Math 171A Homework 5 Solutions
Instructor: Jiawang Nie
February 21, 2012
1. (5 points) Consider the linear program
Minimize
3
x
1

x
2
+2
x
3
subject to

2
x
1
+4
x
2
+4
x
3
≥
6
x
1
+4
x
2
+
x
3
≥
5

2
x
1
+
x
2
+2
x
3
≥
1
2
x
1

2
x
2
≥
0

3
x
2
+
x
3
≥ 
2
x
1
≥
0
x
2
≥
0
x
3
≥
0
and the point ¯
x
= (1
,
1
,
1)
T
. Find the active set at ¯
x
and determine if the point ¯
x
is
optimal. If ¯
x
is not optimal, ﬁnd a direction p such that
c
T
p <
0 and
A
a
p
≥
0.
Solution: Write down the standard form ﬁrst.
c
= [3
,

1
,
2]
>
,
A
=

2
4 4
1
4 1

2
1 2
2

2 0
0

3 1
1
0 0
0
1 0
0
0 1
,
b
=
6
5
1
0

2
0
0
0
.
Calculate the residual at ¯
x
= [1
,
1
,
1]
>
is
r
(¯
x
) =
A
¯
x

b
= [0 1 0 0 0 1 1 1]
>
, so the
active set at ¯
x
to be
{
1
,
3
,
4
,
5
}
. To determine if ¯
x
is optimal, we must invoke Farka’s
Lemma and check the optimality conditions. Farka’s Lemma states that if one has
ANY nonnegative solution to
A
T
a
λ
=
c
, then the point ¯
x
is optimal. This amounts to
solving
A
T
a
λ
=
c
and checking if there are any nonnegative solutions. The problem is
that
A
a
is not square, and it turns out that
A
T
a
λ
=
c
does not have a unique solution.
Use the method of calculating the basic solution, we pick the submatrix of
A
a
to get
the square submatrix, and solve the problem to get
λ
, i.e. solve the following problem:
A
T
134
λ
=
c,
get
λ
= [0
.
7500
,

0
.
5000
,
1
.
7500]
>
A
T
135
λ
=
c,
get
λ
= [1
.
7222
,

3
.
2222
,
1
.
5556]
>
1
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T
145
λ
=
c,
get
λ
= [0
.
5714
,
2
.
0714
,

0
.
2857]
>
A
T
345
λ
=
c,
get
λ
= [1
.
6000
,
3
.
1000
,

1
.
2000]
>
It turns out that we will not be able to ﬁnd a nonnegative solution to any of these four
problems and thus the point ¯
x
is not optimal. To ﬁnd a feasible search direction
p
, take
the solution of the ﬁrst problem. Let
s
denote the component of
λ
which is negative,
pick the second one, if you then solve
A
134
p
=
e
2
,
get
p
= [

0
.
5000
,

0
.
5000
,
0
.
2500]
>
.
, check this direction
p
will have the property that
c
T
p
=

0
.
5000
<
0 and we must
then check if
A
a
p
= [0;1
.
0000;0;1
.
7500]
≥
0, which satisfy the condition.
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 Spring '08
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 Math, Linear Programming, Optimization, Mathematical optimization, Simplex algorithm, feasible descent direction

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