Fall 2010
Lecture 20 Notes
Nicholas Harvey
http://www.math.uwaterloo.ca/~harvey/
1 Finding Neighbouring Vertices in the Simplex Method
Let
x
be a basic feasible solution of the polyhedron
P
=
{
x
:
Ax
≤
b
}
. Let
B
be the subset of the
constraints that are tight at
x
. Let
A
B
denote the submatrix of
A
corresponding to these constraints.
Similarly, let
b
B
denote the portion of
b
corresponding to these constraints. So
A
B
x
=
b
B
holds.
Assume that we have perturbed the matrix
A
such that each vertex of
P
has exactly
n
tight constraints.
Then

B

=
n
, so
A
B
is square. Since
x
is a basic feasible solution, rank
A
B
=
n
, and so
A
B
is invertible.
Since
A
B
is invertible, we can express the objective function
c
as a linear combination of the tight
constraints. That is, there exists a vector
u
such that
c
T
=
u
T
A
B
.
Case 1:
u
≥
0
.
In this case, we have expressed the objective function as a nonnegative linear
combination of the constraints that are tight at
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 Winter '10
 Harvey
 Optimization, basic feasible solution, Polytope

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