Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

Volume 2008, Article ID 598191,10pages 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,[email protected]

Received 1 March 2007; Accepted 27 November 2007

Recommended by H. Bevan Thompson

LetHbe a real Hilbert space,Ωa nonempty closed convex subset ofH, andT :Ω→2Ha maximal monotone operator withT−1_{0/∅. LetP}

Ωbe the metric projection ofH ontoΩ. Suppose that, for any given xn ∈ H,βn > 0, anden ∈ H, there existsxn ∈ Ωsatisfying the following set-valued mapping equation:xnen ∈ xnβnTxnfor alln≥0, where{βn} ⊂ 0,∞withβn → ∞as n→ ∞and{en}is regarded as an error sequence such that∞n0en

2_<∞

. Let{αn} ⊂0,1be a real sequence such thatαn→0 asn→ ∞and∞n0αn∞. For any fixedu∈Ω, define a sequence

{xn}iteratively asxn1αnu 1−αnPΩxn−enfor alln≥0. Then{xn}converges strongly to a pointz∈T−1_{0 asn→ ∞, wherezlim}

t→∞Jtu.

Copyrightq2008 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 setT ⊂H ×H is called amonotone operatoronHifThas the following property:

x−x_{, y−y ≥0, ∀x, y,x, y∈T. 1.1}

ofTby

Jt ItT−1, ∀t >0. 1.2

It is well known thatJt :H →DTis nonexpansive. Also we can define theYosida approxima-tionTtby

Tt 1_tI−Jt, ∀t >0. 1.3

We know thatTtx∈TJtxfor allx∈H,Ttx ≤ |Tx|for allx∈DT,where|Tx|inf{y:y∈

Tx},andT−1_0FJ

tfor allt >0.

Throughout this paper, we assume thatΩ is a nonempty closed convex subset of a real Hilbert spaceHandT :Ω→2H is a maximal monotone operator withT−1_0/∅.

It is well known that, for anyu∈H,there exists uniquelyy0∈Ωsuch that _{u−y0infu−y:y∈Ω}

. 1.4

When a mappingPΩ :H→Ωis defined byPΩuy0in1.4, we callPΩthemetric projectionof

HontoΩ.The metric projectionPΩ ofHontoΩhas the following basic properties: iPΩx−x, x−PΩx ≥0 for allx∈Ωandx ∈H,

iiPΩx−PΩy2≤ x−y, PΩx−PΩy for allx, y ∈H, iiiPΩx−PΩy ≤ x−yfor allx, y ∈H,

ivxn→x0weakly andPxn→y0strongly imply thatPx0y0.

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:

0∈Tz, 1.5

where T : Ω → 2H is a maximal monotone operator, is the proximal point algorithm which, starting with any pointx0∈H,updatesxn1iteratively conforming to the following recursion:

xn∈xn1βnTxn1, ∀n≥0, 1.6

where{βn} ⊂β,∞,β > 0,is a sequence of real numbers. However, as pointed out in1, 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 problem1.5. 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 ofT.

In 1976, Rockafellar2gave an inexact variant of the method

xnen1∈xn1βnTxn1, ∀n≥0, 1.7

where{en}is regarded as an error sequence. This method is called aninexact proximal point algo-rithm. It was shown that if∞_n0en<∞, then the sequence{xn}defined by1.7converges

Question 1. How to modify Rockafellar’s algorithm so that strong convergence is guaranteed?

Xu4gave one solution to Question 1. However, this requires that the error sequence

{en}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{en}given in1.7?

It is our purpose in this paper to give an affirmative answer toQuestion 2under a weaker assumption on the error sequence {en} 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 allx, y∈H andλ∈0,1,

_{λx 1−λy}2_{λx2 1−λy2−λ1−λx−y}2_{. 1.8}

Lemma 1.2see5, Lemma 1. For allu ∈H,limt→∞Jtuexists and it is the point ofT−10nearest tou.

Lemma 1.3see1, Lemma 2. For any givenxn ∈ H,βn > 0, anden ∈ H, there existsxn ∈ Ω conforming to the following set-valued mapping equation (in short, SVME):

xnen∈xnβnTxn, ∀n≥0. 1.9

Furthermore, for anyp∈T−1_{0, one has}

xn−p, xn−xnen ≥xn−xn, xn−xnen , _xn−en−p2 _≤x

n−p2−xn−xn2en2.

