Volume 2011, Article ID 794203, 12 pagesdoi:10.1155/2011/794203 Research Article Algorithms Construction for Variational Inequalities 1 Department of Mathematics, Tianjin Polytechnic Uni
Trang 1Volume 2011, Article ID 794203, 12 pages
doi:10.1155/2011/794203
Research Article
Algorithms Construction for
Variational Inequalities
1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
3 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
Correspondence should be addressed to Shin Min Kang,smkang@gnu.ac.kr
Received 4 October 2010; Accepted 19 February 2011
Academic Editor: Yeol J Cho
Copyrightq 2011 Yonghong Yao 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
We devote this paper to solving the variational inequality of finding x∗with property x∗∈ FixT
such thatA−γfx∗, x−x∗ ≥ 0 for all x ∈ FixT Note that this hierarchical problem is associated
with some convex programming problems For solving the above VI, we suggest two algorithms:
Implicit Algorithmml: x t TPcI − tA − γfx t for all t ∈ 0, 1 and Explicit Algorithm: x n1
β n x n 1 − β n TPc1 − α n A − γfx n for all n ≥ 0 It is shown that these two algorithms converge
strongly to the unique solution of the above VI As special cases, we prove that the proposed
algorithms strongly converge to the minimum norm fixed point of T.
1 Introduction
Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied sciences Several numerical techniques for solving variational inequalities and the related optimization problem have been considered by some authors See, for example,1 16
Our main purpose in this paper is to consider the following variational inequality:
Find x∗∈ FixT such thatA − γf
x∗, x − x∗
≥ 0, ∀x ∈ FixT, 1.1
where T is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H, A : C → H is a strongly positive linear bounded operator, and f : C → H
is a ρ-contraction.
Trang 2At this point, we wish to point out this hierarchical problem associated with some convex programming problems The reader can refer to17–21 and the references therein For solving VI1.1, we suggest two algorithms which converge to the unique solution
of VI1.1 As special cases, we prove that the proposed algorithms strongly converge to the
minimum norm fixed point of T.
2 Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , respectively, and let C be
a nonempty closed convex subset of H Let f : C → H be a ρ-contraction; that is, there exists
a constant ρ ∈ 0, 1 such that
f x − f
y ≤ ρx − y, ∀x,y ∈ C. 2.1
A mapping A is said to be strongly positive on H if there exists a constant γ > 0 such that
Ax, x ≥ γ x 2
Recall that a mapping T : C → C is said to be nonexpansive if
A point x ∈ C is a fixed point of T provided Tx x Denote by FixT the set of fixed points of
T; that is, FixT {x ∈ C : Tx x}.
Remark 2.1 If A : C → H is a strongly positive linear bounded operator and f : C → H is
a ρ-contraction, then for 0 < γ < γ/ρ, the mapping A − γf is strongly monotone In fact, we
have
A − γf
x −
A − γf
y, x − y
Ax − y
, x − y − γ
f x − fy
, x − y
≥ γx − y2− γρx − y2
≥ 0.
2.4
The metricor nearest point projection from H onto C is the mapping P C : H → C which assigns to each point x ∈ C the unique point P C x ∈ C satisfying the property
x − P C x inf
The following properties of projections are useful and pertinent to our purposes
Lemma 2.2 Given x ∈ H and z ∈ C,
a z P Cx if and only if there holds the relation
x − z, y − z
Trang 3b z P Cx if and only if there holds the relation
x − z 2≤x − y2−y − z2
, ∀y ∈ C, 2.7
c there holds the relation
PCx − PC y, x − y
≥PC x − PC y2
, ∀x, y ∈ H. 2.8
Consequently, PC is nonexpansive and monotone.
In the sequel, we will make use of the following for our main results
Lemma 2.3 Demiclosedness Principle for Nonexpansive Mappings, 22 Let C be a
nonempty closed convex subset of a real Hilbert space H and T : C → C be a nonexpansive mapping with Fix T / ∅ If {x n } is a sequence in C weakly converging to x and if {I − Tx n } converges
strongly to y, then I − Tx y; in particular, if y 0, then x ∈ FixT.
