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Math 171A Homework 2 Solutions
Instructor: Jiawang Nie
February 1, 2012
1. (8 points) Consider the feasible set
F
de±ned by the following constraints
x
1
+
x
2
≥
4
,
x
1
+ 3
x
2
≥
6
,
6
x
1

x
2
≤
18
,
3
≤
x
2
≤
6
,
x
1
≥ 
1
.
(a) Express
F
in the standard form
Ax
≥
b
. Write down
A
and
b
explicitly.
Solution:
A
=
1
1
1
3

6
1
0
1
0

1
1
0
b
=
4
6

18
3

6

1
(b) Draw the set
F
graphically in the plane, and ±nd all the corner points of
F
.
Solution: Like the previous homework, we draw all of the lines and ±nd which
side of the lines the feasible region is on, see Figure 1. Then by ±nding which
lines intersect we can solve for the corners. The corners that lie in the feasible
region should be
(

1
,
5)
,
(1
,
3)
,
(3
.
5
,
3)
,
(4
,
6)
,
(

1
,
6)
(c) Solve the following LP
minimize 2
x
1

3
x
2
subject to
Ax
≥
b
where
A
and
b
are from part (a).
Solution: Since we know the corner points of the feasible region from part (b),
we need to plug those points into the objective function and compare. At ±ve
corner points, the objective function values are 17, 7, 2, 10, 20 respectively.
The point which minimizes 2
x
1

3
x
2
is the point (

1
,
6) with a value of

20.
1
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0
1
2
3
4
5
6
25
20
15
10
5
0
5
10
15
20
C6
Feasible Region
C5
C4
C3
C2
C1
Figure 1: Feasible Region of Question 1.
(d) Compute the residual vector
r
(
x
) for all the constraints at the point ¯
x
= (2
,
4),
and ±nd the constraints whose residuals would decrease after a positive step
α
along the direction
p
= (1
,

2) emanating from the point ¯
x
.
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.
 Spring '08
 staff
 Math

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