Báo cáo hóa học: "TOWARDS VISCOSITY APPROXIMATIONS OF HIERARCHICAL FIXED-POINT PROBLEMS" pptx

The limit attained by these curves is the solution of the general variational inequality, 0∈I − Qx ∞+NFixPx ∞, whereNFixPdenotes the normal cone to the set of fixed point of the original

HIERARCHICAL FIXED-POINT PROBLEMS

A MOUDAFI AND P.-E MAING ´E

Received 10 February 2006; Revised 14 September 2006; Accepted 18 September 2006

We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems The limit attained by these curves is the solution of the general variational inequality, 0∈(I − Q)x ∞+NFixP(x ∞), whereNFixPdenotes the normal cone to the set of fixed point of the original nonexpan-sive mappingP and Q a suitable nonexpansive mapping criterion The link with other

approximation schemes in this field is also made

Copyright © 2006 A Moudafi and P.-E Maing´e This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In nonlinear analysis, a common approach to solving a problem with multiple solutions

is to replace it by a family of perturbed problems admitting a unique solution, and to obtain a particular solution as the limit of these perturbed solutions when the perturba-tion vanishes Here, we will introduce a more general approach which consists in finding

a particular part of the solution set of a given fixed-point problem, that is, fixed points which solve a variational inequality “criterion.” More precisely, the main purpose of this note consists in building methods which hierarchically lead to fixed points of a nonex-pansive mappingP with the aid of a nonexpansive mapping Q, in the following sense:

findx∈Fix(P) such that



x − Q( x),x −  x

≥0 ∀x ∈Fix(P), (1.1) where Fix(P) = {x ∈ C; x = P(x)}is the set of fixed points ofP and C is a closed convex

subset of a real Hilbert spaceᏴ

It is not hard to check that solving (1.1) is equivalent to the fixed-point problem

findx ∈ C such that x=projFix(P) ◦Q( x), (1.2)

Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2006, Article ID 95453, Pages 1 10

DOI 10.1155/FPTA/2006/95453

where projFix(P)stands for the metric projection on the convex set Fix(P), and by using

the definition of the normal cone to Fix(P), that is,

NFixP:x −→

⎧

⎨

⎩



u ∈Ᏼ; (∀y ∈FixP) y − x,u 0

, ifx ∈FixP,

we easily obtain that (1.1) is equivalent to the variational inequality

0∈(I − Q) x + N FixP(x). (1.4)

It is worth mentioning that when the solution set,S, of (1.1) is a singleton (which is the case, e.g., whenQ is a contraction) the problem reduces to the viscosity fixed-point

solution introduced in [6] and further developed in [3,8]

Throughout,Ᏼ is a real Hilbert space, ·, denotes the associated scalar product, and stands for the corresponding norm To begin with, let us recall the following concepts are of common use in the context of convex and nonlinear analysis, see, for example, Rockafellar-Wets [7] An operator is said to be monotone if

u − v,x − y 0 wheneveru ∈ A(x), v ∈ A(y). (1.5)

It is said to be maximal monotone if, in addition, the graph, gphA := {(x, y) ∈Ᏼ×Ᏼ :

y ∈ A(x)}, is not properly contained in the graph of any other monotone operator It is well known that the single-valued operatorJ A

λ :=(I + λA) −1, called the resolvent ofA of

parameterλ, is a nonexpansive mapping which is everywhere defined Recall also that a

mappingP is nonexpansive if for all x, y, one has

and finally that, a sequenceA nis said to be graph convergent toA, if

lim sup

n→+∞ gphA n ⊂gphA ⊂lim inf

where the lower limit of the sequence{gphA n }is the subset defined by

lim inf

n→+∞ gphA n =(x, y) ∈Ᏼ× Ᏼ/ ∃x n,y n

−→(x, y),

x n,y n

∈gphA n n ∈ N ∗

(1.8) and the upper limit of the sequence{gphA n }is the closed subset defined by

lim sup

n→+∞ gphA n =(x, y)/∃n ν

ν∈N,∃x ν,y ν

−→(x, y),

x ν,y ν

∈gphA n ν ν ∈ N ∗

.

