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Spring 2008
Introduction to Optimization
May 2, 2008
Prof. Ben Van Roy
Homework 4 Solutions
Part A
True or false:
a) TRUE. Let
x
1
be a solution to
Ax
=
b
. If
x
2
is a second distinct solution, then it must
be that
x
2

x
1
∈
N
(
A
). But here
dim
(
N
(
A
)) = 1. Let
{
¯
x
}
be a basis for
N
(
A
). It
follows that we can write
x
2

x
1
as
λ
¯
x
for a suitable scalar
λ
. We may thus conclude
that every solution of
Ax
=
b
is of the form
x
1
+
λ
¯
x
. Thus,
P
is constrained to a
line. Recall that every basic feasible solution is also a vertex. Thus to have more than
two basic feasible solutions, we would need to have more than two vertices on the line
given by
x
1
+
λ
¯
x
. But it is impossible to have more than two vertices on a line (say
we have three, then it is clear that one can be written as a convex combination of the
remaining two). Hence, we cannot have more than two basic feasible solutions.
b) FALSE. Consider min
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 Spring '06
 UNKNOWN
 Optimization

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