Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 717341, 6 pages
doi:10.1155/2010/717341
Research Article
An Iterative Algorithm of Solution for Quadratic
Minimization Problem in Hilbert Spaces
Li Liu,1 Guanghui Gu,1 and Yongfu Su2
1
2
Department of Mathematics, Cangzhou Normal University, Hebei, Cangzhou 061001, China
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Yongfu Su, suyongfu@tjpu.edu.cn
Received 24 October 2009; Accepted 28 December 2009
Academic Editor: Jong Kim
Copyright q 2010 Li Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is to introduce an iterative algorithm for finding a solution of quadratic
minimization problem in the set of fixed points of a nonexpansive mapping and to prove a strong
convergence theorem of the solution for quadratic minimization problem. The result of this article
improved and extended the result of G. Marino and H. K. Xu and some others.
1. Introduction and Preliminaries
Iterative methods for nonexpansive mappings have recently been applied to solve convex
minimization problems; see, for example, 1, 2 and the references therein. A typical problem
is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping
on a real Hilbert space H:
1
min Ax, x − x, u,
x∈C 2
1.1
where C is the fixed point set of a nonexpansive mapping T defined on H, and u is a given
point in H. Let A be a strongly positive operator defined on H, that is, there is a constant
γ > 0 with the property
Ax, x ≥ γx2 ,
∀x ∈ H.
1.2
Then minimization 1.1 has a unique solution x∗ ∈ C which satisfies the optimality condition
Ax∗ − u, x − x∗ ≥ 0,
∀x ∈ C.
1.3
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Journal of Inequalities and Applications
In 1, 2 it is proved that the sequence {xn } generated by the following algorithm
xn1 I − αn AT xn αn u,
n≥0
1.4
converges in norm to the solution x∗ of 1.1 provided that the sequence {αn } in 0, 1 satisfies
conditions
n→∞
lim αn 0,
C1 ∞
αn ∞,
C2 n1
and additionally, either the condition
∞
|αn1 − αn | < ∞,
C3 |αn1 − αn |
0.
αn1
C4 n1
or the condition
lim
n→∞
The purpose of this paper is to introduce the following iterative algorithm:
xn1 I − αn Ayn αn u,
yn βn xn 1 − βn T xn ,
1.5
and to prove that the iterative sequence {xn } defined by 1.5 converges strongly to the
solution x∗ of 1.1 under the conditions C1 , C2 and 0 < a ≤ βn ≤ b < 1 for some constants
a, b.
Lemma 1.1 see 3, 4. Let {xn } and {yn } be bounded sequences in a Banach space X such that
xn1 λn xn 1 − λn yn ,
n ≥ 0,
1.6
where {λn } is a sequence in 0, 1 such that
0 < lim inf λn ≤ lim sup λn < 1.
1.7
lim sup yn1 − yn − xn1 − xn ≤ 0.
1.8
n→∞
n→∞
Assume that
n→∞
Then limn → ∞ yn − xn 0.
Journal of Inequalities and Applications
3
Lemma 1.2 see 1. Assume that A is a strongly positive linear bounded operator on a real Hilbert
space H with coefficient γ > 0 and 0 < α ≤ A−1 . Then I − αA ≤ 1 − αγ.
Lemma 1.3 see 5. Let H be a Hilbert space, K a closed convex subset of H, and T : K → K
a nonexpansive mapping with nonempty fixed point set FT . If {xn } is a sequence in K weakly
converging to x and if xn − T xn converges strongly to 0, then x T x.
Lemma 1.4 see 6. Assume that {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an γn δn ,
1.9
where {γn } is a sequence in 0, 1 and {δn } is a real sequence such that
i ∞
n1 γn ∞,
ii lim supn → ∞ δn ≤ 0 or ∞
n1 γn |δn | < ∞.
Then limn → ∞ an 0.
2. Main Results
Theorem 2.1. Suppose that A is strongly positive operator with coefficient γ > 0 as given in 1.2.
Suppose that the sequences {αn }, {βn } satisfy the conditions C1 , C2 and 0 < a ≤ βn ≤ b < 1 for
some constants a, b. Then the sequence {xn } generated by algorithm 1.5 converges strongly to the
unique solution x∗ of the minimization problem 1.1.