(1.9)

2 Convergence of approximating curves

2.1 A hierarchical fixed-point method Let P,Q : C → C be two nonexpansive

map-pings on a closed convex setC and assume that Fix(P) and the solution set S of (1.1) are nonempty

Given a real numbert ∈(0, 1), we define a mapping

P t Q:C −→ C by P Q t (x) = tQ(x) + (1 − t)P(x). (2.1)

For simplicity we will writeP tforP Q t It is clear thatP tis nonexpansive onC Throughout

the paper we will also assume that

Fix

P t

this is the case for instance ifQ is a contraction or under a compactness condition on C.

Now, let us state two preliminary results which will be needed in the sequel

Lemma 2.1 Let A be a maximal monotone operator, then (t −1A) graph converges to N A −1 (0)

as t → 0 provided that A −1(0)= ∅.

Proof It is well known, see [4, Proposition 2], that if A −1(0)= ∅, then for any x ∈

Ᏼ, J A

t −1(x) pointwise converges to proj A −1 (0)x Since J A

t −1(x) = J t −1

A

1 (x) and proj A −1 (0)x =

J N A −1 (0)

1 (x), thanks to the fact that the pointwise convergence of the resolvents is

equiv-alent to the graph convergence of the corresponding operators (see, e.g., [7, Theorem 12.32]), we easily deduce thatt −1A graph converges to N A −1 (0)ast →0  The following lemma contains stability and closure results of the class of maximal monotone operators under graph convergence, see, for example, [1] or [2]

Lemma 2.2 Let ( A t ) be a sequence of maximal monotone operators If B is a Lipschitz maximal monotone operator, then A t+B is maximal monotone Furthermore, if (A t ) graph converges to A, then A is maximal monotone and (A t+B) graph converges to A + B.

Now, we are in position to study the convergence of an arbitrary curve{x t }in Fix(P t)

ast →0

Proposition 2.3 Every weak-cluster point x ∞ of {x t } is solution of ( 1.1 ), or equivalently a fixed point of ( 1.2 ) or equivalently a solution of the variational inequality

find x ∞ ∈ C; 0∈(I − Q)x ∞+N S

x ∞

N S being the normal cone to the closed convex set S.

Proof {x t }is assumed to be bounded, so are{P(x t)}and{Q(x t)} As a result,

lim

t→0 x t − P

t→0t P

x t

− Q

Letx ∞be a weak cluster point of{x t }, say{x t ν }weakly converges tox ∞, we will show that

x ∞is a solution of the variational inequality (1.1)

x t ν ∈FixP t ν can be rewritten as

I − Q +1− t ν

t ν (I − P)

x t ν

Now, in the light ofLemma 2.2the family (I − Q + ((1 − t ν)/t ν)(I − P)) graph converges

to (I − Q) + NFixP, because ((1− t ν)/t ν)(I − P) graph converges to the normal cone of

(I − P) −1(0)=FixP according toLemma 2.1and the operatorI − Q is a Lipschitz

con-tinuous maximal monotone operator

By passing to the limit in the equality (2.5) ast ν →0, and by taking into account the fact that the graph of (I − Q) + NFixPis weakly-strongly closed, we obtain 0∈(I − Q)x ∞+

NFixP(x ∞) By using the definition of the normal cone, this amounts to writing x ∞ − Q(x ∞),x ∞ − x 0∀x ∈FixP, that is, x ∞solves the variational inequality (1.1)  Now, we would like to mention some interesting particular cases

Example 2.4 (monotone inclusions) By setting Q = I − γᏲ, where Ᏺ is κ-Lipschitzian

andη-strongly monotone with γ ∈(0, 2κ/η2), (1.1) reduces to

findx∈FixP such that

x −  x,Ᏺ( x) 

≥0 ∀ x ∈FixP, (2.6)

a variational inequality studied in Yamada [9]

On the other hand, if we setC = Ᏼ, P = J λ A, andQ = J λ Bwith A, B two maximal mono-tone operators andJ A

λ,J B

λ the corresponding resolvent mappings, the variational inequal-ity (1.1) reduces to

findx∈Ᏼ; 0∈I − J λ B

(x) + N A −1 (0)(x), (2.7) whereN A −1 (0) denotes the normal cone to,A −1(0)=FixJ A

λ, the set of zeroes ofA The

inclusion (2.7) can be rewritten as findx; 0 ∈ B λ(x) + N A −1 (0)(x), B λ:=(λI + B −1)−1being the Yosida approximate ofB.

