# Algorithmic Approach to a Minimization Problem

Authors: Yonghong Yao; Shin Min Kang; Yeong-Cheng Liou; Zhitao Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We first construct an implicit algorithm for solving the minimization problemminx∈Ω∥x∥
, where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. We prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping A:C→H is called α-inverse-stronglymonotone if there exists a constant α>0 such that
(1.1)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈C.
A mapping S:C→C is said to be nonexpansive if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C. Denote the set of fixed points of S by F(S).LetB:C→H be a nonlinear mapping and F:C×C→R be a bifunction. Now we concern the following equilibrium problem is to find z∈C such that
(1.2)F(z,y)+〈Bz,y-z〉≥0,∀y∈C.
The solution set of (1.2) is denoted by EP(F,B). If B=0, then (1.2) reduces to the following equilibrium problem of finding z∈C such that
(1.3)F(z,y)≥0,∀y∈C.
The solution set of (1.3) is denoted by EP(F). If F=0, then (1.2) reduces to the variational inequality problem of finding z∈C such that
(1.4)〈Bz,y-z〉≥0,∀y∈C.
The solution set of variational inequality (1.4) is denoted by VI(C,B).Equilibrium problems which were introduced by Blum and Oettli [1] in 1994 have had a great impact and influence in pure and applied sciences. It has been shown that the equilibrium problems theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Equilibrium problems include variational inequalities, fixed point, Nash equilibrium, and game theory as special cases. The equilibrium problems and the variational inequality problems have been investigated by many authors. Please see [2–35] and the references therein. The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others.On the other hand, we also notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subsetC of a Hilbert space H1 and a bounded linear operator R:H1→H2, where H2 is another Hilbert space. The C-constrained pseudoinverse of R, RC†, is then defined as the minimum-norm solution of the constrained minimization problem:
(1.5)RC†(b):=argmin⁡x∈C‖Rx-b‖,
which is equivalent to the fixed point problem:
(1.6)x=PC(x-λR*(Rx-b)),
where PC is the metric projection from H1 onto C, R* is the adjoint of R, λ>0 is a constant, and b∈H2 is such that PR(C)¯(b)∈R(C).It is therefore an interesting problem to invent some algorithms that can generate schemes which converge strongly to the minimum-norm solution of a given problem.In this paper, we focus on the following minimization problem: findx*∈Ω such that
(1.7)x*=argmin⁡x∈Ω‖x‖,
where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. We will suggest and analyze two very simple algorithms for solving the above minimization problem.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper, we assume that a bifunction F:C×C→R satisfies the following conditions:(H1)
F(x,x)=0 for all x∈C;(H2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C;(H3)
for eachx,y,z∈C, limt↓0F(tz+(1-t)x,y)≤F(x,y);(H4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.The metric (or nearest point) projection from H onto C is the mapping PC:H→C which assigns to each point x∈C the unique point PCx∈C satisfying the property:
(2.1)‖x-PCx‖=inf⁡y∈C‖x-y‖=:d(x,C).
It is well known that PC is a nonexpansive mapping and satisfies
(2.2)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,∀x,y∈H.
We need the following well-known lemmas for proving our main results.Lemma 2.1 (see [13]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C→R be a bifunction which satisfies conditions (H1)–(H4). Let μ>0 and x∈C. Then there exists z∈C such that
(2.3)F(z,y)+1μ〈y-z,z-x〉≥0,∀y∈C.
Further, if Tμ(x)={z∈C:F(z,y)+(1/μ)〈y-z,z-x〉≥0,forally∈C}, then the following hold:(a)
Tμ is single-valued and Tμ is firmly nonexpansive, that is, for any x,y∈C, ∥Tμx-Tμy∥2≤〈Tμx-Tμy,x-y〉;(b)
EP(F) is closed and convex and EP(F)=F(Tμ).Lemma 2.22.2 (see [27]).
Let{xn} and {vn} be bounded sequences in a Banach space X  and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)vn+βnxn for all n≥0 and limsupn→∞(∥vn+1-vn∥-∥xn+1-xn∥)≤0. Then limn→∞∥vn-xn∥=0.Lemma 2.3 (see [29]).
LetC be a closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping. Then the mapping I-S is demiclosed. That is, if {xn} is a sequence in C such that xn→x* weakly and (I-S)xn→y strongly, then (I-S)x*=y.Lemma 2.4 (see [29]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.4)an+1≤(1-γn)an+δnγn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that(a)
∑n=1∞‍γn=∞;(b)
limsupn→∞δn≤0 or ∑n=1∞‍|δnγn|<∞.Then limn→∞an=0.


