A Class of Hybrid Projection Methods with Perturbations - Advanced Research on Computer Education, Simulation and Modeling - page 222

In the following, we prove the convergence property of Algorithm 2.1. We first
assume that

{}

k x is an infinite sequence generated by Algorithm 2.1. The following
assumptions are satisfied.

(H 5 ) The objective function f ( x ) has a lower bound on

n
R .

(H 6 ) The gradient g ( x ) is uniformly continuous on an open convex set D that

{}

contains

k x .
(H 7 ) There exists

2

m > 0 , for any k , P

T

B P ≥ m P .

k k k k

The assumption (H 5 ) is very mild. Because the objective function f ( x ) can be

replaced by

f
e , if the assumption is not satisfied.
According to the related reference, we have the following Lemma.

( x )

Lemma 2.1 [5] . Suppose that (H 2 ), (H 3 ) and (H 4 ) hold for

ω γ

s , and . Then when

k k k

k ∈ I is sufficiently large, we have

g ≤ c γ ,

k 3 k

where c is a constant.

Analogous to the proof of the corresponding Lemmas in [5], we can obtain the
following Lemmas.

> 0

3

{}

Lemma 2.2. Let

k x be an infinite iteration sequence generated by Algorithm 2.1. If

{}

g ( x ) is uniformly continuous on an open convex set D containing

k x , then there

{ }

exists a constant σ > 0 such that f + σ T is monotone and non-increasing.

1 l ( k ) 1 k

{} )

Further, the sequence is convergent (which converges to a finite value or −∞ ).

f

l ( k

Without loss of generality, we assume that lim

f k
l

=

f

*

.

( )

k → ∞

{}

Lemma 2.3. Let

k x be an infinite iteration sequence generated by Algorithm 2.1. If

{}
k

g ( x ) is uniformly continuous on an open convex set D containing

x

and

lim

f k

=

f

*

( 0 ≤

i

≤

M

− 1 ), then we have

k ∈ K , k → ∞

i − 1

(1) li ,

1

g

= 0

k

k ∈ K k → ∞

i −

(2) lim

f k

=

f

*

.

k ∈ K , k → ∞

i

Proof. Now we show that (1) holds in two cases.

Case 1: When

I ∩ K

is an infinite index set, it follows from Lemma 2.1 that

i

lim ,

g

= 0 .

k

k ∈ I ∩ K k → ∞ i