Example 2.5 (convex programming) By setting

P =proxλϕ:=arg min



ϕ(y) + 1

2 

ϕ a lower semicontinuous convex function and Q = I − γ∇ψ, ψ a convex function such

that∇ψ is κ-strongly monotone and η-Lipschitzian (which is equivalent to the fact that

∇ψ is η −1 cocoercive) with γ ∈(0, 2/η), and thanks to the fact that Fix(prox λϕ)=

(∂ϕ) −1(0)=arg minϕ, (1.1) reduces to the hierarchical minimization problem:

min

On the other hand, if we set in (2.7), A = ∂ϕ and B = ∂ψ, subdifferential operators of

lower semicontinuous convex functionsϕ and ψ, the inclusion (1.1) reduces to the fol-lowing hierarchical minimization problem: minx∈arg minϕ ψ λ(x), where ψ λ(x)=infy {ψ(y)+

(1/2λ) x − y 2}, is the Moreau-Yosida approximate ofψ.

Example 2.6 (minimization on a fixed-point set) By setting Q = I − γ∇ϕ, ϕ a convex

function;∇ϕ is κ-strongly monotone and η-Lipschitzian (thus η −1cocoercive) withγ ∈

(0, 2/η], (1.1) reduces to minx∈FixP ϕ(x), a problem studied in Yamada [9] On the other hand, whenP is a nonexpansive mapping and Q =I − γ(A − γ f ), A being a linear bounded

γ-strongly monotone operator, f a given α-contraction, and γ > 0 withγ ∈(0,/ A +γ),

(1.1) reduces to the problem of minimizing a quadratic function over the set of fixed points of a nonexpansive mapping studied in Marino and Xu [5], namely,

 (A − γ f )x,x − x

which is the optimality condition for the minimization problem

min

x∈FixP

1

whereh is a potential function for γ f , that is, h (x) = γ f (x), for x ∈Ᏼ

Fort ∈(0, 1) let{x t }be a fixed point ofP t Our interest now is to show that any net

{x t } obtained in this way is an approximate fixed-point net for P.

Proposition 2.7 Assume that Fix Q = ∅ Then, for any t ∈ (0, 1),

Qx t − Px t 2 inf

Moreover, the net {x t } is an approximate fixed-point net for the mapping P, that is,

lim

Proof Consider any p ∈Fix(P) and q ∈Fix(Q) and let p t :=ProjΔt(p) and q t :=

ProjΔt(q) be the metric projections of p and q onto Δ t, respectively, where the closed convex setΔtis defined byΔt:= {λ(Px t − Qx t) +x t;λ ∈ R}

Now, suppose that conditionPx t = Qx tis satisfied It is then immediate thatx t = Px t

andx t = Qx tprovided thatt ∈(0, 1) Seta t:=(1/2)(x t+Px t) andb t:=(1/2)(x t+Qx t), it

is then easily checked that



Qx t − b t,q − b t



=1

4 x t − q 2− Qx t − q 2

,



Px t − a t,p − a t

=1

4 x t − p 2− Px t − p 2

.

(2.14)

Thanks to the nonexpansiveness ofQ and P, we deduce that



Qx t − b t,q − b t



≥0, 

Px t − a t,p − a t



Furthermore, it is obvious that there exist two real numbersλ tandμ tsuch thatq t = b t+

λ t(Qx t − b t) and p t = a t+μ t(Px t − a t) In the light of the metric projection properties,

we can write

0=q t − q,Qx t − b t



=b t − q,Qx t − b t

 +λ t Qx t − b t 2, (2.16) hence

λ t =



q − b t,Qx t − b t



Qx t − b t 2

In a similar way, we get

0=p t − p,Px t − a t

=a t − p,Px t − a t

+μ t Px t − a t 2, (2.18) and we obtain

μ t =



p − a t,Px t − a t

Note also thatb t − a t =(1/2)(Qx t − Px t) and, according to the fact thatx t ∈FixP t, that

x t − Px t = t(Qx t − Px t) and x t − Qx t =(1− t)(Px t − Qx t) Hence, we get x t − Px t =