Lemma 2.4 see 14 Let C be a nonempty closed convex subset of a real Hilbert space H Assume
that the mapping F : C → H is monotone and weakly continuous along segments, that is, Fx ty → Fx weakly as t → 0 Then the variational inequality
x∗∈ C, Fx∗, x − x∗ ≥ 0, ∀x ∈ C 2.9
is equivalent to the dual variational inequality
x∗∈ C, Fx, x − x∗ ≥ 0, ∀x ∈ C. 2.10
Lemma 2.5 see 23 Let {x n } and {y n } be bounded sequences in a Banach space X and {β n } be a
sequence in 0, 1 with
0 < lim inf
n → ∞
Suppose that xn1 1 −β n y n β nxn for all n ≥ 0 and lim sup n → ∞ y n1 −y n − x n1 −x n ≤ 0.
Then lim n → ∞ y n − x n 0.
Lemma 2.6 see 24 Assume that {a n } is a sequence of nonnegative real numbers such that
an1≤1− γ n
an γ nδn, ∀n ≥ 0, 2.12
where {γ n } is a sequence in 0, 1 and {δ n } is a sequence inÊsuch that
a∞n0 γn ∞,
b lim supn → ∞ δn ≤ 0 or∞
n0 |δ nγn | < ∞.
Then limn → ∞an 0.
Trang 43 Main Results
In this section, we first consider an implicit algorithm and prove its strong convergence for solving variational inequality1.1
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H
be a strongly positive linear bounded operator and f : C → H be a ρ-contraction Let T : C → C
be a nonexpansive mapping with Fix T / ∅ Let γ > 0 be a constant satisfying γ − 1/ρ < γ < γ/ρ.
For each t ∈ 0, 1, let the net {xt } be defined by
xt TP C
I − t
Then the net {x t } converges in norm, as t → 0, to x∗ ∈ FixT which is the unique solution of VI
1.1.
Proof First, we note that the net {x t} defined by 3.1 is well-defined As a matter of fact, we have, for sufficiently small t,
TPC
I − t
A − γf x − TPC
I − t
A − γf y
≤ I − t
A − γf x −
I − t
A − γf y
≤ tγf x − f
y I − tA x − y
≤ tγρx − y 1 − tγx − y
1−γ − γρt x − y, ∀x, y ∈ C,
3.2
C I −tA−γfx is a contractive from C into C Using the Banach contraction principle, there exists a unique point x t ∈ C satisfying the following
fixed point equation:
x TPC
I − t
this is,
xt TP C
I − t
which is exactly3.1
Next, we show that the net{x t } is bounded Take an x∗ ∈ FixT to derive that
x t − x∗ TPC
I − t
A − γf xt − TP Cx∗
≤ I − t
A − γf xt − x∗
≤ tγf x t − fx∗ tγfx∗ − Ax∗ I − tAx t − x∗
≤1− γt x t − x∗ tγρ x t − x∗ tγf x∗ − Ax∗.
3.5
Trang 5This implies that
x t − x∗ ≤ γ − γρ1 γf x∗ − Ax∗. 3.6
It follows that{x t } is bounded, so are the nets {fx t } and {Ax t}
From3.1, we get
x t − Tx t TPC
I − t
A − γf xt − TP Cxt
≤ tA − γf
xt
−→ 0.
3.7
Set y t P C I − tA − γfx t for all t ∈ 0, 1 It follows that
yt − x t ≤ tA − γfx t −→ 0. 3.8
At the same time, we note that
x t − x∗ ≤yt − x∗. 3.9 From3.1 and the property of the metric projection, we have
yt − x∗2PC
I − t
A − γf xt− I − t
A − γf xt, yt − x∗
I − t
A − γf xt − x∗, yt − x∗
≤ I − t
A − γf xt − x∗, yt − x∗
tγfx t − Ax∗, yt − x∗ I − tAx t − x∗, y t − x∗
≤1− tγ x t − x∗ yt − x∗ tγfx t − Ax∗, yt − x∗
≤1− tγyt − x∗2 tγf x t − Ax∗, yt − x∗
.
3.10
It follows that
yt − x∗2≤ 1γγf x t − Ax∗, yt − x∗
1γ γ
f x t − fx∗, y t − x∗
γf x∗ − Ax∗, yt − x∗
≤ 1γ γρyt − x∗2A − γf
x∗, x∗− y t
.
