ISE 536–Fall03: Linear Programming and Extensions
September 17, 2003
Lecture 6: Simplex Method, Moving in the Polyhedra
Lecturer: Fernando Ord´o˜nez
1
Conceptual Algorithm to solve LP
Consider the following to solve min
c
t
x

x
∈
P
, where
P
=
{
x
∈ <
n

Ax
=
b, x
≥
0
}
.
1.
Compute Cone(
P
) if Cone(
P
) =
∅
, then
P
is bounded goto 3.
2.
Find
w
j
extreme rays in Cone(
P
). If
c
t
w
j
<
0 then problem unbounded, STOP.
3.
Find
x
i
extreme points of
P
, move toward the one with smallest
c
t
x
i
If
A
is
m
×
n
, How many extreme rays and extreme points do we have to ﬁnd in this
algorithm??
2
Feasible Directions and moving from a BFS
Deﬁnition
For a convex set
P
, and a point
x
∈
P
, a vector
d
is a feasible direction if
x
+
θd
∈
P
for some
θ >
0.
The simplex algorithm moves from BFS to adjacent BFS, that is solutions that have all but
one of the same basic variables.
1
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View Full DocumentConsider a BFS
x
= (
x
B
,x
N
) and a feasible direction
d
= (
d
B
,d
N
) such that the direction
brings nonbasic variable
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 Spring '05
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 Linear Programming, Cone, Lecturer, Simplex algorithm

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