Tutorial 5: Outline of Solutions
Q1. Consider a minimization linear programming problem in standard form. Let
x
be a basic
feasible solution associated with the basis
B
. Prove the following:
(a) If the reduced cost of every nonbasic variable is positive, then
x
is the unique optimal
solution.
(b) If
x
is the unique optimal solution and is nondegenerate, then the reduced cost of every
nonbasic variable is positive.
Solution
:
Consider a standard form linear program
min
c
0
x
s.t.
Ax
=
b
x
≥
0
.
(a) Suppose
¯
c
N
>
0
, where
N
is the index set of all nonbasic variables at
x
and
B
is the index
set of the basic variables.
For any feasible solution
y
, let
d
=
y

x
.
Then
Ad
=
0
implies
d
B
=

B

1
Nd
N
where
A
is partitioned using the basic and nonbasic variables as (
B

N
). The
change in cost is equal to
c
0
d
=
c
0
B
d
B
+
c
0
N
d
N
= (
c
0
N

c
0
B
B

1
N
)
d
N
=
¯
c
0
N
d
N
.
Because
¯
c
N
>
0
and
d
N
=
y
N

x
N
=
y
N
≥
0, we have
¯
c
0
N
d
N
≥
0
.
This shows that
c
0
x
≤
c
0
y
for any feasible
y
, i.e.,
x
is a minimum solution.
To show uniqueness, assume
y
is also a minimum solution, then
c
0
y
=
c
0
x
, i.e.,
c
0
d
= 0. This
implies
¯
c
0
N
d
N
= 0.
Because
¯
c
N
>
0
, we have
d
N
= 0.
Then
d
B
=

B

1
Nd
N
=
0
.
Thus,
d
=
0
, i.e.,
y
=
x
. This shows that
x
is the unique optimal solution.
(b) By contradiction, we assume that ¯
c
j
= 0 for some
j
∈
N
.
Construct a direction
d
with
d
j
= 1,
d
i
= 0 for
i
∈
N
\ {
j
}
and
d
B
=

B

1
A
j
. Then
Ad
=
0
. Because
x
is nondegenerate,
x
B
>
0
.
Thus, there exists a
θ >
0 such that
x
B
+
θ
d
B
≥
0
.
Since
d
N
≥
0
, we have
x
N
+
θ
d
N
≥
0
. Thus,
ˆx
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 Spring '10
 TanBanPin
 Linear Programming, Optimization, X1, Simplex algorithm

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