3.11
Trang 6That is,
yt − x∗2≤ 1
γ − γρ
A − γf
x∗, x∗− y t
Therefore,
x t − x∗ 2≤yt − x∗ ≤ 1γ − γρ
A − γf
x∗, x∗− y t
In particular,
x n − x∗ 2 ≤ γ − γρ1 A − γf
x∗, x∗− y n
Next, we show that{x t } is relatively norm-compact as t → 0 Assume{t n } ⊂ 0, 1 is such that t n → 0as n → ∞ Put x n: xt n and y n: yt n From3.7, we have
x n − Tx n −→ 0. 3.15
Since{x n } is bounded, without loss of generality, we may assume that {x n} converges weakly
to a point x ∈ C and hence y n also converges weakly to x Noticing 3.15, we can use
Lemma 2.3to get x ∈ FixT Therefore, we can substitute x for x∗in3.14 to get
x n − x 2 ≤ γ − γρ1 A − γf
x, x − y n
Consequently, the weak convergence of{y n } to x actually implies that x n → x strongly This
has proved the relative norm-compactness of the net{x t } as t → 0
Now, we return to3.14 and take the limit as n → ∞ to get
x − x∗ 2≤ γ − γρ1 A − γf
x∗, x∗− x, ∀x∗∈ FixT. 3.17 Hencex solves the following VI:
A − γf
x∗, x∗− x ≥ 0, ∀x∗∈ FixT 3.18
or the equivalent dual VIseeRemark 2.1andLemma 2.4
A − γf
x, x∗− x≥ 0, ∀x∗∈ FixT. 3.19
From the strong monotonicity of A − γf, it follows the uniqueness of a solution of the above
VIsee 11, Theorem 3.2, x PFixTI −Aγfx That is, x is the unique fixed point in FixT
Trang 7of the contraction PFixTI − A γf Clearly this is sufficient to conclude that the entire net {x t } converges in norm to x as t → 0 This completes the proof
Next, we suggest an explicit algorithm and prove its strong convergence
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let A : C → H
be a strongly positive linear bounded operator and f : C → H be a ρ-contraction Let T : C → C
be a nonexpansive mapping with Fix T / ∅ Let γ > 0 be a constant satisfying γ − 1/ρ < γ < γ/ρ.
For x0∈ C, let the sequence {x n } be generated iteratively by
xn1 β nxn1− β n
TPC
I − αn
where the sequences {α n } ⊂ 0, 1 and {β n } ⊂ 0, 1 satisfy the following control conditions:
C1 limn → ∞αn 0,
C2 limn → ∞αn ∞,
C3 0 < lim inf n → ∞βn≤ lim supn → ∞ βn < 1.
Then {x n } converges strongly to x∗∈ FixT which is the unique solution of the variational inequality
VI1.1.
Proof First we show that {x n } is bounded Set y n TP C un and u n I − α n A − γfx n for
all n ≥ 0 For any p ∈ FixT, we have
yn − p TP Cun − TP C p
≤ I − αn
A − γf xn − p
≤ α nγf x n − γf
p α nγf
p− Ap I − α nA xn − p
≤ α nγρxn − p α nγf
p− Ap 1 − α n γxn − p
1−γ − γραn xn − p α nγf
p
− Ap.
3.21
It follows that
xn1 − p ≤ β nxn − p 1 − β nyn − p
≤ β nxn − p 1 − β n
1−γ − γραn xn − p
α n
1− β nγf
p
− Ap
1−γ − γραn
1− β n xn − p
γ − γραn
1− β n
γf
p
− Ap
γ − γρ ,
3.22
which implies that
xn − p ≤ max x0− p,γf
p
− Ap
γ − γρ
, ∀n ≥ 0. 3.23
Trang 8Hence{x n } is bounded and so are {y n }, {u n }, {Ax n }, and {fx n}.
From3.20, we observe that
yn1 − y n TP Cun1 − TP Cun
≤ I − αn1
A − γf xn1− I − αn
A − γf xn
αn1γ
f x n1 − fx n α n1 − α n γfx n
I − α n1A x n1 − x n α n − α n1 Ax n
≤ α n1γf x n1 − fx n 1 − αn1 γ x n1 − x n
|α n1 − α n|γf x n Axn
≤ α n1γρ x n1 − x n 1− α n1 γ x n1 − x n
|α n1 − α n|γf x n Axn
1−γ − γραn1 x n1 − x n |α n1 − α n|γf x n Axn .
