CO350 L
INEAR
O
PTIMIZATION
 S
OLUTION
HW7
Exercise 4.8.9
Claim.
The LP (P):
min
e
T
x
subject to
Ax
=
0
x
≥
0
is unbounded, if and only if, the system
Ax
= 0
x
≥
0
x
6
= 0
(i)
has a solution.
Proof.
Suppose (P) is unbounded. Then there exists a nontrivial solution to
{
Ax
= 0
, x
≥
0
}
which is a
solution to (i). Suppose (i) has a nonzero solution
x
*
, then for any
λ
≥
0
,
λx
*
is a feasible solution to (P) of
value
λe
T
x
*
where
e
T
x
*
>
0
. Hence (P) is unbounded.
/
The dual of (D) is given by
min
0
T
y
subject to
A
T
y
≥
e
Suppose (i) has a solution. Then by the Claim, (P) is unbounded. It follows from weak duality that (D) is
infeasible, so (ii) has no solution. Suppose (ii) has a solution. Clearly, (D) cannot be unbounded, since its
objective function is
0
. It follows from strong duality that (P) has an optimal solution. By the claim, (i) has
no solution.
Exercise 1
(a) Suppose that
C
is not convex. Then there exist two elements
x
1
, x
2
∈
C
and a real number
λ
∈
[0
,
1]
such that
λx
1
+ (1

λ
)
x
2
6∈
C.
(1)
Thus, for points
x
1
, x
2
, x
1
and
λ
1
=
λ, λ
2
= 1

λ
1
, λ
3
= 0
, (
λ
1
+
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 Winter '07
 S.Furino,B.Guenin
 Vector Space, Optimization, ax

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