FOR MAXIMAL MONOTONE OPERATORS IN A BANACH SPACE
FUMIAKI KOHSAKA AND WATARU TAKAHASHI Received 12 March 2003
We first introduce a modified proximal point algorithm for maximal monotone opera- tors in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.
1. Introduction
LetEbe a real Banach space and letT⊂E×E∗be a maximal monotone operator. Then we study the problem of finding a pointv∈Esatisfying
0∈Tv. (1.1)
Such a problem is connected with theconvex minimization problem. In fact, if f :E→ (−∞,∞] is a proper lower semicontinuous convex function, then Rockafellar’s theorem [14,15] ensures that the subdifferential mapping∂ f ⊂E×E∗of f is a maximal mono- tone operator. In this case, the equation 0∈∂ f(v) is equivalent to f(v)=minx∈Ef(x).
In 1976, Rockafellar [17] proved the following weak convergence theorem.
Theorem1.1 (Rockafellar [17]). LetHbe a Hilbert space and letT⊂H×Hbe a maximal monotone operator. LetIbe the identity mapping and letJr=(I+rT)−1for allr >0. Define a sequence{xn}as follows:x1=x∈Hand
xn+1=Jrnxn (n=1, 2,...), (1.2) where{rn} ⊂(0,∞)satisfieslim infn→∞rn>0. IfT−10= ∅, then the sequence{xn}con- verges weakly to an element ofT−10.
This is called theproximal point algorithm, which was first introduced by Martinet [11].
IfT=∂ f, where f :H→(−∞,∞] is a proper lower semicontinuous convex function,
Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:3 (2004) 239–249 2000 Mathematics Subject Classification: 47H05, 47J25 URL:http://dx.doi.org/10.1155/S1085337504309036
then (1.2) is reduced to
xn+1=arg min
y∈H
f(y) + 1 2rn
xn−y2 (n=1, 2,...). (1.3)
Later, many researchers studied the convergence of the proximal point algorithm in a Hilbert space; see Br´ezis and Lions [4], Lions [10], Passty [12], G¨uler [7], Solodov and Svaiter [19] and the references mentioned there. In particular, Kamimura and Takahashi [8] proved the following strong convergence theorem.
Theorem1.2 (Kamimura and Takahashi [8]). LetHbe a Hilbert space and letT⊂H×H be a maximal monotone operator. LetJr=(I+rT)−1for allr >0and let{xn}be a sequence defined as follows:x1=x∈Hand
xn+1=αnx+1−αn
Jrnxn (n=1, 2,...), (1.4) where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfylimn→∞αn=0,∞n=1αn= ∞, andlimn→∞rn=
∞. IfT−10= ∅, then the sequence{xn}converges strongly toPT−10(x), wherePT−10is the metric projection fromHontoT−10.
Recently, using the hybrid method in mathematical programming, Kamimura and Takahashi [9] obtained a strong convergence theorem for maximal monotone operators in a Banach space, which extended the result of Solodov and Svaiter [19] in a Hilbert space. On the other hand, Censor and Reich [6] introduced a convex combination which is based on Bregman distance and studied some iterative schemes for finding a common asymptotic fixed point of a family of operators in finite-dimensional spaces.
In this paper, motivated by Censor and Reich [6], we introduce the following itera- tive sequence for a maximal monotone operatorT⊂E×E∗in a smooth and uniformly convex Banach space:x1=x∈Eand
xn+1=J−1αnJx+1−αn JJrnxn
(n=1, 2,...), (1.5) where{αn} ⊂[0, 1],{rn} ⊂(0,∞),J is the duality mapping from EintoE∗, andJr= (J+rT)−1Jfor allr >0. Then we extend Kamimura-Takahashi’s theorem to the Banach space (Theorem 3.3). It should be noted that we do not assume the weak sequential con- tinuity of the duality mapping [1,5,13]. Finally, we apply Theorem 3.3to the convex minimization problem and the variational inequality problem.
