Volume 2008, Article ID 598191, 10 pagesdoi:10.1155/2008/598191 Research Article Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces
Trang 1Volume 2008, Article ID 598191, 10 pages
doi:10.1155/2008/598191
Research Article
Approximate Proximal Point Algorithms for
Finding Zeroes of Maximal Monotone Operators
in Hilbert Spaces
Yeol Je Cho, 1 Shin Min Kang, 2 and Haiyun Zhou 3
1 Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, South Korea
2 Department of Mathematics and the RINS, Gyeongsang National University,
Chinju 660-701, South Korea
3 Department of Mathematics, Shijiazhuang Mechanical Engineering College,
Shijiazhuang 050003, China
Correspondence should be addressed to Haiyun Zhou, witman66@yahoo.com.cn
Received 1 March 2007; Accepted 27 November 2007
Recommended by H Bevan Thompson
LetH be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2 Ha maximal monotone operator withT−1 0/ ∅ Let PΩ be the metric projection ofH onto Ω Suppose that, for
any given x n ∈ H, β n > 0, and e n ∈ H, there exists x n ∈ Ω satisfying the following set-valued mapping equation:x n e n ∈ x n β n Tx n for all n ≥ 0, where {β n } ⊂ 0, ∞ with β n → ∞ as
n → ∞ and {e n} is regarded as an error sequence such that∞n0 e n 2< ∞ Let {α n } ⊂ 0, 1 be a
real sequence such thatα n → 0 as n → ∞ and∞
n0 α n ∞ For any fixed u ∈ Ω, define a sequence {x n } iteratively as x n1 α n u 1 − α n PΩx n − e n for all n ≥ 0 Then {x n} converges strongly to a pointz ∈ T−1 0 asn → ∞, where z lim t→∞ J t u.
Copyright q 2008 Yeol Je Cho 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.
1 Introduction and preliminaries
LetH be a real Hilbert space with the inner product ·, · and norm · A set T ⊂ H × H is
called a monotone operator on H if T has the following property:
A monotone operatorT on H is said to be maximal monotone if it is not properly contained in
ifRI tT H for all t > 0 For a maximal monotone operator T, we can define the resolvent
Trang 2ofT by
It is well known thatJ t :H → DT is nonexpansive Also we can define the Yosida approxima-tion T tby
T t 1
t
We know thatT t x ∈ TJ t x for all x ∈ H, T t x ≤ |Tx| for all x ∈ DT, where |Tx| inf{y : y ∈ Tx}, and T−10 FJ t for all t > 0.
Hilbert spaceH and T : Ω → 2 H is a maximal monotone operator withT−10/ ∅.
It is well known that, for anyu ∈ H, there exists uniquely y0∈ Ω such that
When a mappingPΩ :H → Ω is defined by PΩu y0in1.4, we call PΩthe metric projection of
H onto Ω The metric projection PΩ ofH onto Ω has the following basic properties:
i PΩx − x, x − PΩx ≥ 0 for all x ∈ Ω and x ∈ H,
ii PΩx − PΩy2≤ x − y, PΩx − PΩy for all x, y ∈ H,
iii PΩx − PΩy ≤ x − y for all x, y ∈ H,
iv x n → x0weakly andPx n → y0strongly imply thatPx0 y0.
Finding zeroes of maximal monotone operators is the central and important topic in nonlinear functional analysis A classical method to solve the following set-valued equation:
starting with any pointx0∈ H, updates x n1iteratively conforming to the following recursion:
x n ∈ x n1 β n Tx n1 , ∀n ≥ 0, 1.6 where{β n } ⊂ β, ∞, β > 0, is a sequence of real numbers However, as pointed out in 1, the ideal form of the algorithm is often impractical since, in many cases, solving the problem1.6 exactly is either impossible or as difficult as the original problem 1.5 Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of T
In 1976, Rockafellar2 gave an inexact variant of the method
x n e n1 ∈ x n1 β n Tx n1 , ∀n ≥ 0, 1.7 where{e n } is regarded as an error sequence This method is called an inexact proximal point
algo-rithm It was shown that if∞
n0 e n < ∞, then the sequence {x n} defined by 1.7 converges weakly to a zero ofT G ¨uler 3 constructed an example showing that Rockafellar’s proximal point algorithm1.7 does not converge strongly, in general This gives rise to the following question
Trang 3Question 1 How to modify Rockafellar’s algorithm so that strong convergence is guaranteed?