2t(b t − a t) andx t − Qx t = −2(1− t)(b t − a t) Moreover, we immediately haveQx t − b t =

(1/2)(Qx t − x t) andPx t − a t =(1/2)(Px t − x t), so that

q t − b t = λ t

Qx t − b t

= λ t(1− t)

b t − a t

 ,

a t − p t = −μPx t − a t

= μt

b t − a t

Consequently, we obtain

q t − p t =q t − b t

 +

b t − a t

 +

a t − p t

=λ t(1− t) + 1 + tμ t

b t − a t

=1

2

λ t(1− t) + 1 + tμ t

Qx t − Px t

.

(2.21)

Thus

q t − p t 1

2

λ t(1− t) + 1 + tμ t Qx t − Px t (2.22) Finally, by nonexpansiveness of the projection mapping, we have

q t − p t ProjΔt(p) −ProjΔt(q) p − q , (2.23) which by (0.1) leads to

2

λ t(1− t) + 1 + tμ t Qx t − Px t 1

2 Qx t − Px t (2.24)

By taking the infimum overp in Fix P and q in Fix Q, we obtain the desired formula The

latter combined with the fact thatx t − Px t = t(Qx t − Px t) leads to the fact that{x t }is an

2.2 Coupling the hierarchical fixed-point method with viscosity approximation To

begin with, we will assume that

S ⊂ s −lim inf

t→0 FixP t, s standing for the strong topology, (2.25) which is satisfied, for example, whenQ is a contraction.

Now, given a real numbers ∈(0, 1) and a contraction f : C → C Define another

map-ping

P t,s f(x) = s f (x) + (1 − s)P t(x), (2.26)

for simplicity we will writeP t,sforP t,s f

It is not hard to see thatP t,sis a contraction onC Indeed, for x, y ∈ C, we have

P t,s(x) − P t,s(y) s

f (x) − f (y)

+ (1− s)

P t(x) − P t(y)

≤ αs x − y + (1− s) x − y

=1− s(1 − α)

x − y

(2.27)

Letx t,sbe the unique solution of the fixed point ofP t,s, that is,x t,sis the unique solution

of the fixed-point equation

x t,s = s f

x t,s + (1− s)P t

x t,s

The purpose of this section is to study the convergence of{x t,s }ast,s →0

Let us first recall the following diagonal lemma (see, e.g., [1])

Lemma 2.8 Let ( X,d) be a metric space and (a n,m ) a “double” sequence in X satisfying

∀n ∈ N lim

m→+∞ a n,m = a n, lim

Then, there exists a nondecreasing mapping k : N → N which to m associates k(m) and such that lim m→+∞ a k(m),m = a.

Now, we are able to give our main result

Theorem 2.9 The net {x t,s } strongly converges, as s → 0, to x t , where x t satisfies x t =

projFixP t ◦ f (x t ) or equivalently x t is the unique solution of the quasivariational inequality

0∈(I − f )x t+NFixP t

x t

Moreover, the net {x t } in turn weakly converges, as t → 0, to the unique solution x ∞ of the fixed-point equation x ∞ =projS ◦ f (x ∞ ) or equivalently x ∞ ∈ S is the unique solution of the variational inequality

0∈(I − f )x ∞+N S

x ∞

Furthermore, if dim Ᏼ < ∞, then there exists a subnet {x s ν s n } of {x t n,s n } which converges

to x ∞

Proof We first show that {x t,s }is bounded Indeed takex t ∈FixP tto derive

x t,s −  x t s f (x t,s)−  x t + (1− s) P t(x t,s)− P t(x t) . (2.32)

It follows

x t,s −  x t f

x t,s

−  x t f

x t,s

− f



x t + f



x t

−  x t

≤ αs x t,s −  x t + f



x t

Hence

x t,s x t +1

α f



x t

This ensures that{x t,s }is bounded, since { x t }and{ f ( xt)}are bounded Now, we will show that{x t,s n }contains a subnet converging tox t, wherex t ∈FixP tis the unique solu-tion of the quasivariasolu-tional inequality