2. Preliminaries
LetEbe a (real) Banach space with norm · and letE∗denote the Banach space of all continuous linear functionals onE. For allx∈Eandx∗∈E∗, we denotex∗(x) by x,x∗. We denote by Rand Nthe set of all real numbers and the set of all positive integers, respectively. Theduality mappingJfromEintoE∗is defined by
J(x)=
x∗∈E∗:x,x∗ = x2=x∗2 (2.1)
for allx∈E. IfEis a Hilbert space, thenJ=I, whereIis the identity mapping. We some- times identify a set-valued mappingA:E→2E∗ with its graph G(A)= {(x,x∗) :x∗∈ Ax}. An operatorT⊂E×E∗with domainD(T)= {x∈E:Tx= ∅}and rangeR(T)= {Tx:x∈D(T)}is said to bemonotoneif x−y,x∗−y∗ ≥0 for all (x,x∗), (y,y∗)∈ T. We denote the set{x∈E: 0∈Tx}byT−10. A monotone operatorT⊂E×E∗is said to bemaximalif its graph is not properly contained in the graph of any other monotone operator. IfT⊂E×E∗is maximal monotone, then the solution setT−10 is closed and convex. A proper function f :E→(−∞,∞] (which means that f is not identically∞) is said to beconvexif
fαx+ (1−α)y≤α f(x) + (1−α)f(y) (2.2) for allx,y∈Eandα∈(0, 1). The function f is also said to belower semicontinuousif the set{x∈E:f(x)≤r}is closed inEfor allr∈R. For a proper lower semicontinuous convex function f :E→(−∞,∞], thesubdifferential∂ f of f is defined by
∂ f(x)=
x∗∈E∗:f(x) +y−x,x∗ ≤ f(y)∀y∈E (2.3) for all x∈E. It is easy to verify that 0∈∂ f(v) if and only if f(v)=minx∈Ef(x). It is known that the subdifferential of the functionf defined byf(x)= x2/2 for allx∈Eis the duality mappingJ. The following theorem is also well known (see Takahashi [21] for details).
Theorem2.1. LetEbe a Banach space, let f :E→(−∞,∞]be a proper lower semicontin- uous convex function, and letg:E→Rbe a continuous convex function. Then
∂(f+g)(x)=∂ f(x) +∂g(x) (2.4) for allx∈E.
A Banach spaceEis said to bestrictly convexif x = y =1, x=y=⇒
x+y 2
<1. (2.5)
Also,Eis said to beuniformly convexif for eachε∈(0, 2], there existsδ >0 such that x = y =1, x−y ≥ε=⇒
x+y 2
≤1−δ. (2.6)
It is also said to besmoothif the limit limt→0
x+ty − x
t (2.7)
exists for allx,y∈ {z∈E:z =1}. We know the following (see Takahashi [20] for de- tails):
(1) ifEis smooth, thenJis single-valued;
(2) ifEis strictly convex, thenJis one-to-one and x−y,x∗−y∗>0 holds for all (x,x∗), (y,y∗)∈Jwithx=y;
(3) ifEis reflexive, thenJis surjective;
(4) ifEis uniformly convex, then it is reflexive;
(5) ifE∗is uniformly convex, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
LetEbe a smooth Banach space. We use the following function studied in Alber [1], Kamimura and Takahashi [9], and Reich [13]:
φ(x,y)= x2−2 x,J y+y2 (2.8) for allx,y∈E. It is obvious from the definition ofφthat (x − y)2≤φ(x,y) for all x,y∈E. We also know that
φ(x,y)=φ(x,z) +φ(z,y) + 2 x−z,Jz−J y (2.9) for allx,y,z∈E. The following lemma was also proved in [9].
Lemma2.2 (Kamimura-Takahashi [9]). LetEbe a smooth and uniformly convex Banach space and let{xn}and{yn}be sequences inEsuch that either{xn}or{yn}is bounded. If limn→∞φ(xn,yn)=0, thenlimn→∞xn−yn =0.
LetEbe a strictly convex, smooth, and reflexive Banach space, and letT⊂E×E∗be a monotone operator. ThenT is maximal if and only ifR(J+rT)=E∗for allr >0 (see Barbu [2] and Takahashi [21]). IfT⊂E×E∗is a maximal monotone operator, then for eachr >0 andx∈E, there corresponds a unique elementxr∈D(T) satisfying
J(x)∈Jxr
+rTxr. (2.10)
We define theresolventofTbyJrx=xr. In other words,Jr=(J+rT)−1Jfor allr >0. The resolventJris a single-valued mapping fromEintoD(T). IfEis a Hilbert space, thenJris nonexpansive, that is,Jrx−Jry ≤ x−yfor allx,y∈E(see Takahashi [20]). It is easy to show thatT−10=F(Jr) for allr >0, whereF(Jr) denotes the set of all fixed points of Jr. We can also define, for eachr >0, theYosida approximationofTbyAr=(J−JJr)/r.
