Math 171A: Linear Programming
Lecture 12
Finding a Vertex
Philip E. Gill
c 2011
http://ccom.ucsd.edu/~peg/math171a
Wednesday, February 2nd, 2011
Recap: Vertices (aka corner points)
A
vertex
is a
feasible
point at which there are at least
n
linearly
independent
constraints active.
i.e., the activeconstraint matrix
A
a
has rank
n
at a vertex.
UCSD Center for Computational Mathematics
Slide 2/30, Wednesday, February 2nd, 2011
x
2
x
1
#2
#1
#4
#3
#5
A
a
=
1 1
1 2
0 1
A
a
=
1 1
1 0
(nondegenerate vertex)
(degenerate vertex)
A
=
1
1
1
0

1
0
0
1
1
2
When can we
guarantee
that
Ax
≥
b
has a vertex?
How do we
compute
a vertex?
Theorem (Existence of a vertex)
Suppose that
F
=
{
x
:
Ax
≥
b
}
has at least one point. If
rank(
A
) =
n then
F
has at least one vertex.
UCSD Center for Computational Mathematics
Slide 4/30, Wednesday, February 2nd, 2011
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Proof: (Constructive)
The assumptions are:
There is at least one point
x
0
(say) such that
Ax
0
≥
b
.
There is at least one subset of
n
independent rows of
A
.
Consider any feasible point
x
0
. Define the active set at
x
0
, i.e.,
A
0
x
0
=
b
0
Assume that rank(
A
0
)
<
n
(otherwise there is nothing to prove!)
A
0
may be
any shape
(because it may have dependent rows).
UCSD Center for Computational Mathematics
Slide 5/30, Wednesday, February 2nd, 2011
We construct a sequence of feasible points, labeled as
x
0
,
x
1
,
. . . ,
x
k
,
x
k
+1
,
. . .
with activeset matrices
A
0
,
A
1
,
. . . ,
A
k
,
A
k
+1
,
. . .
such that
rank(
A
k
+1
)
>
rank(
A
k
)
Next we describe how to construct
x
k
+1
from
x
k
(i.e., the
k
th
iteration)
UCSD Center for Computational Mathematics
Slide 6/30, Wednesday, February 2nd, 2011
k
th iteration: Step 1
The previous step gives
x
k
,
A
k
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 staff
 Linear Programming, UCSD Center for Computational Mathematics, UCSD Center, Computational Mathematics

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