A Class of Hybrid Projection Methods with Perturbations - Advanced Research on Computer Education, Simulation and Modeling - page 224

Taking limits on the both sides of the above inequality as k ∈ K

i ,

k → ∞ , according

to lim f = f

*

= lim f , we obtain

k l ( k )

k ∈ K , k → ∞ k → ∞ i

l i , α λ g = 0 . (2.8)

k k k

k ∈ K k → ∞ i

By (2.3), we have

li , α λ = 0 . (2.9)

k k

k ∈ K k → ∞ i

α λ

′

It follows from (2.9), Armijo rule and NNLS that if ψ =

k

and ψ =

k

at least

k k

γ γ

one of the following three inequalities holds for k ∈ K sufficiently large.

i

g ( x + ψ d ), d > μ g , d .

k k k k 0 k k

2

g ( x + ψ d ), g < μ g ( x + ψ d ) .

k k k k 1 k k k

1

′ ′ ′

f ( x + ψ P ) > max f + σ [ g

T
k

P + ψ P

T

B P ].

k k k k − j k k k k k k

2

0 ≤ j ≤ m ( k )

But whichever holds, similar to the proof of Lemma 2.4 in [5], by using (2.3),(2.8)

1

and (2.9), we can obtain the corresponding μ ≥ 1 i = 0 ), σ > , which is a

i

2

contradiction. Therefore we have

li , g = 0

k

k ∈ K k → ∞

i

(2) Similar to the proof of (2) in Lemma 2.4 [5] , we can obtain the conclusion
easily.
Analogous to the proof of Theorem 2.1 in [5], it is easy to prove the following
theorem.

{}

Theorem 2.1. Suppose that

k x is an infinite iteration sequence generated by
Algorithm 2.1. If there exists an open convex set D containing

{}

k x such that

g ( x )

{}
k

is uniformly continuous on D , then either lim g = −∞ or f converges to a finite

k

k → ∞

value and

lim g = 0

k

k → ∞

By Theorem 2.1, the following corollary is obvious.

Corollary 2.1. Suppose that the assumption conditions of Theorem 2.1 hold. If the
infinite iteration sequence

{}

k x generated by Algorithm 2.1 has cluster point

*
x , then

x

*

∈ Ω

*

.