Xu4 gave one solution toQuestion 1 However, this requires that the error sequence
{e n} is summable, which is too strong This gives rise to the following question
Question 2 Is it possible to establish some strong convergence theorems under the weaker
assumption on the error sequence{e n} given in 1.7?
It is our purpose in this paper to give an affirmative answer toQuestion 2under a weaker assumption on the error sequence {e n} in Hilbert spaces For this purpose, we collect some lemmas that will be used in the proof of the main results in the next section
The first lemma is standard and it can be found in some textbooks on functional analysis
Lemma 1.1 For all x, y ∈ H and λ ∈ 0, 1,
Lemma 1.2 see 5, Lemma 1 For all u ∈ H, limt→∞ J t u exists and it is the point of T−10 nearest
to u.
Lemma 1.3 see 1, Lemma 2 For any given xn ∈ H, β n > 0, and e n ∈ H, there exists x n ∈ Ω
conforming to the following set-valued mapping equation (in short, SVME):
x n e n ∈ x n β n Tx n , ∀n ≥ 0. 1.9
Furthermore, for any p ∈ T−10, one has
x n − p, x n − x n e n
≥x n − x n , x n − x n e n
,
x n − e n − p2 ≤x n − p2−x n − x n2e n2. 1.10
Lemma 1.4 see 6, Lemma 1.1 Let {an }, {b n }, and {c n } be three real sequences satisfying
a n1≤1− t na n b n c n , ∀n ≥ 0, 1.11
where {t n } ⊂ 0, 1,∞n0 t n ∞, b n ◦t n , and∞n0 c n < ∞ Then a n → 0 as n → ∞.
2 The main results
Now we give our main results in this paper
Theorem 2.1 Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2 H
a maximal monotone operator with T−10/ ∅ Let PΩbe the metric projection of H onto Ω Suppose that, for any given x n ∈ H, β n > 0, and e n ∈ H, there exists x n ∈ Ω conforming to the SVME 1.9, where {β n } ⊂ 0, ∞ with β n → ∞ as n → ∞ and∞
n0 e n2< ∞ Let {α n } be a real sequence in 0, 1
such that
i α n → 0 as n → ∞,
ii∞
n0 α n ∞.
Trang 4For any fixed u ∈ Ω, define the sequence {x n } iteratively as follows:
x n1 α n u 1− α n
PΩ
x n − e n
Then {x n } converges strongly to a fixed point z of T, where z lim t→∞ J t u.
Proof
Claim 1 {x n} is bounded
Fixp ∈ T−10 and setM max{u − p2, x0− p2} First, we prove that
x n − p2≤ M n−1
j0
Whenn 0, 2.2 is true Now, assume that 2.2 holds for some n ≥ 0 We will prove that 2.2 holds forn 1 By using the iterative scheme 2.1 and Lemmas1.1and1.3, we have
x n1 − p2
α n u − p21− α n PΩ
x n − e n
− p2− α n
1− α n u − PΩ
x n − e n2
≤ α n M 1− α n x n − e n − p2≤ α n M 1− α n x n − p2e n2
≤ α n M 1− α n
M n
j0
e j2 M n
j0
e j2.
2.3
By induction, we assert that
x n − p2≤ M n−1
j0
e j2< M ∞
j0
e j2< ∞, ∀n ≥ 0. 2.4
This implies that{x n } is bounded and so is {J β n x n }.
