On an algorithm for finding integer points on perfect ellipsoids

2021 ◽  
Author(s):  
Otabek Gulomov ◽  
Sadulla Shodiev
Keyword(s):  
Integer Points

scholarly journals Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

4OR ◽  
2020 ◽  
Author(s):  
Michele Conforti ◽  
Marianna De Santis ◽  
Marco Di Summa ◽  
Francesco Rinaldi
Keyword(s):  
Integer Point ◽  
Cutting Plane ◽  
Integer Program ◽  
Compact Set ◽  
Cutting Plane Method ◽  
Integer Points ◽  
Gomory Cuts ◽  
The Family ◽  
Split Cuts ◽  
Number Of Iterations

AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

scholarly journals Density of integer points on affine homogeneous varieties

1993 ◽  
Vol 71 (1) ◽  
pp. 143-179 ◽  
Author(s):  
W. Duke ◽  
Z. Rudnick ◽  
P. Sarnak
Keyword(s):  
Integer Points ◽  
Homogeneous Varieties

scholarly journals On the number of integer points in the displaced circles

1968 ◽  
Vol 14 (2) ◽  
pp. 141-152 ◽  
Author(s):  
A. Yudin
Keyword(s):  
Integer Points

scholarly journals Integer points close to convex surfaces

2009 ◽  
Vol 138 (1) ◽  
pp. 1-23 ◽  
Author(s):  
M. C. Lettington
Keyword(s):  
Convex Surfaces ◽  
Integer Points

scholarly journals On the number of large integer points on elliptic curves

2009 ◽  
Vol 138 (4) ◽  
pp. 317-327 ◽  
Author(s):  
P. G. Walsh
Keyword(s):  
Elliptic Curves ◽  
Large Integer ◽  
Integer Points

scholarly journals Cyclic polygons of integer points

2009 ◽  
Vol 138 (2) ◽  
pp. 109-136 ◽  
Author(s):  
M. N. Huxley ◽  
S. V. Konyagin
Keyword(s):  
Integer Points
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