0∈(I − f )x t+NFixP t

x t

Since{x t,s n }is bounded, it admits a weak cluster pointx t, that is, there exists a subnet

{x t,s ν }of{x t,s n }which weakly converges tox t On the other hand,

(I − f ) +1− s

s

I − P t

 graph converges to (I − f ) + NFixP t ass −→0. (2.36)

By passing to the limit in the following equality:

 (I − f ) +

1− s ν

s ν P t



x t,s ν

we obtain thatx tis the unique solution of the quasivariational inequality

0∈(I − f ) + NFixP t

x t

or equivalentlyx tsatisfiesx t =projFixP t ◦ f (x t) It should be noticed that in contrast with the first section{x t }is unique (a select approximating curve in FixP t) Hence the whole net{x t,s n }weakly converges tox t In fact the convergence is strong Indeed, since

x t,s − x t = s

f

x t,s

− x t + (1− s)

P t

x t,s

− x t

we successively have

x t,s − x t 2=(1− s)

P t

x t,s

− x t,x t,s − x t

 +s

f

x t,s

− x t,x t,s − x t



≤(1− s) x t,s − x t 2+s

f

x t,s

− x t,x t,s − x t

.

(2.40)

Hence

x t,s − x t

2

≤f

x t,s

− x t,x t,s − x t



=f

x t,s

− f

x t ,x t,s − x t

+

f

x t

− x t,x t,s − x t

≤ α x t,s −  x t 2+

f

x t

− x t,x t,s − x t

.

(2.41)

This implies that

x t,s n − x t 2≤ 1

1− α



f

x t

− x t,x t,s n − x t



But{x t,s n }weakly converges tox t, by passing to the limit in (2.31), it follows that{x t,s n }

strongly converges tox t

According to the first section,{x t }is bounded and w −lim supt→0FixP t ⊂ S which

together with (2.25) is nothing but (FixP t) converges toS in the sense of Mosco, which

in turn amounts to saying, thanks to [7, Proposition 7.4(f)], that the indicator function (δFixP t) Mosco converges toδ S In the light of Attouch’s theorem (see [7, Theorem 12.35]), this implies the graph convergence of (NFixP t) toN S Now, by taking a subnet{x t ν }which weakly converges to somex ∞and by passing to the limit in

0∈(I − f ) + NFixP tν

x t ν

we obtain

0∈(I − f ) + N S

x ∞

becauseI − f is a Lipschitz continuous maximal monotone operator which ensures, by

virtue ofLemma 2.2, the fact that the graph convergence of (NFixP t) toN Simplies that

of ((I − f ) + NFixP t) to (I − f ) + N S and also that the graph of the operator (I − f ) +

N Sis weakly-strongly closed The weak cluster pointx ∞being unique, we infer that the whole net{x t }weakly converges tox ∞which solves (2.28) We conclude by applying the

Conclusion The convergence properties of new types of approximating curves for fixed

point problems are investigated relying on the graph convergence The limits attained by these curves are solutions of variational or quasivariational inequalities involving fixed-point sets Approximating curves are also relevant to numerical methods since under-standing their properties is central in the analysis of parent continuous and discrete dy-namical systems, so we envisage to study the related iterative schemes in a forthcoming paper

Acknowledgment

The authors thank the anonymous referees for their careful reading of the paper

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Analysis and Applications 241 (2000), no 1, 46–55.

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A Moudafi: GRIMAAG, D´epartement Scientifique interfacultaires, Universit´e Antilles Guyane,

97200 Schelcher, Martinique, France

E-mail address:abdellatif.moudafi@martinique.univ-ag.fr

P.-E Maing´e: GRIMAAG, D´epartement Scientifique interfacultaires, Universit´e Antilles Guyane,

97200 Schelcher, Martinique, France

E-mail address:paul-emile.mainge@martinique.univ-ag.fr

of the fixed-point equation

x t,s... closure results of the class of maximal monotone operators under graph convergence, see, for example, [1] or [2]

Lemma 2.2 Let ( A t ) be a sequence of maximal monotone... x ∞ of {x t } is solution of ( 1.1 ), or equivalently a fixed point of ( 1.2 ) or equivalently a solution of the variational inequality

find x ∞