Claim 2 lim n→∞ u − z, x n1 − z ≤ 0, where z lim t→∞ J t u, which is guaranteed byLemma 1.2 Noting that T is maximal monotone, u − J t u tT t u, T t u ∈ TJ t u, x n − J β n x n β n T β n x n,
T β n x n ∈ TJ β n x n, andβ n → ∞ n → ∞, we have
u − J t u, J β n x n − J t u −tT t u, J t u − J β n x n
−tT t u − T β n x n , J t u − J β n x n
− tT β n x n , J t u − J β n x n
≤ −β t
n
x n − J β n x n , J t u − J β n x n
−→ 0 n −→ ∞, ∀t > 0
2.5
and hence
lim
n→∞
u − J t u, J β n x n − J t u ≤ 0. 2.6 Note thatJ β n x n e n − J β n x n ≤ e n → 0 as n → ∞, and so it follows from 2.6 that
lim
n→∞
u − J t u, J β n
x n e n
Trang 5Note thatPΩx n − e n − J β n x n e n ≤ e n → 0 as n → ∞ and so it follows from 2.7 that
lim
n→∞
u − J t u, PΩ
x n − e n
Sinceα n → 0 as n → ∞, from 2.1 we have
x n1 − PΩ
x n − e n
It follows from2.8 and 2.9 that
lim
n→∞
u − J t u, x n1 − J t u ≤ 0, ∀t > 0, 2.10 and so, fromz lim t→∞ J t u and 2.10, we have
lim
n→∞
Claim 3 x n → z as n → ∞.
Observe that
1− α n
PΩ
x n − e n
and so
1− α n2PΩ
x n − e n
− PΩz2≥x n1 − z2− 2α n
u − z, x n1 − z , 2.13 which implies that
x n1 − z2≤1− α n x n − e n − z2 2α n
u − z, x n1 − z . 2.14
It follows fromLemma 1.3and2.14 that
x n1 − z2≤1− α n x n − z2−1− α n x n − x n2e n2 2α n
u − z, x n1 − z
≤1− α n x n − z2 2α n
u − z, x n1 − z e n2. 2.15
Setσ n max{u − z, x n1 − z , 0} Then σ n → 0 as n → ∞ Indeed, by the definition of σ n, we see that σ n ≥ 0 for all n ≥ 0 On the other hand, by 2.11, we know that for arbitrary > 0,
there exists some fixed positive integer N such that u − z, x n1 − z ≤ for all n ≥ N This
implies that 0 ≤ σ n ≤ for all n ≥ N, and the desired conclusion follows Set a n x n − z2,
b n 2α n σ n, andc n e n2 Then 2.15 reduces to
a n1≤1− α n
a n b n c n , ∀n ≥ 0, 2.16
n0 α n ∞, b n ◦α n, and∞n0 c n < ∞ Thus it follows fromLemma 1.4thata n → 0
asn → 0, that is, x n → z ∈ T−10 asn → ∞ This completes the proof.
Trang 6Remark 2.2 The maximal monotonicity of T is only used to guarantee the existence of solutions
to the SVME1.9 for any given x n ∈ H and β n > 0 If we assume that T : Ω → 2 H is monotone
need not be maximal and T satisfies the range condition
DT Ω ⊂
r>0
then for any givenx n ∈ Ω and β n > 0, we may find x n ∈ Ω and e n ∈ H satisfying the SVME
1.9 Furthermore,Lemma 1.2also holds foru ∈ Ω, and henceTheorem 2.1still holds true for monotone operators which satisfy the range condition
Following the proof lines ofTheorem 2.1, we can prove the following corollary
Corollary 2.3 Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and S : Ω → Ω
a continuous and pseudocontractive mapping with a fixed point in Ω Suppose that, for any given
x n ∈ Ω, β n > 0, and e n ∈ H, there exists x n ∈ Ω such that
x n e n1 β n
x n − β n Sx n , ∀n ≥ 0, 2.18
where β n → ∞ n → ∞ and {e n } satisfies the condition∞n0 e n2< ∞ Let {α n } ⊂ 0, 1 be a real
sequence such that α n → 0 as n → ∞ and∞n0 α n ∞ For any fixed u ∈ Ω, define the sequence {x n}
iteratively as follows:
x n1 α n u 1− α nPΩx n − e n, ∀n ≥ 0. 2.19
Then {x n } converges strongly to a fixed point z of S, where z lim t→∞ J t u, and J t I tI − S−1
for all t > 0.
