6.32. Let k be a positive integer and let P = conv.hull({(0,0), (1,0), (1/2, k)}). Show that the Chvátal rank of P is at least k. Chvátal Rank Gomory-Chvátal cutting planes have an interesting connection with the general problem of finding linear descriptions of combinatorial convex hulls. In this context, we do not think of the cuts as coming sequentially, as in cutting-plane proofs, but rather in waves that provide successively tighter approximations to P1, the convex hull of the integral vectors in P. This ap- proximation process gives a finite procedure for obtaining a linear description of Pr. We start by taking all possible Gomory-Chvátal cuts for P in a first wave. Although there appear to be infinitely many such cutting-planes, actually a finite subset imply the rest. A nice way to describe this is to let P' denote the set of all vectors in P that satisfy every Gomory-Chvátal cut for P. Then we have the following result of Schrijver [1980].
6.32. Let k be a positive integer and let P = conv.hull({(0,0), (1,0), (1/2, k)}). Show that the Chvátal rank of P is at least k. Chvátal Rank Gomory-Chvátal cutting planes have an interesting connection with the general problem of finding linear descriptions of combinatorial convex hulls. In this context, we do not think of the cuts as coming sequentially, as in cutting-plane proofs, but rather in waves that provide successively tighter approximations to P1, the convex hull of the integral vectors in P. This ap- proximation process gives a finite procedure for obtaining a linear description of Pr. We start by taking all possible Gomory-Chvátal cuts for P in a first wave. Although there appear to be infinitely many such cutting-planes, actually a finite subset imply the rest. A nice way to describe this is to let P' denote the set of all vectors in P that satisfy every Gomory-Chvátal cut for P. Then we have the following result of Schrijver [1980].
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:6.32. Let k be a positive integer and let
P = conv.hull({(0,0), (1,0), (1/2, k)}).
Show that the Chvátal rank of P is at least k.
![Chvátal Rank
Gomory-Chvátal cutting planes have an interesting connection with the
general problem of finding linear descriptions of combinatorial convex hulls.
In this context, we do not think of the cuts as coming sequentially, as in
cutting-plane proofs, but rather in waves that provide successively tighter
approximations to P1, the convex hull of the integral vectors in P. This ap-
proximation process gives a finite procedure for obtaining a linear description
of Pr.
We start by taking all possible Gomory-Chvátal cuts for P in a first wave.
Although there appear to be infinitely many such cutting-planes, actually a
finite subset imply the rest. A nice way to describe this is to let P' denote
the set of all vectors in P that satisfy every Gomory-Chvátal cut for P. Then
we have the following result of Schrijver [1980].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb650aa62-eb05-422e-b3d0-f796a05ace46%2Fa7eef47d-1b62-4948-83b2-458c4498a11b%2Fk230sza_processed.png&w=3840&q=75)
Transcribed Image Text:Chvátal Rank
Gomory-Chvátal cutting planes have an interesting connection with the
general problem of finding linear descriptions of combinatorial convex hulls.
In this context, we do not think of the cuts as coming sequentially, as in
cutting-plane proofs, but rather in waves that provide successively tighter
approximations to P1, the convex hull of the integral vectors in P. This ap-
proximation process gives a finite procedure for obtaining a linear description
of Pr.
We start by taking all possible Gomory-Chvátal cuts for P in a first wave.
Although there appear to be infinitely many such cutting-planes, actually a
finite subset imply the rest. A nice way to describe this is to let P' denote
the set of all vectors in P that satisfy every Gomory-Chvátal cut for P. Then
we have the following result of Schrijver [1980].
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