Proof. First we show that {xn } is bounded. As a matter of fact, take p ∈ FT and use
Lemma 1.2 to obtain
xn1 − p I − αn A βn xn 1 − βn T xn − p αn u − Ap ≤ 1 − γαn xn − p αn u − Ap.
2.1
By induction we can get
xn − p ≤ max x0 − p, 1 u − Ap ,
γ
n ≥ 0.
2.2
Hence, {xn } is bounded and so is {yn }. Next rewrite xn1 in the form
xn1 1 − λn xn λn zn ,
2.3
λn 1 − 1 − αn βn ,
2.4
where
zn αn βn
1 − βn
αn
u.
I − Axn I − αn AT xn λn
λn
λn
2.5
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Journal of Inequalities and Applications
Since αn → 0 and 0 < a ≤ βn ≤ b < 1, then
0 < lim inf λn ≤ lim sup λn < 1.
n→∞
n→∞
2.6
Next some manipulations give us that
βn1 αn1
βn αn
αn1 αn
−
u
zn1 − zn I − Axn1 −
I − Axn λn1
λn
λn1 λn
1 − βn1 αn1
1 − βn1
AT xn1 − T xn T xn1 − T xn −
λn1
λn1
1 − βn1 1 − βn
αn1 αn −
−
T xn −
1 − βn AT xn
λn1
λn
λn1 λn
αn1
−
βn − βn1 AT xn .
λn1
2.7
Therefore,
αn1 αn βn1 αn1
βn αn
u
−
I − Axn1 I − Axn zn1 − zn − xn1 − xn ≤
λn1
λn
λn1 λn 1 − βn1 αn1
1 − βn1
− 1 xn1 − xn Axn1 − xn λn1
λn1
1 − βn1 1 − βn T xn αn1 − αn 1 − βn AT xn −
λn1
λn
λn1 λn
αn1 βn − βn1 AT xn .
λn1
2.8
Since λn 1 − 1 − αn βn and αn → 0, then
1 − βn
αn βn
lim
lim 1 −
1.
n → ∞ λn
n→∞
λn
2.9
Then last inequality implies that
lim supzn1 − zn − xn1 − xn ≤ 0,
n→∞
2.10
and so an application of Lemma 1.1 asserts that
lim zn − xn 0.
n→∞
2.11
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By 2.5 we have that
αn βn
zn − T xn I − Axn λn
1 − βn αn
1 − βn
αn
− 1 T xn −
AT xn u.
λn
λn
λn
2.12
Again since αn → 0, {xn } is bounded, and λn 1 − 1 − αn βn , then we deduce from 2.12
that
lim zn − T xn 0.
2.13
lim xn − T xn 0.
2.14
n→∞
This together with 2.11 yields
n→∞
By using Lemma 1.3, we obtain ωw xn ⊂ FT , where ωw xn {z : ∃ xnk z} is the set of
weak ω-limit points of sequence {xn }.
Let x∗ be the unique solution to the minimization 1.1. Then by the definition of
algorithm 1.5, we can write
xn1 − x∗ I − αn A βn xn 1 − βn T xn − x∗ αn u − Ax∗ .
2.15
Since H is a Hilbert space, then we have that
2
xn1 − x∗ 2 ≤ I − αn Aβn xn 1 − βn T xn − x∗ 2αn u − Ax∗ , xn1 − x∗ ≤ 1 − γαn xn − x∗ 2αn u − Ax∗ , xn1 − x∗ .
2.16
However, we can take a subsequence {xnk } of {xn } such that
lim supu − Ax∗ , xn − x∗ lim u − Ax∗ , xnk − x∗ ,
n→∞
k→∞
2.17
and also {xnk } converges weakly to a fixed point p ∈ FT . It follows from optimality
condition 1.3 that
lim supu − Ax∗ , xn − x∗ u − Ax∗ , p − x∗ ≤ 0.
n→∞
2.18
Therefore, by using Lemma 1.4 and noticing 2.18, we conclude that xn → x∗ . This
completes the proof.
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Journal of Inequalities and Applications
References
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