Spring 2008
Introduction to Optimization
April 18, 2008
Prof. Ben Van Roy
Homework Assignment 2 : Solutions
Solve Questions 1, 2, 3, 8 and 10 from Chapter 3.
3.1.
Adding
x
+
y
≥
0 reduces the feasible region to
{
[0
,
0]
}
.
3.2.
The polyhedron is the unit simplex, which has one face and three vertices. It is a
triangle with vertices (1
,
0
,
0), (0
,
1
,
0) and (0
,
0
,
1). The maximum of
x
+ 2
y
+ 3
z
is 3 and is
attained in the vertex (0
,
0
,
1).
3.3
Infeasibility is evident from sketching the feasible region. Here we show this algebraically:
Let
x,y
be feasible. Then, from the third constraint, we have that
x,y
must satisfy
3
2
(2
x
+
5
y
)
≤
9
2
, and from the last constraint,
x,y
must satisfy

3
x
+ 8
y
≤ 
5. Adding these
constraints, we have that
x,y
must satisfy
31
2
y
≤ 
1
2
. But the second constraint requires
y
≥
0. Since we cannot simultaneously have
y
≥
0 and
31
2
y
≤ 
1
2
, the feasible region must
be empty.
3.8