Proof Let T I − S Then T : Ω → 2 H is continuous and monotone and satisfies the range condition
DT Ω ⊂
r>0
Gx r
1 r Sx
1
ThenG : Ω → Ω is continuous and strongly pseudocontractive By Kamimura et al 7, Corol-lary 1, G has a unique fixed point x in Ω, that is, x r/1rSx1/1ry, which implies thaty ∈ RI rT for all r > 0 In particular, for any given x n ∈ Ω and β n > 0, there exist x n∈ Ω ande n ∈ H such that
x n e n x n β n T x n , ∀n ≥ 0, 2.22 which means that
x n e n 1 β n
x n − β n S x n , ∀n ≥ 0, 2.23 and the relation2.18 follows The reminder of proof is the same as in the corresponding part
ofTheorem 2.1 This completes the proof
Trang 7Remark 2.4 In Corollary 2.3, we do not know wether the continuity assumption onS can be
dropped or not
Remark 2.5 InTheorem 2.1, if the operatorT is defined on the whole space H, then the metric
projection mappingPΩis not needed
Remark 2.6 Our convergence results are different from those results obtained by Kamimura
et al.7
Theorem 2.7 Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2 H
a maximal monotone operator with T−10/ ∅ Suppose that, for any given x n ∈ H, β n > 0, and e n ∈ H,
there exists x n ∈ Ω conforming to the following relation:
x n e n ∈ x n β n T x n , ∀n ≥ 0, 2.24
where lim n→∞ β n > 0 and∞n0 e n2 < ∞ Let {α n } be a sequence in 0, 1 with lim n→∞ α n < 1 and define the sequence {x n } iteratively as follows:
x0∈ Ω
x n1 α n x n1− α n
PΩ
x n − e n
Then {x n } converges weakly to a point p ∈ T−10.
Proof
Claim 1 {x n} is bounded
SinceT−10/ ∅, we can take some w ∈ T−10 By using2.25 and Lemmas1.1and1.3, we obtain
x n1 − w2
α n x n − w21− α n PΩ
x n − e n
− w2− α n
1− α n x n − PΩ
x n − e n2
≤ α n x n − w21− α n x n − e n − w2
≤ α n x n − w21− α n x n − w2−1− α n x n − x n2e n2
x n − w2−1− α n x n − x n2e n2
≤x n − w2e n2
2.26
and so2.26 together with∞n0 e n2 < ∞ implies that lim n→∞ x n − w2 exists Therefore,
{x n} is bounded
Claim 2 x n − J β n x n → 0 as n → ∞.
It follows from2.26 that
1− α n x n − x n2≤x n − w2−x n1 − w2e n2 2.27 and so2.26 together with limn→∞ α n < 1 implies that
Trang 8Sincex n J β n x n e n and J β nis nonexpansive, we have
x n − J β n x n ≤ x n − x n x n − J β n x n ≤ x n − x n e n −→ 0 2.29
asn → ∞ and consequently, x n − J β n x n → 0 as n → ∞.
Claim 3 {x n } converges weakly to a point p ∈ T−10 asn → ∞.
Set y n J β n x n and let p ∈ H be a weak subsequential limit of {x n } such that {x n j} converges weakly to a pointp as j → ∞ Thus it follows that {y n j } converges weakly to p as
j → ∞ Observe that
y n − J1y n I − J1
y n T1y n ≤ infz : z ∈ Ty n
T β n x n x n − y n
β n
2.30
By assumption limn→∞ β n > 0, we have
Since J1 is nonexpansive, by Browder’s demiclosedness principle, we assert thatp ∈ FJ1
T−10 Now Opial’s condition of H guarantees that {x n } converges weakly to p ∈ T−10 as
n → ∞ This completes the proof.
