22M:174 Optimization Techniques
Homework 4 — due Monday Mar 19
1. Suppose that
A
is a symmetric matrix.
This problem is about finding a
number
τ
that guarantees that
A
+
τ
I
is positive definite. The idea is based
on an estimate for eigenvalues called the
Gershgorin circle theorem
. Show
that
λ
min
(
A
)
≥
min
i
a
ii

∑
j
6
=
i

a
i j

!
.
[
Hint:
Suppose
v
6
=
0 and
A
v
=
λ
v
where
λ
=
λ
min
(
A
)
.
Then
λ
v
i
=
∑
n
j
=
1
a
i j
v
j
for all
i
. Choose
i
so that

v
i

=
max
j
v
j
. Then use
(
λ

a
ii
)
v
i
=
∑
j
6
=
i
a
i j
v
j
. Take absolute values and use the usual rules for absolute values
and
v
j
≤ 
v
i

for all
j
to get the bound on

λ

a
ii

.]
This bound is often useful for giving a starting point for more refined algo
rithms.
2. Implement the dogleg trust region method with
m
k
(
p
) =
f
(
x
k
)+
p
T
∇
f
(
x
k
)+
1
2
p
T
∇
2
f
(
x
k
)
p
: set
p
*
k
to be the minimizer of
m
k
on the line segment joining
the Cauchy point
p
C
k
to the Newton point
p
N
k
=

∇
2
f
(
x
k
)

1
∇
f
(
x
k
)
. Apply
this method to the problem of minimizing
f
(
x
,
y
) =
x
4

x
2
y
+
2
y
2

2
y
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 Spring '12
 DavidStewart
 Orthogonal matrix, Euclidean space, Gradient descent, Quadratic form

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