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Fall 2007
Linear and Nonlinear Optimization
Oct 18, 2007
Prof. Yinyu Ye
Homework Assignment 3: Sample Solution
Problem 1.
a) The dual of the problem:
minimize
4
y
1
+ 11
y
2
subject to
2
y
1
+ 5
y
2
≥
9
3
y
1
+ 4
y
2
≥
8
y
1
+ 3
y
2
≥
5
y
1
, y
2
≥
0
b) After we change b to
b
0
(6:11), then we need to check whether
A

1
B
b
0
is greater or less
than 0:
A

1
B
b
0
=
3

1

5
2
6
11
=
7

8
Hence, the optimal basis will change.
c) Since
x
2
is a nonbasic variable, we need to check whether
r
N
+
4
c
2
e
j
is greater or less
than 0:
r
N
+
4
c
2
e
j
= 2 + (

3)
*
1
≤
0
Hence, the optimal basis will change.
d) Since the optimal solution still satisﬁes the new constraint, the optimal solution won’t
change after the new constraint is added.
Problem 2.
a) We need to use the complementary slackness conditions to ﬁnd the dual
solutions. First, note that based on the primal solutions, the constraints in the primal corre
sponding to
λ
2
,
λ
3
and
λ
5
have slack. This indicates that
λ
2
= 0,
λ
3
= 0 and
λ
5
= 0. Now,
since
x
1
, x
3
and
x
4
are all greater than zero. This indicates that the ﬁrst, third and fourth
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View Full Documentconstraints of the dual will be binding. Now, just substitute in the value
y
= 0
.
7 and solve
for the
λ
0
s
. We ﬁnd (
λ
1
, λ
2
, λ
3
, λ
4
, λ
5
)=(0
.
2
,
0
,
0
,
0
.
5
,
0).
b) The value of y represents the lowest ratio of
p
i
r
i
of all the companies that you have invested
in. It is basically your threshold ratio value for investing in a company. This value also
represents the shadow price of your budget constraint. So, it indicates how much money you
could make if your budget was increased by 1.
c) The value of
λ
i
represents the diﬀerence in return between an incremental amount of
ownership in company
i
instead of your lowest ratio investment. If you were allowed to
invest more in company
i
, the
λ
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 Fall '07
 YINYUYE

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