Math 171A: Linear Programming
Lecture 18
Finding a Feasible Point
Philip E. Gill
c
±
2011
http://ccom.ucsd.edu/~peg/math171a
Wednesday, February 16th, 2011
Recap: the simplex method
One step moves from a
vertex
to an
adjacent vertex
.
Each step requires the solution of two sets of equations:
A
T
k
λ
k
=
c
and
A
k
p
k
=
e
s
where
A
k
is the nonsingular
workingset
matrix.
Given a nonsingular
A
0
, all subsequent
A
k
are nonsingular.
If there is a tie in the choice of constraint to enter or leave the
working set, the simplex method may
stall
at a vertex.
The method
cycles inﬁnitely
if there is a repeat of a sequence
of constraint changes at a stalled vertex.
UCSD Center for Computational Mathematics
Slide 2/33, Wednesday, February 16th, 2011
Recap: Bland’s Leastindex rule
Apply the simplex method with the leastindex rules:
w
s
= min
{
w
i
: (
λ
k
)
i
<
0
}
t
= min
{
j
:
σ
j
=
α
k
}
Bland’s rule is not useful
computationally
.
it usually needs more iterations than the Dantzig rule.
Better anticycling rules are based on
constraint perturbation
.
Nevertheless, Bland’s rule is useful as a
theoretical tool
.
UCSD Center for Computational Mathematics
Slide 3/33, Wednesday, February 16th, 2011
Theorem
A vertex x
0
with activeset matrix A
a
is a solution of an LP
if and only if
there is a
nonnegative basic solution
of A
T
a
λ
a
=
c.
Proof: If
x
0
is a nondegenerate vertex then
A
a
is nonsingular,
λ
a
is
unique and the result follows.
Assume that
x
0
is a degenerate vertex, with
A
T
a
=
UCSD Center for Computational Mathematics
Slide 4/33, Wednesday, February 16th, 2011
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View Full DocumentConsider the auxiliary problem:
minimize
x
∈
R
n
c
T
x
subject to
A
a
x
≥
A
a
x
0
x
0
is an initial vertex for this problem.
Starting at
x
0
, run the simplex method with Bland’s rule.
The simplex method solves a sequence of square systems
A
T
w
λ
w
=
c
A
w
p
=
e
s
and must
terminate
or declare the problem
unbounded
.
UCSD Center for Computational Mathematics
Slide 5/33, Wednesday, February 16th, 2011
EITHER (A)
x
0
is optimal
OR
(B)
there is an unbounded direction
p
In other words:
EITHER (A)
x
0
is optimal
⇒
c
=
A
T
w
λ
w
for some
λ
w
≥
0
OR
(B)
∃
p
such that
c
T
p
<
0,
A
w
p
=
e
s
and
A
a
p
≥
0
UCSD Center for Computational Mathematics
Slide 6/33, Wednesday, February 16th, 2011
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 Spring '08
 staff
 Linear Programming, Equations, Sets, Optimization, Simplex algorithm, UCSD Center for Computational Mathematics, UCSD Center, Computational Mathematics

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