A Class of Hybrid Projection Methods with Perturbations - Advanced Research on Computer Education, Simulation and Modeling - page 223

Case 2: When

J ∩ K

is an infinite index set, suppose, on the contrary, that there

i

exist an infinite subset

K

⊆

J

∩

K

and ε > 0 such that

i i 0

. (2.3)

g

≥ ε , ∀

k

∈

K

k 0 i

Utilizing (2.1),(2.2),(H 2 ),(H 3 ) and (2.3), we have

v

,

x

−

y

-

g

,

P

=

k k k

g

,

v

≥ μ

v

,

x

−

y

= − μ α

g

(

y

),

d

k k 2 k k 1 k k k 1 k k k

v

k

≥ − μ μ α

g

,

d

= − μ μ α [

g

,

s

+

g

, ω ]

0 1 k k k 0 1 k k k k k

2

≥ μ μ α [

c g

−

g

ω ] (2.4)

0 1 k 2 k k k

2

≥ μ μ α [

c g

− γ

g

(

q

+

p g

)]

0 1 k 2 k k k k

2

≥ μ μ (

c

− γ

p

− γ

q

ε ) α

g

0 1 2 k k 0 k k

for

k ∈
On the other hand, by Algorithm 2.1, (H 1 ), (H 3 ) and (2.3), we have

K

.

i

v , x − y

P =

k k k

≤ x − y = α d

k k k k k

v

k

≤ α ( c g + γ ( q + p g )) (2.5)

k 1 k k k

≤ α ( c + γ p + γ q ε ) g

k 1 k k 0 k

for

Note that for every

k ∈ K .

i

k ∈ K , there exists an integer k ′ such that k = l ( k ′ ) − i − 1 ,

i

that is, k + 1 = l ( k ′ ) − ( i − 1 − 1 ∈ K .

i − 1

Therefore by (1.6), for ∀ k ∈ K , we have

i

1

f − f ≥ - σλ g

T
k

P − σλ

2

P

T
k

B P

l ( k ) k + 1 k k

2

k k k

1

≥ - σλ g

T
k

P − σλ s P

T

B P

k k k k k k k

2

(2.6)

1 − δ g

T
k

P

= - σλ g

T
k

P − σλ P

T

B P

k k

2

k

P

T
k

B P

k k k

k k

σ ( 2 − δ )

= - λ g

T
k

P ,

k k

2

which, together with (2.4) and σ > 0 , δ < 2 , implies that

σ ( 2 − δ ) σ ( 2 − δ ) 2

f − f ≥ - λ g

T
k

P ≥ - μ μ ( c − γ p − γ q ε ) α λ g . (2.7)

l ( k ) k + 1

2

k k

2

0 1 2 k k 0 k k k