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2 Hybrid Projection Methods with Perturbations
In this paper, the algorithms have the following iterative scheme
x
=
x
+
α
d
k
k
+
1
k
k
In the above-mentioned two formulae, the main direction
d
=
s
+
ω
.
k
k
k
s
satisfies the following
conditions.
(H
1
)
s
≤
c
g
.
k
1
k
2
(H
2
)
g
,
d
≤
−
c
g
.
k
k
2
k
Perturbation term
ω
satisfies
k
(H3)
ω
≤ γ
(
q
+
p
g
)
and
γ
>
0
satisfies
k
k
k
k
∑
∞
=
(H4)
γ
2
<
+∞
,
k
k
1
where
c
,
2
c
,
q
and
p
are positive constants.
1
{
}
{
}
Let
J
= .
The hybrid projection methods with perturbations are described as follows.
N
=
1
2
,
I
=
k
∈
N
g
k
s
,
+
ω
≥
0
and
N
I
K
k
k
1
≥
1
∈
∈
(
0
)
Algorithm 2.1.
Given a nonnegative integer
M
,
x
R
n
,
β
∈
(
0
,
,
σ
0
2
and
,
μ
μ
γ
∈
0
1
and a symmetric positive definite matrix
B
. Set
δ
∈
[
0
5
,
2
)
0
1
k
:
=
0
.
Step 1.
If
g
=
0
, then stop.
x
is a stationary point. Else, goto step 2.
k
Step 2.
If
k
∈
I
, then let
x
=
x
+
γ
d
,
k
=
k
+
1
return to step 1. Else, goto
k
+
1
k
k
k
step 3.
Step 3.
Let
α
=
γ
m
, where
m
is the smallest nonnegative integer satisfying
k
k
g
(
x
+
α
d
),
d
≤
μ
g
,
d
(2.1)
k
k
k
k
0
k
k
and
2
g
(
x
+
α
d
),
d
≥
μ
g
(
x
+
α
d
)
(2.2)
k
k
k
k
1
k
k
k
Step 4.
Set
δ
=
x
−
x
,
γ
=
g
−
g
and modify
B
as
k
B
by using BFGS or
k
k
+
1
k
k
k
+
1
k
+
1
DFP formula or other quasi-Newton formulae.
v
,
x
−
y
Step 5.
Let
y
=
x
+
α
d
,
v
=
g
(
y
)
,
P
=
−
k
k
k
v
.
i
>
k
k
k
k
k
k
k
k
2
v
k
2
−
P
T
B
P
P
,
B
=
B
+
iI
.
x
=
x
+
λ
P
,
where
λ
is defined by the NNLS.
k
k
k
k
k
k
k
+
1
k
k
k
k
Step 6.
Set
k
=
k
+
1
, return to step 1.
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