Spring 2010
CS530 – Analysis of Algorithms
Homework 7
H
OMEWORK
7,
DUE
M
ARCH
24
You must prove your answer to every question.
Problems with a
(
*
)
in place of a score may be a little too advanced, or
too challenging to most students, so I do not assign a score to them. But I
will still note if you solve them.
Problem 1.
(15pts) Prove the separating hyperplane theorem as formulated
in class.
[
Hint: To apply our version of the Farkas Lemma, it helps to turn
the inequality
v
T
x
>
z
in the formulation into an inequality with
.
]
Solution.
In class, we introduced the separating hyperplane theorem, here it
is again. Let
u
1
,...,
u
m
be some vectors in an
n
dimensional space. Let
L
be
the set of convex linear combinations of these points. Thus, a point
v
is in
L
if the following inequalities have a solution:
j
u
i
·
y
i
=
v
,
i
y
i
=
1,
y
0.
(1)
The theorem says that if
v
is not in
L
then there is a separating hyperplane
between
L
and
v
, namely, the following set of inequalities has a solution for
x
:
u
T
i
x
z
(
i
=
1,...,
m
)
,
v
T
x
>
z
.
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 Spring '09
 Linear Algebra, Algorithms, Vector Space, Euclidean space, Convex combination

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