1.10

Lemma 1.4see6, Lemma 1.1. Let{an},{bn}, and{cn}be three real sequences satisfying

an1≤

1−tnanbncn, ∀n≥0, 1.11

where{tn} ⊂0,1,∞n0tn∞,bn◦tn, and _∞

n0cn<∞.Thenan→0asn→ ∞.

2. The main results

Now we give our main results in this paper.

Theorem 2.1. LetHbe a real Hilbert space,Ωa nonempty closed convex subset ofH, andT :Ω→2H

a maximal monotone operator withT−1_0/∅.LetP

Ωbe the metric projection ofHontoΩ. Suppose that, for any givenxn∈H,βn >0, anden ∈H, there existsxn ∈Ωconforming to the SVME1.9, where

{βn} ⊂0,∞withβn→ ∞asn→ ∞and∞n0en2<∞. Let{αn}be a real sequence in0,1 such that

iαn→0asn→ ∞,

ii∞

For any fixedu∈Ω, define the sequence{xn}iteratively as follows:

xn1αnu1−αnPΩxn−en, ∀n≥0. 2.1

Then{xn}converges strongly to a fixed pointzofT,wherezlimt→∞Jtu.

Proof

Claim 1. {xn}is bounded.

Fixp∈T−1_{0 and setMmax{u−p}2_,x

0−p2}.First, we prove that _xn−p2_≤Mn−1

j0

_ej2_{, ∀n≥}

0. 2.2

Whenn0,2.2is true. Now, assume that2.2holds for somen≥0. We will prove that2.2 holds forn1. By using the iterative scheme2.1and Lemmas1.1and1.3, we have

_xn1−p2

αnu−p21−αnPΩxn−en−p2−αn1−αnu−PΩxn−en2

≤αnM1−αnxn−en−p2≤αnM1−αnxn−p2en2

≤αnM1−αnM n

j0

_ej2_Mn j0

_ej2_.

2.3

By induction, we assert that

_xn−p2_≤Mn−1 j0

_ej2_{< M}∞ j0

_ej2_{<∞, ∀n≥}

0. 2.4

This implies that{xn}is bounded and so is{Jβnxn}.

Claim 2. limn→∞u−z, xn1−z ≤0, wherezlimt→∞Jtu,which is guaranteed byLemma 1.2.

Noting that T is maximal monotone, u−Jtu tTtu,Ttu ∈ TJtu, xn−Jβnxn βnTβnxn, Tβnxn∈TJβnxn, andβn→∞ n→ ∞,we have

u−Jtu, Jβnxn−Jtu −t

Ttu, Jtu−Jβnxn

−tTtu−Tβnxn, Jtu−Jβnxn −t

Tβnxn, Jtu−Jβnxn

≤ −_βt

n

xn−Jβnxn, Jtu−Jβnxn

−→0 n−→ ∞, ∀t >0

2.5

and hence

lim

n→∞

u−Jtu, Jβnxn−Jtu ≤0. 2.6

Note thatJβnxnen−Jβnxn ≤ en →0 asn→ ∞, and so it follows from2.6that

lim

n→∞

u−Jtu, Jβn

Note thatPΩxn−en−Jβnxnen ≤ en →0 asn→ ∞and so it follows from2.7that

lim

n→∞

u−Jtu, PΩxn−en−Jtu ≤0. 2.8

Sinceαn → 0 asn→ ∞, from2.1we have

xn1−PΩxn−en−→0 n−→ ∞. 2.9

It follows from2.8and2.9that

lim

n→∞

u−Jtu, xn1−Jtu ≤0, ∀t >0, 2.10

and so, fromzlimt→∞Jtuand2.10, we have

lim

n→∞

u−z, xn1−z ≤0. 2.11

Claim 3. xn→zasn→ ∞.

Observe that

1−αnPΩxn−en−zxn1−z

−αnu−z 2.12

and so

1−αn2PΩxn−en−PΩz2≥xn1−z2−2αnu−z, xn1−z , 2.13

which implies that

_xn1−z2_≤

1−αnxn−en−z22αnu−z, xn1−z . 2.14

It follows fromLemma 1.3and2.14that

_xn1−z2_≤

1−αnxn−z2−1−αnxn−xn2en22αnu−z, xn1−z

≤1−αnxn−z22αnu−z, xn1−z en2.

