper and lower bounds but not a probability distribution, with no correlation between the two periods. We minimize the maximum total regret over the two periods that results from the pricing decisions. We consider a dynamic model where the decision maker chooses the price for each period contingent on the remaining inventory at the beginning of the period, and a static model where the decision maker chooses the prices for both periods at the beginning of the first period. Both models can be solved by a polynomial time algorithm that solves systems of linear inequalities. Our computational study demonstrates that the prices generated by both our models are insensitive to errors in estimating the demand intervals. Our dynamic model outperforms our static model and two classical approaches that do not use demand probability distributions, when evaluated by maximum regret, average relative regret, variability, and risk measures. Further, our dynamic model generates a total expected revenue which closely approximates that of a maximum expected revenue approach which requires demand probability distributions.

A new shielding analysis method for used fuel transport and storage casks

To provide a cask with the largest possible loading capacity of spent fuel assemblies with the largest practicable burnup and shortest cooling time within all safety requirements, AREVA TN has adapted its design process and developed a more elaborated shielding analysis method. Taking advantage of the potential heterogeneities between sources of fuel assemblies to be loaded, and the self-shielding of assemblies loaded at the basket centre by the assemblies loaded at the basket periphery, the result of this method is expressed under the shape of a linear inequalities system allowing to optimize the cask capacity and performance.

Dissipative Switched Linear Differential Systems

The authors develop a dissipativity theory for switched systems whose dynamical modes are described by systems of higher order linear differential equations. They give necessary and sufficient conditions for dissipativity based on systems of linear matrix inequalities (LMIs), constructed from the coefficient matrices of the differential equations describing the modes. The relationship between dissipativity and stability is also discussed, and an application to the stabilization of power converters is provided.

A generalized Farkas lemma with a numerical certificate and linear semi-infinite programs with SDP duals

In this paper, we establish a new non-homogeneous Farkas lemma for a linear semi-infinite inequality system, where the dual statement is given in terms of linear matrix inequalities and thus, it can be numerically verified by solving a semidefinite linear program. It is achieved by way of employing a special variable transformation together with a strong separation theorem. Consequently, we establish a duality theorem for a class of parametric linear semi-infinite programs which admit semi-definite linear programming (SDP) dual problems. A numerical example is given to show how a parametric linear semi-infinite optimization problem can be solved by way of solving its SDP dual problem using the Matlab toolbox CVX.

A Sampling Kaczmarz--Motzkin Algorithm for Linear Feasibility

We combine two iterative algorithms for solving large-scale systems of linear inequalities: the relaxation method of Agmon, Motzkin, et al. and the randomized Kaczmarz method. We obtain a family of algorithms that generalize and extend both projection-based techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.

Solving a Linear Optimization Word Problems by Using GeoGebra

Abstract Today's modern age is characterized by the rapid development of information and communication technologies, which is also reflected in the educational process. It is therefore necessary to prepare the young generation of even at primary and secondary schools to solve problems from real life. Using the appropriate motivation, innovative methods and application of modern information and communication technologies into the teaching process, we can succeed. Constructions created using by dynamic geometry systems bring new opportunities to learning. The aim of this article is to introduce the possibility of using GeoGebra in graphics solution of system of linear inequalities and also in the geometric interpretation of solutions to word problems leading to the linear optimization. The article describes the observations of experimental teaching, where was used GeoGebra. The experimental teaching was conducted at a grammar school in Košice. GeoGebra offers suitable tools to create graphical models in solving the optimization problems.

On the resolution of bipolar max-min equations

This paper investigates bipolar max-min equations which can be viewed as a generalization of fuzzy relational equations with max-min composition. The relation between the consistency of bipolar max-min equations and the classical boolean satisfiability problem is revealed. Consequently, it is shown that the problem of determining whether a system of bipolar max-min equations is consistent or not is NP-complete. Moreover, a consistent system of bipolar max-min equations, as well as its solution set, can be fully characterized by a system of integer linear inequalities.

Submodularity and its application to some global constraints

Submodularity defines a general framework rich of important theoretical properties while accommodating numerous applications. Although the notion has been present in the literature of Combinatorial Optimization for several decades, it has been overlooked in the analysis of global constraints. The current work illustrates the potential of submodularity as a powerful tool for such an analysis. In particular, we show that the cumulative constraint, when all tasks are identical, has the submodular/supermodular representation property, i.e., it can be represented by a submodular/supermodular system of linear inequalities. Motivated by that representation, we show that the system of any two (global) constraints not necessarily of the same type, each bearing the above-mentioned property, has an integral relaxation given by the conjunction of the linear inequalities representing each individual constraint. This result is obtained through the use of the celebrated polymatroid intersection theorem.

Injective Convex Polyhedra

We characterize the convex polyhedra P in $${\mathbb {R}}^n$$Rn for which any family of n-dimensional axis-parallel hypercubes centered in P and intersected with P has the binary intersection property. The characterization is effective, concrete and convex geometric. As an application, we prove that the convex polyhedra determined by a finite linear system of inequalities with at most two variables per inequality are of this type. This provides in particular new examples of injective (or equivalently hyperconvex) metric spaces.

Functional observer design using linear matrix inequalities

It is shown that the problem of observer design for estimating a set of linear combinations of state variables of a plant can be formulated in terms of linear matrix inequalities. An algorithm for constructing functional observers is proposed based on a nonsingular transformation of a plant model in the state space by matrix canonization with subsequent solution of the system of linear matrix inequalities.

The width and integer optimization on simplices with bounded minors of the constraint matrices

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.

Recognizing Parameters of the Information Security Unit Implemented by the Threshold K-Valued Function

This article discusses some approaches to the recognition of the parameters of the threshold k-valued functions, which can be used for building information processing and security units. The main focus is put on the issue of proving k-valued function belonging to the threshold class. For solving this problem it is proposed to use the input coefficients of expansion and increase. With the help of the latter, the coefficients of linear forms of the k-valued threshold function are procedurally approximated. Along with the proposed analytical approach, the article discusses an algorithmic method based on reducing the problem of finding a threshold representation of k-valued functions to the system of linear inequalities, for the solution of which the ellipsoid method, modified by Khachiyan, is applied. The comparative analysis of the proposed methods is carried out based on experiments.

A novel scheme for current sensor faults diagnosis in the stator of a DFIG described by a T-S fuzzy model

Abstract Diagnosis of current sensor faults (CSF) for doubly fed induction generators (DFIGs) is of paramount importance for the reliable power generation of DFIG-based wind turbines (WT). In this paper, a new scheme is developed for current sensors faults diagnosis in the stator of a DFIG-based WT. The nonlinear model of the DFIG is first transformed into an equivalent Takagi-Sugeno (T-S) fuzzy model. Secondly, using this model, a novel fault detection and isolation (FDI) algorithm is proposed. This algorithm is based on a bank of Luenberger observers for residuals generation combined with a new proposed residual vector. Furthermore, a new binary decision logic is used for CSF isolation. Stability analysis of the observer bank is analyzed using a Lyapunov theorem, which allows deriving sufficient stability conditions by solving a system of Linear Matrix Inequalities (LMIs). A simulation study is carried out to assess the performance and the effectiveness of the new FDI scheme.