We know that (Jrx,Arx)∈Tfor allr >0 andx∈E. LetCbe a nonempty closed convex subset of the spaceE. By Alber [1] or Kamimura and Takahashi [9], for eachx∈E, there corresponds a unique elementx0∈C(denoted byPC(x)) such that
φx0,x=min
y∈Cφ(y,x). (2.11)
The mappingPCis called thegeneralized projectionfromEontoC. IfEis a Hilbert space, thenPCis coincident with the metric projection fromEontoC. We also know the fol- lowing lemma.
Lemma2.3 ([1], see also [9]). LetEbe a smooth Banach space, letCbe a nonempty closed convex subset ofE, and letx∈Eandx0∈C. Then the following are equivalent:
(1)φ(x0,x)=miny∈Cφ(y,x);
(2) y−x0,Jx−Jx0 ≤0for ally∈C.
3. Strong convergence theorem
The resolvents of maximal monotone operators have the following property, which was proved in the case of the resolvents of normality operators in Kamimura and Takahashi [9].
Lemma3.1. LetEbe a strictly convex, smooth, and reflexive Banach space, letT⊂E×E∗ be a maximal monotone operator withT−10= ∅, and letJr=(J+rT)−1J for eachr >0.
Then
φu,Jrx+φJrx,x≤φ(u,x) (3.1) for allr >0,u∈T−10, andx∈E.
Proof. Letr >0,u∈T−10, andx∈Ebe given. By the monotonicity ofT, we have φ(u,x)=φu,Jrx+φJrx,x+ 2u−Jrx,JJrx−Jx
=φu,Jrx+φJrx,x+ 2ru−Jrx,−Arx
≥φu,Jrx+φJrx,x.
(3.2) LetEbe a strictly convex, smooth, and reflexive Banach space, and letJbe the duality mapping fromEintoE∗. ThenJ−1is also single-valued, one-to-one, and surjective, and it is the duality mapping fromE∗intoE. We make use of the following mappingVstudied in Alber [1]:
Vx,x∗= x2−2x,x∗ +x∗2 (3.3) for allx∈Eand x∗∈E∗. In other words, V(x,x∗)=φ(x,J−1(x∗)) for allx∈Eand x∗∈E∗. For eachx∈E, the mappinggdefined byg(x∗)=V(x,x∗) for allx∗∈E∗is a continuous and convex function fromE∗intoR. We can prove the following lemma.
Lemma3.2. LetEbe a strictly convex, smooth, and reflexive Banach space, and letV be as in (3.3). Then
Vx,x∗+ 2J−1x∗−x,y∗ ≤Vx,x∗+y∗ (3.4) for allx∈Eandx∗,y∗∈E∗.
Proof. Letx∈Ebe given. Defineg(x∗)=V(x,x∗) and f(x∗)= x∗2 for allx∗∈E∗. SinceJ−1is the duality mapping fromE∗intoE, we have
∂gx∗=∂−2 x,·+fx∗= −2x+ 2J−1x∗ (3.5) for allx∗∈E∗. Hence, we have
gx∗+ 2J−1x∗−x,y∗ ≤gx∗+y∗, (3.6)
that is,
Vx,x∗+ 2J−1x∗−x,y∗ ≤Vx,x∗+y∗ (3.7)
for allx∗,y∗∈E∗.
Now we can prove the following strong convergence theorem, which is a generalization of Kamimura-Takahashi’s theorem (Theorem 1.2).
Theorem3.3. LetEbe a smooth and uniformly convex Banach space and letT⊂E×E∗be a maximal monotone operator. LetJr=(J+rT)−1Jfor allr >0and let{xn}be a sequence defined as follows:x1=x∈Eand
xn+1=J−1αnJx+1−αn JJrnxn
(n=1, 2,...), (3.8) where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfylimn→∞αn=0,∞n=1αn= ∞, andlimn→∞rn=
∞. IfT−10= ∅, then the sequence{xn}converges strongly toPT−10(x), wherePT−10is the generalized projection fromEontoT−10.