2.15

Setσnmax{u−z, xn1−z ,0}.Thenσn → 0 asn → ∞. Indeed, by the definition ofσn, we

see that σn ≥ 0 for alln ≥ 0. On the other hand, by2.11, we know that for arbitrary > 0,

there exists some fixed positive integer N such thatu−z, xn1−z ≤ for alln ≥ N. This

implies that 0 ≤ σn ≤ for alln ≥ N, and the desired conclusion follows. Setan xn−z2,

bn 2αnσn, andcnen2.Then2.15reduces to

an1≤

1−αnanbncn, ∀n≥0, 2.16

where∞_n0αn∞,bn◦αn, and∞n0cn<∞. Thus it follows fromLemma 1.4thatan → 0

Remark 2.2. The maximal monotonicity ofTis only used to guarantee the existence of solutions to the SVME1.9for any givenxn∈Handβn>0. If we assume thatT :Ω→2H is monotone

need not be maximalandT satisfies the range condition

DT Ω⊂

r>0

RIrT, 2.17

then for any givenxn ∈ Ωandβn > 0, we may findxn ∈ Ωanden ∈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. LetHbe a real Hilbert space,Ωa nonempty closed convex subset ofH, andS:Ω→Ω

a continuous and pseudocontractive mapping with a fixed point in Ω. Suppose that, for any given

xn∈Ω,βn>0, anden∈H, there existsxn∈Ωsuch that

xnen1βnxn−βnSxn, ∀n≥0, 2.18

whereβn→ ∞n→ ∞and{en}satisfies the condition∞n0en2<∞. Let{αn} ⊂0,1be a real sequence such thatαn→0asn→ ∞and∞n0αn∞. For any fixedu∈Ω, define the sequence{xn} iteratively as follows:

xn1αnu1−αnPΩxn−en, ∀n≥0. 2.19

Then{xn}converges strongly to a fixed pointzofS, wherez limt→∞Jtu,andJt ItI−S−1 for allt >0.

Proof. LetT I−S. Then T : Ω → 2H is continuous and monotone and satisfies the range condition

DT Ω⊂

r>0

RIrT. 2.20

Now we only need to verify the last assertion. For any y ∈ Ω and r > 0, define an operatorG :Ω→Ωby

Gx r 1rSx

1

1ry. 2.21

ThenG:Ω →Ωis continuous and strongly pseudocontractive. By Kamimura et al.7, Corol-lary 1,Ghas a unique fixed pointxinΩ, that is,x r/1rSx1/1ry,which implies thaty∈RIrTfor allr >0. In particular, for any givenxn∈Ωandβn>0, there existxn∈Ω

anden∈H such that

xnenxnβnT xn, ∀n≥0, 2.22

which means that

xnen 1βnxn−βnS xn, ∀n≥0, 2.23

Remark 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 spaceH, 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. LetHbe a real Hilbert space,Ωa nonempty closed convex subset ofH, andT :Ω→2H

a maximal monotone operator withT−10/∅.Suppose that, for any givenxn ∈H,βn>0, anden∈H, there existsxn∈Ωconforming to the following relation:

xnen∈xnβnT xn, ∀n≥0, 2.24

wherelim_n→∞βn> 0and∞n0en2 <∞. Let{αn}be a sequence in0,1withlimn→∞αn <1and define the sequence{xn}iteratively as follows:

x0∈Ω

xn1αnxn1−αnPΩxn−en, ∀n≥0.

2.25

Then{xn}converges weakly to a pointp∈T−10.

Proof

Claim 1. {xn}is bounded.

SinceT−1_{0/∅,we can take somew∈T}−1_{0. By using2.25and Lemmas1.1and1.3, we}

obtain

_xn1−w2

αnxn−w21−αnPΩxn−en−w2−αn1−αnxn−PΩxn−en2

≤αnxn−w21−αnxn−en−w2

≤αnxn−w21−αnxn−w2−1−αnxn−xn2en2

xn−w2−1−αnxn−xn2en2

≤xn−w2en2

2.26

and so2.26together with∞_n0en2 < ∞ implies that limn→∞xn−w2 exists. Therefore,

{xn}is bounded.

