Method for finding a general expression for the integer-valued solutions of a system of linear inequalities

Cybernetics ◽  
1981 ◽  
Vol 16 (4) ◽  
pp. 552-558
Author(s):  
Nguen Ngos Chu
Keyword(s):  
General Expression ◽  
Linear Inequalities ◽  
System Of Linear Inequalities

scholarly journals Greedy systems of linear inequalities and lexicographically optimal solutions

2019 ◽  
Vol 53 (5) ◽  
pp. 1929-1935
Author(s):  
Satoru Fujishige
Keyword(s):  
Convex Functions ◽  
Linear Inequalities ◽  
Convex Polyhedra ◽  
Optimal Solutions ◽  
Integral Points ◽  
System Of Linear Inequalities ◽  
Discrete Convexity ◽  
The Greedy Algorithm

The present note reveals the role of the concept of greedy system of linear inequalities played in connection with lexicographically optimal solutions on convex polyhedra and discrete convexity. The lexicographically optimal solutions on convex polyhedra represented by a greedy system of linear inequalities can be obtained by a greedy procedure, a special form of which is the greedy algorithm of J. Edmonds for polymatroids. We also examine when the lexicographically optimal solutions become integral. By means of the Fourier–Motzkin elimination Murota and Tamura have recently shown the existence of integral points in a polyhedron arising as a subdifferential of an integer-valued, integrally convex function due to Favati and Tardella [Murota and Tamura, Integrality of subgradients and biconjugates of integrally convex functions. Preprint arXiv:1806.00992v1 (2018)], which can be explained by our present result. A characterization of integrally convex functions is also given.

On Finding the Maximum Feasible Subsystem of a System of Linear Inequalities

2018 ◽  
Vol 28 (2) ◽  
pp. 169-173
Author(s):  
N. N. Katerinochkina ◽  
V. V. Ryazanov ◽  
A. P. Vinogradov ◽  
Liping Wang
Keyword(s):  
Linear Inequalities ◽  
System Of Linear Inequalities

On the Number of Vertices of a Convex Polytope

1964 ◽  
Vol 16 ◽  
pp. 701-720 ◽  
Author(s):  
Victor Klee
Keyword(s):  
Sharp Estimate ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Bounded Subset ◽  
Empty Interior ◽  
System Of Linear Inequalities ◽  
System Of Inequalities ◽  
Basic Solutions ◽  
Upper Estimate

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

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