FromTheorem 2.7and the same proof ofCorollary 2.3, we have the following corollary
Corollary 2.8 Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and U : Ω → Ω
a continuous and pseudocontractive mapping with a fixed point Set T I − U Suppose that, for any given x n ∈ Ω, β n > 0, and e n ∈ H, there exists x n ∈ Ω such that
x n e n1 β n
x n − β n Ux n , ∀n ≥ 0. 2.32
Define the sequence {x n } iteratively as follows:
x0∈ Ω,
x n1 α n x n1− α n
PΩ
x n − e n
where {α n } ⊂ 0, 1 with lim n→∞ α n < 1, {β n } ⊂ 0, ∞ with lim n→∞ β n > 0, and {e n } ⊂ H with
n0 e n2< ∞ Then {x n } converges weakly to a fixed point p of U.
3 Applications
We can apply Theorems2.1and2.7to find a minimizer of a convex functionf Let H be a real
subdifferential ∂f of f is defined as follows:
∂fz v∗∈ H : fy ≥ fz y − z, v∗
, y ∈ H, ∀ z ∈ H. 3.1
Trang 9Theorem 3.1 Let H be a real Hilbert space and f : H → −∞, ∞ a proper convex lower
semicon-tinuous function Suppose that, for any x n ∈ H, β n > 0, and e n ∈ H, there exists x n conforming to
x n e n ∈ x n β n ∂fx n, ∀n ≥ 0, 3.2
where {β n } is a sequence in 0, ∞ with β n → ∞ n → ∞ and ∞n0 e n2 < ∞ Let {α n } be a
sequence in 0, 1 such that α n → 0 n → ∞ and∞n0 α n ∞ For any fixed u ∈ H, let {x n } be the
sequence generated by
u, x0∈ H,
x n arg min
z∈H
fz 1
2β n z − x n − e n2 ,
x n1 α n u 1− α n
x n − e n
, ∀n ≥ 0.
3.3
If ∂f−10/ ∅, then {x n } converges strongly to the minimizer of f nearest to u.
Proof Since f : H → −∞, ∞ is a proper convex lower semicontinuous function, by 2, the subdifferential ∂f of f is a maximal monotone operator Noting that
x n arg min
z∈H
fz 1
is equivalent to
0∈ ∂fx n
β1
n
x n − x n − e n
we have
x n e n ∈ x n β n ∂fx n
Therefore, usingTheorem 2.1, we have the desired conclusion This completes the proof
Theorem 3.2 Let H be a real Hilbert space and f : H → −∞, ∞ a proper convex lower
semicon-tinuous function Suppose that, for any given x n ∈ H, β n > 0, and e n ∈ H, there exists x n ∈ H such
that
x n e n ∈ x n β n ∂fx n
where {β n } is a sequence in 0, ∞ with lim n→∞ β n > 0 and∞n0 e n2< ∞ Let {α n } be a sequence
in 0, 1 with lim n→∞ α n < 1 and let {x n } be the sequence generated by
x0∈ H,
x n arg min
z∈H
fz 1
2β n z − x n − e n2 ,
x n1 α n x n1− α n
x n − e n
, ∀n ≥ 0.
3.8
If ∂f−10/ ∅, then {x n } converges weakly to the minimizer of f nearest to u.
Trang 10Proof As shown in the proof lines ofTheorem 3.1,∂f : H → H is a maximal monotone
opera-tor, and so the conclusion ofTheorem 3.2follows fromTheorem 2.7
Acknowledgment
The authors are grateful to the anonymous referee for his helpful comments which improved the presentation of this paper
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Proof Since f : H → −∞, ∞ is a proper convex lower semicontinuous function, by 2, the subdifferential ∂f of f is a maximal monotone operator