Claim 2. xn−Jβnxn→0 asn→ ∞. It follows from2.26that

1−αnxn−xn2≤xn−w2−xn1−w2en2 2.27

and so2.26together with limn→∞αn<1 implies that

SincexnJβnxnenandJβnis nonexpansive, we have

_{xn−Jβnxn≤xn−xnxn−Jβnxn≤xn−xnen−→0 2.29}

asn → ∞and consequently,xn−Jβnxn → 0 asn → ∞.

Claim 3. {xn}converges weakly to a pointp∈T−10 asn→ ∞.

Set yn Jβnxn and let p ∈ H be a weak subsequential limit of {xn} such that {xnj} converges weakly to a pointpas j → ∞. Thus it follows that{ynj}converges weakly topas j→ ∞. Observe that

_{yn−J1ynI−J1}

ynT1yn≤inf

z:z∈TynTβnxn

xn−yn

βn

. 2.30

By assumption lim_n→∞βn>0,we have

yn−J1yn−→0 n−→ ∞. 2.31

Since J1 is nonexpansive, by Browder’s demiclosedness principle, we assert thatp∈FJ1

T−1_{0. Now Opial’s condition ofH guarantees that {x}

n} converges weakly to p ∈ T−10as

n→ ∞. This completes the proof.

FromTheorem 2.7and the same proof ofCorollary 2.3, we have the following corollary.

Corollary 2.8. LetHbe a real Hilbert space,Ωa nonempty closed convex subset ofH, andU:Ω→Ω

a continuous and pseudocontractive mapping with a fixed point. SetT I−U.Suppose that, for any givenxn∈Ω,βn>0, anden∈H, there existsxn∈Ωsuch that

xnen1βnxn−βnUxn, ∀n≥0. 2.32

Define the sequence{xn}iteratively as follows:

x0∈Ω,

xn1αnxn1−αnPΩxn−en, ∀n≥0, 2.33

where{αn} ⊂ 0,1withlimn→∞αn < 1,{βn} ⊂ 0,∞withlim_n→∞βn > 0, and{en} ⊂ Hwith _∞

n0en2<∞. Then{xn}converges weakly to a fixed pointpofU.

3. Applications

We can apply Theorems2.1and2.7to find a minimizer of a convex functionf. LetHbe a real Hilbert space andf:H →−∞,∞a proper convex lower semicontinuous function. Then the

subdifferential∂f offis defined as follows:

Theorem 3.1. LetH be a real Hilbert space andf : H → −∞,∞a proper convex lower semicon-tinuous function. Suppose that, for any xn ∈ H, βn > 0, and en ∈ H, there existsxn conforming to

xnen∈xnβn∂fxn, ∀n≥0, 3.2

where {βn} is a sequence in 0,∞with βn → ∞n → ∞and ∞n0en2 < ∞. Let {αn}be a sequence in0,1such thatαn →0 n→ ∞andn∞0αn ∞. For any fixedu∈H,let{xn}be the sequence generated by

u, x0∈H,

xnarg min z∈H

fz 1 2βn

_z−xn−en2_,

xn1 αnu1−αnxn−en, ∀n≥0.

3.3

If ∂f−1_{0/∅, then{x}

n}converges strongly to the minimizer offnearest tou.

Proof. Sincef : H → −∞,∞is a proper convex lower semicontinuous function, by2, the subdifferential∂f offis a maximal monotone operator. Noting that

xnarg min z∈H

fz 1 2βn

_z−xn−en2 _3.4

is equivalent to

0∈∂fxn _β1 n

xn−xn−en, 3.5

we have

xnen∈xnβn∂fxn, ∀n≥0. 3.6

Therefore, usingTheorem 2.1, we have the desired conclusion. This completes the proof.

Theorem 3.2. LetH be a real Hilbert space andf : H → −∞,∞a proper convex lower semicon-tinuous function. Suppose that, for any givenxn∈H,βn >0, anden ∈H, there existsxn ∈Hsuch that

xnen∈xnβn∂fxn, ∀n≥0, 3.7

where{βn}is a sequence in0,∞withlim_n→∞βn> 0and∞n0en2< ∞. Let{αn}be a sequence in0,1withlimn→∞αn<1and let{xn}be the sequence generated by

x0∈H,

xnarg min z∈H

fz 1 2βn

_z−xn−en2_,

xn1αnxn1−αnxn−en, ∀n≥0.

3.8

Proof. 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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