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Let
ε < γ
1
1
αγ
ð
Þ. Then the asymptotic convergence rate of Algorithm 6.2
satisfies
2
kk
1
ε
1
γαþ ε
21
αγ
ð
c
Þγ þ γ
2
γ
KðÞKkk
1
|
{z
}
¼:δ
Þ
1
αγ
ρ
ε þ
:
1
αγ
ð
1
(1
αγ
If the stronger condition
ε < γ
)
holds, we obtain
2
kk
1
ε
1
γαþ ε
γ
δ
Þ
1
αγ
:
ð
Proof Let
:¼ I LRAL
1
A
:¼ I γP
,
K
RA
,
Q
:¼ I A
:
In the following, we shall establish an expression for
KQ
in terms of
K
and
Q
.
To this end, notice that we have for some
B
mm
and
nn
,
∈
R
δ
A
∈
R
K ¼ I LRAL
1
þ BRA
¼ I LRAL
1
RA LBR
A
|
{z
}
¼:
B
¼ I LRAL
1
RA LRAL
1
R
δ
A BA
¼ K
11
αγ
A B
|
{z
}
LR
δ
A
:
¼: δ
K
Moreover, we have
KQ ¼ K Q
|{z}
¼ O
þK
δ
Q þ δ
KQ
:¼ Q Q
, and the identity
K Q ¼ O
follows from the fact that, as an
immediate consequence of the definition of
P
,
K
is an oblique projector along the
range of
P
(cf. Lemma 3.2.7 in [Pap11]). Combining the above expressions, we
obtain
where
δ
Q
LR
1
αγ
δ
KQ ¼ K
Q
|
{z
}
¼:
δ
QQ
|
{z
}
¼:
þ BAQ
|
{z
}
¼:
:
E
ðÞ
E
ðÞ
E
ðÞ