Proof. Putyn=Jrnxnfor alln∈N. We denote the mappingPT−10byP. We first prove that {xn}is bounded. FromLemma 3.1, we have
φPx,xn+1
=φPx,J−1αnJx+1−αn J yn
=VPx,αnJx+1−αn J yn
≤αnV(Px,Jx) +1−αn
VPx,J yn
=αnφ(Px,x) +1−αn
φPx,Jrnxn
≤αnφ(Px,x) +1−αn
φPx,xn
(3.9)
for all n∈N. Hence, by induction, we haveφ(Px,xn)≤φ(Px,x) for alln∈N. Since (u − v)2≤φ(u,v) for allu,v∈E, the sequence{xn}is bounded. Sinceφ(Px,yn)= φ(Px,Jrnxn)≤φ(Px,xn) for alln∈N,{yn}is also bounded. We next prove
lim sup
n→∞
xn−Px,Jx−JPx ≤0. (3.10)
Putzn=xn+1for alln∈N. Since{zn}is bounded, we have a subsequence{zni}of{zn} such that
ilim→∞
zni−Px,Jx−JPx =lim sup
n→∞
zn−Px,Jx−JPx (3.11)
and{zni}converges weakly to somev∈E. From the definition of{xn}, we have Jzn−J yn=αn
Jx−J yn
(3.12) for alln∈N. Since{yn}is bounded andαn→0 asn→ ∞, we have
nlim→∞Jzn−J yn=nlim
→∞αnJx−J yn=0. (3.13)
SinceEis uniformly convex,E∗is uniformly smooth, and hence the duality mappingJ−1 fromE∗intoEis uniformly norm-to-norm continuous on each bounded subset ofE∗. Therefore, we obtain from (3.13) that
nlim→∞zn−yn=nlim
→∞J−1Jzn
−J−1J yn=0. (3.14) This implies thatynivasi→ ∞, whereimplies the weak convergence. On the other hand, fromrn→ ∞asn→ ∞, we have
nlim→∞Arnxn=nlim
→∞
1 rn
Jxn−J yn=0. (3.15)
If (z,z∗)∈T, then it holds from the monotonicity ofTthat
z−yni,z∗−Arnixni ≥0 (3.16)
for alli∈N. Lettingi→ ∞, we get z−v,z∗ ≥0. Then, the maximality ofT implies v∈T−10. ApplyingLemma 2.3, we obtain
lim sup
n→∞
zn−Px,Jx−JPx =lim
i→∞
zni−Px,Jx−JPx = v−Px,Jx−JPx ≤0. (3.17)
Finally, we prove thatxn→Px as n→ ∞. Let ε >0 be given. From (3.10), we have m∈Nsuch that
xn−Px,Jx−JPx ≤ε (3.18)
for alln≥m. Ifn≥m, then it holds from (3.18) and Lemmas3.1and3.2that φPx,xn+1
=VPx,αnJx+1−αn J yn
≤VPx,αnJx+1−αnJ yn−αn(Jx−JPx)
−2J−1αnJx+1−αn J yn
−Px,−αn(Jx−JPx)
=VPx,1−αnJ yn+αnJPx+ 2xn+1−Px,αn(Jx−JPx)
≤ 1−αn
VPx,J yn
+αnV(Px,JPx) + 2αn
xn+1−Px,Jx−JPx
≤ 1−αn
φPx,yn
+αnφ(Px,Px) + 2αnε
= 1−αn
φPx,Jrnxn + 2αnε
≤ 1−αn
φPx,xn + 2αnε
=2ε1−
1−αn
+1−αn
φPx,xn .
(3.19)
Therefore, we have φPx,xn+1
≤2ε1−
1−αn
+1−αn
2ε1− 1−αn−1
+1−αn−1
φPx,xn−1
=2ε1−
1−αn 1−αn−1
+1−αn 1−αn−1
φPx,xn−1
≤ ··· ≤2ε
1− n i=m
1−αi +
n i=m
1−αi
φPx,xm
(3.20)
for alln≥m. Since∞i=1αi= ∞, we have∞i=m(1−αi)=0 (see Takahashi [21]). Hence, we have
lim sup
n→∞ φPx,xn
=lim sup
l→∞ φPx,xm+l+1
≤lim sup
l→∞
2ε
1−
m+l
i=m
1−αi +
m+l
i=m
1−αi
φPx,xm
=2ε. (3.21) This implies lim supn→∞φ(Px,xn)≤0 and hence we get
nlim→∞φPx,xn
=0. (3.22)
ApplyingLemma 2.2, we obtain
nlim→∞Px−xn=0. (3.23)
Therefore,{xn}converges strongly toPT−10(x).
