ISE 536–Fall03: Linear Programming and Extensions
September 15, 2003
Lecture 5: Geometry, Optimality of BFS
Lecturer: Fernando Ord´
o˜nez
1
Existence of a BFS
Theorem 1.
P
=
{
x
∈
n

Ax
≤
b
}
=
∅
.
P
has a BFS iff
P
does not contain a line
Proof: pg. 6364.
Check that this result implies the following
•
Bounded polyhedra must have a BFS
•
A polyhedra in standard form must have a BFS
2
Optimality of a BFS
Theorem 2.
If
min
{
c
t
x

Ax
=
b, x
≥
0
}
has an optimal solution then there is an optimal
BFS
proof:
Consider
Q
=
{
x

c
t
x
=
z
*
, Ax
=
b, x
≥
0
}
1
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3
Unbounded Polyhedra and Extreme Rays
Consider
P
=
{
x

Ax
=
b, x
≥
0
}
an unbounded polyhedra. Therefore there is a direction
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 Spring '05
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 Operations Research, 1 J, Polytope, Fernando Ord´nez

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