strict inequalities
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Author(s):  
Ирина Александровна Шарая ◽  
Сергей Петрович Шарый

В работе рассматриваются интервальные линейные включения Cx ⊆ d в полной интервальной арифметике Каухера. Вводится количественная мера выполнения этого включения, названная “резервом включения”, исследуются ее свойства и приложения. Показано, что резерв включения оказывается полезным инструментом при изучении АЕ-решений и кванторных решений интервальных линейных систем уравнений и неравенств. В частности, использование резерва включения помогает при определении положения точки относительно множества решений, при исследовании пустоты множества решений или его внутренности и т.п In this paper, we consider interval linear inclusions Cx ⊆ d in the Kaucher complete interval arithmetic. These inclusions are important both on their own and because they provide equivalent and useful descriptions for the so-called quantifier solutions and AE-solutions to interval systems of linear algebraic relations of the form Ax σ b , where A is an interval m × n -matrix, x ∈ R , b is an interval m -vector, and σ ∈ {= , ≤ , ≥} . In other words, these are interval systems in which equations and non-strict inequalities can be mixed. Considering the inclusion Cx ⊆ d in the Kaucher complete interval arithmetic allows studing simultaneously and in a uniform way all the different special cases of quantifier solutions and AE-solutions of interval systems of linear relations, as well as using interval analysis methods. A quantitative measure, called the “inclusion reserve”, is introduced to characterize how strong the inclusion Cx ⊆ d is fulfilled. In our work, we investigate its properties and applications. It is shown that the inclusion reserve turns out to be a useful tool in the study of AE-solutions and quantifier solutions of interval linear systems of equations and inequalities. In particular, the use of the inclusion reserve helps to determine the position of a point relative to a solution set, in investigating whether the solution set is empty or not, whether a point is in the interior of the solution set, etc


Author(s):  
María D. Fajardo ◽  
Miguel A. Goberna ◽  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez

Author(s):  
María D. Fajardo ◽  
Miguel A. Goberna ◽  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez

2017 ◽  
Vol 173 (1) ◽  
pp. 131-154
Author(s):  
Margarita M. L. Rodríguez ◽  
José Vicente-Pérez

Author(s):  
Maroua Maalej ◽  
Vitor Paisante ◽  
Pedro Ramos ◽  
Laure Gonnord ◽  
Fernando Magno Quintao Pereira
Keyword(s):  

2014 ◽  
Vol 24 (05) ◽  
pp. 569-607 ◽  
Author(s):  
Xavier Allamigeon ◽  
Uli Fahrenberg ◽  
Stéphane Gaubert ◽  
Ricardo D. Katz ◽  
Axel Legay

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier–Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).


2014 ◽  
Vol 28 (3) ◽  
pp. 1306-1333 ◽  
Author(s):  
Geoffrey R. Grimmett ◽  
Zhongyang Li

2011 ◽  
Vol 142 (3) ◽  
pp. 460-486 ◽  
Author(s):  
Massimo Franceschetti ◽  
Mathew D. Penrose ◽  
Tom Rosoman

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