3.24
It follows that
yn1 − y n − x n1 − x n ≤γ − γραn1 x n1 − x n |α n1 − α n|γf x n Axn ,
3.25 which implies, fromC1 and the boundedness of {x n }, {fx n } and {Ax n}, that
lim sup
n → ∞
yn1 − y n − x n1 − x n ≤ 0. 3.26
Hence, byLemma 2.5, we have
lim
Consequently, it follows that
lim
n → ∞ x n1 − x n lim
n → ∞
1− β nyn − x n 0. 3.28
On the other hand, we have
x n − Tx n ≤ x n1 − x n x n1 − Tx n
x n1 − x n β x n − Tx n
1− β n
yn − Tx n
≤ x n1 − x n β n x n − Tx n 1− β nyn − TP Cxn
≤ x n1 − x n β n x n − Tx n 1− β n
αnA − γf
xn,
3.29
Trang 9that is,
x n − Tx n ≤ 1
1− β n x n1 − x n α nA − γf
xn. 3.30
This together withC1, C3, and 3.28 implies that
lim
Next, we show that, for any x∗∈ FT,
lim sup
n → ∞
un − x∗, γf x∗ − Ax∗
Now we take a subsequence{x n k } of {x n} such that
lim sup
n → ∞
xn − x∗, γf x∗ − Ax∗
lim
k → ∞
xn k − x∗, γf x∗ − Ax∗
. 3.33
Since{x n } is bounded, we may assume that x n k → z weakly Note that z ∈ FixT by virtue
ofLemma 2.3and3.31 Therefore,
lim sup
n → ∞
xn − x∗, γf x∗ − Ax∗
z − x∗, γf x∗ − Ax∗
≤ 0. 3.34
We notice that
u n − x n ≤ α nA − γf
xn −→ 0. 3.35
Hence, we get
lim sup
n → ∞
un − x∗, γf x∗ − Ax∗
Finally, we prove that{x n } converges to the point x∗ We observe that
u n − x∗ ≤ x n − x∗ α nA − γf
xn. 3.37
Trang 10Therefore, from3.20, we have
x n1 − x∗ 2≤ β n x n − x∗ 21− β nyn − x∗2
≤ β n x n − x∗ 21− β n
u n − x∗ 2
β n x n − x∗ 21− β nαn
γf x n − Ax∗
I − α nA x n − x∗2
≤ β n x n − x∗ 21− β n
×
1− α n γ2 x n − x∗ 2 2α n
γf x n − Ax∗, un − x∗
1− 2α n γ 1− β n
α2n γ2
x n − x∗ 2
2α n
γf x n − γfx∗, u n − x∗
2α n
γf x∗ − Ax∗, un − x∗
≤ 1− 2α n γ 1− β n
α2n γ2
x n − x∗ 2
2α nγρ x n − x∗ u n − x∗ 2α n
γf x∗ − Ax∗, un − x∗
≤ 1− 2α n
γ − γρ x n − x∗ 21− β n
α2n γ2 x n − x∗ 2
2α2
n γρ x n − x∗ A − γf
xn 2α n
γf x∗ − Ax∗, un − x∗
.
3.38
Since{x n }, {fx n }, and {Ax n } are all bounded, we can choose a constant M > 0 such that
sup
n
1
γ − γρ
1− β n
γ2
2 x n − x∗ 2 γρ x n − x∗ A − γf
xn≤ M. 3.39
It follows that
x n1 − x∗ 2≤ 1− 2γ − ργαn x n − x∗ 2 2γ − ργαnδn, 3.40 where
δn α nM γ − γρ1
γf x∗ − Ax∗, un − x∗
ByC1 and 3.36, we get
lim sup
n → ∞
Now, applying Lemma 2.6 and 3.40, we conclude that x n → x∗ This completes the proof
Trang 11From Theorems3.1and3.2, we can deduce easily the following corollaries.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C
be a nonexpansive mapping with Fix T / ∅ For each t ∈ 0, 1, let the net {x t } be defined by
xt TP C 1 − tx t, ∀t ∈ 0, 1. 3.43
Then the net {x t } defined by 3.43 converges in norm, as t → 0, to the minimum norm element
x∗∈ FixT.
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → C be
a nonexpansive mapping with Fix T / ∅ For x0 ∈ C, let the sequence {x n } be generated iteratively
by
xn1 β nxn1− β n
TPC 1 − α n x n, ∀n ≥ 0, 3.44
where the sequences {α n } ⊂ 0, 1 and {β n } ⊂ 0, 1 satisfy the following control conditions:
C1 limn → ∞αn 0,
C2 limn → ∞αn ∞,
C3 0 < lim inf n → ∞βn≤ lim supn → ∞ βn < 1.
Then the sequence {x n } generated by 3.44 converges strongly to the minimum norm element x∗ ∈ FixT
Acknowledgments
The authors thank the referees for their comments and suggestions which improved the presentation of this paper The first author was supported in part by Colleges and Universities, Science and Technology Development Foundation 20091003 of Tianjin and NSFC 11071279 The second author was supported in part by NSC 99-2221-E-230-006
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