4. Applications
In this section, we first study the problem of finding a minimizer of a proper lower semi- continuous convex function in a Banach space.
Theorem 4.1. Let Ebe a smooth and uniformly convex Banach space and let f :E→ (−∞,∞]be a proper lower semicontinuous convex function such that(∂ f)−1(0)= ∅. Let {xn}be a sequence defined as follows:x1=x∈Eand
yn=arg min
y∈E
f(y) + 1
2rny2− 1 rn
y,Jxn (n=1, 2,...),
xn+1=J−1αnJx+1−αnJ yn (n=1, 2,...),
(4.1)
where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfylimn→∞αn=0,∞n=1αn= ∞, andlimn→∞rn=
∞. Then the sequence{xn}converges strongly toP(∂ f)−1(0)(x).
Proof. By Rockafellar’s theorem [14, 15], the subdifferential mapping∂ f ⊂E×E∗ is maximal monotone (see also Borwein [3], Simons [18], or Takahashi [21]). Fixr >0, z∈E, and letJrbe the resolvent of∂ f. Then we have
Jz∈JJrz+r∂ fJrz (4.2)
and hence,
0∈∂ fJrz+1
rJJrz−1
rJz=∂f+ 1
2r · 2− 1
rJzJrz. (4.3) Thus, we have
Jrz=arg min
y∈E
f(y) + 1
2ry2−1
r y,Jz
. (4.4)
Therefore, yn=Jrnxn for all n∈N. Using Theorem 3.3, {xn} converges strongly to
P(∂ f)−1(0)(x).
We next study the problem of finding a solution of a variational inequality. LetCbe a nonempty closed convex subset of a Banach spaceEand letA:C→E∗be a single-valued monotone operator which ishemicontinuous,that is, continuous along each line segment inCwith respect to the weak∗topology ofE∗. Then a pointv∈Cis said to be a solution of thevariational inequalityforAif
y−v,Av ≥0 (4.5)
holds for ally∈C. We denote byVI(C,A) the set of all solutions of the variational in- equality forA. We also denote byNC(x) thenormal coneforCat a pointx∈C, that is,
NC(x)=
x∗∈E∗:y−x,x∗ ≤0∀y∈C. (4.6) Theorem4.2. LetCbe a nonempty closed convex subset of a smooth and uniformly con- vex Banach spaceEand letA:C→E∗be a single-valued, monotone, and hemicontinuous operator such thatVI(C,A)= ∅. Let{xn}be a sequence defined as follows:x1=x∈Eand
yn=VIC,A+ 1 rn
J−Jxn
(n=1, 2,...), xn+1=J−1αnJx+1−αn
J yn
(n=1, 2,...),
(4.7)
where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfylimn→∞αn=0,∞n=1αn= ∞, andlimn→∞rn=
∞. Then, the sequence{xn}converges strongly toPVI(C,A)(x).
Proof. By Rockafellar’s theorem [16], the mappingT⊂E×E∗defined by Tx=
A(x) +NC(x), ifx∈C,
∅, otherwise, (4.8)
is maximal monotone andT−10=VI(C,A). Fixr >0,z∈E, and letJr be the resolvent ofT. Then we have
Jz∈JJrz+rTJrz (4.9)
and hence,
−AJrz+1 r
Jz−JJrz∈NCJrz. (4.10) Thus, we have
y−Jrz,AJrz+1 r
JJrz−Jz≥0 (4.11)
for ally∈C, that is,
Jrz=VIC,A+1 r
J−Jz. (4.12)
Therefore, yn=Jrnxn for all n∈N. Using Theorem 3.3, {xn} converges strongly to
PVI(C,A)(x).
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Fumiaki Kohsaka: Department of Mathematical and Computing Sciences, Tokyo Institute of Tech- nology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan
E-mail address:[email protected]
Wataru Takahashi: Department of Mathematical and Computing Sciences, Tokyo Institute of Tech- nology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan
E-mail address:[email protected]
