Calculating a minimal sphere containing a polytope defined by a system of linear inequalities

1994 ◽  
Vol 3 (2) ◽  
pp. 181-191 ◽  
Author(s):  
Hiroshi Konno ◽  
Yasutoshi Yajima ◽  
Ayumi Ban
Keyword(s):  
Linear Inequalities ◽  
System Of Linear Inequalities ◽  
Minimal Sphere

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
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