## 3. Main Results

In this section we will introduce two algorithms (one implicit and one explicit) for finding the minimum norm elementx* of Ω:=EP(F,B)∩VI(C,A)∩F(S). Namely, we want to find a point x* which solves the following minimization problem:
(3.1)x*=argmin⁡x∈Ω‖x‖.
Let S:C→C be a nonexpansive mapping and A,B:C→H be α-inverse-strongly monotone and β-inverse-strongly monotone mappings, respectively. Let F:C×C→R be a bifunction which satisfies conditions (H1)–(H4). In order to solve the minimization problem (3.1), we first construct the following implicit algorithm by using the projection method:
(3.2)xt=SPC[(1-t)(I-λA)Tμ(I-μB)xt],∀t∈(0,1),
where Tμ is defined as Lemma 2.1 and λ,μ are two constants such that λ∈(0,2α) and μ∈(0,2β). We will show that the net {xt} defined by (3.2) converges to a solution of the minimization problem (3.1). First, we show that the net {xt} is well defined. As matter of fact, for each t∈(0,1), we consider the following mapping Wt given by
(3.3)Wtx=SPC[(1-t)(I-λA)Tμ(I-μB)x],∀x∈C.
Since the mappings S, PC, I-λA, Tμ and I-μB are nonexpansive, then we can check easily that ∥Wtx-Wty∥≤(1-t)∥x-y∥ which implies that Wt is a contraction. Using the Banach contraction principle, there exists a unique fixed point xt of Wt in C, that is, xt=Wtxt which is exactly (3.2).Next we show the first main result of the present paper.Theorem 3.1.
Suppose thatΩ≠∅. Then the net {xt} generated by the implicit method (3.2) converges in norm, as t→0, to a solution of the minimization problem (3.1).Proof.
Takez∈Ω. First we need use the following facts:(1)
z=Sz=Tμ(z-μBz)=PC(I-λA)z=SPC[(I-λA)Tμ(I-μB)z] for all λ>0, μ>0. In particular, for all t∈(0,1),
(3.4)z=PC[z-(1-t)λAz]=PC[tz+(1-t)(I-λA)z]=SPC[tz+(1-t)(I-λA)Tμ(I-μB)z];(2)
I-λA and I-μB are nonexpansive and for all x,y∈C(3.5)∥(I-λA)x-(I-λA)y∥2≤∥x-y∥2+λ(λ-2α)∥Ax-Ay∥2;‖(I-μB)x-(I-μB)y‖2≤‖x-y‖2+μ(μ-2β)‖Bx-By‖2.
Setut=Tμ(I-μB)xt and yt=(I-λA)ut for all t∈(0,1). It follows that
(3.6)‖ut-z‖2=‖Tμ(xt-μBxt)-Tμ(z-μBz)‖2≤‖(I-μB)xt-(I-μB)z‖2≤‖xt-z‖2+μ(μ-2β)‖Bxt-Bz‖2.
From (3.2), we have
(3.7)‖xt-z‖=‖SPC[(1-t)(I-λA)Tμ(I-μB)xt]-SPC[tz+(1-t)(I-λA)Tμ(I-μB)z]‖≤‖[(1-t)(I-λA)Tμ(I-μB)xt]-[tz+(1-t)(I-λA)Tμ(I-μB)z]‖≤(1-t)‖(I-λA)Tμ(I-μB)xt-(I-λA)Tμ(I-μB)z‖+t‖z‖≤(1-t)‖xt-z‖+t‖z‖,
that is,
(3.8)‖xt-z‖≤‖z‖.
So, {xt} is bounded. Hence {ut}, {Axt}, and {Bxt} are also bounded. Next we will use M>0 to denote some possible constant appearing in the following.
From (3.7), we have
(3.9)‖xt-z‖2≤‖[(1-t)(I-λA)Tμ(I-μB)xt]-[tz+(1-t)(I-λA)Tμ(I-μB)z]‖2=‖(1-t)[(I-λA)Tμ(I-μB)xt-(I-λA)Tμ(I-μB)z]-tz‖2=(1-t)2‖(I-λA)ut-(I-λA)z‖2-2t(1-t)〈z,(I-λA)ut-(I-λA)z〉+t2‖z‖2≤‖ut-z‖2+λ(λ-2α)‖Aut-Az‖2+tM≤‖xt-z‖2+μ(μ-2β)‖Bxt-Bz‖2+λ(λ-2α)‖Aut-Az‖2+tM,
that is,
(3.10)λ(2α-λ)‖Aut-Az‖2+μ(2β-μ)‖Bxt-Bz‖2≤tM⟶0.
Since λ(2α-λ)>0 and μ(2β-μ)>0, we derive
(3.11)lim⁡t→0‖Aut-Az‖=lim⁡t→0‖Bxt-Bz‖=0.
From Lemma 2.1 and (2.2), we obtain
(3.12)‖ut-z‖2=‖Tμ(xt-μBxt)-Tμ(z-μBz)‖2≤〈(xt-μBxt)-(z-μBz),ut-z〉=12(‖(xt-μBxt)-(z-μBz)‖2+‖ut-z‖2-‖(xt-z)-μ(Bxt-Bz)-(ut-z)‖2)≤12(‖xt-z‖2+‖ut-z‖2-‖(xt-ut)-μ(Bxt-Bz)‖2)=12(‖xt-z‖2+‖ut-z‖2-‖xt-ut‖2+2μ〈xt-ut,Bxt-Bz〉-μ2‖Bxt-Bz‖2).
It follows that
(3.13)‖ut-z‖2≤‖xt-z‖2-‖xt-ut‖2+2μ〈xt-ut,Bxt-Bz〉-μ2‖Bxt-Bz‖2≤‖xt-z‖2-‖xt-ut‖2+2μ‖xt-ut‖‖Bxt-Bz‖≤‖xt-z‖2-‖xt-ut‖2+M‖Bxt-Bz‖.
Set zt=PC[(1-t)yt] for all t∈(0,1). By Lemma 2.1 and (2.2), we have
(3.14)‖zt-z‖2=‖PC[(1-t)(I-λA)ut]-PC[(I-λA)z]‖2≤〈(1-t)(I-λA)ut-(I-λA)z,zt-z〉=12{‖(I-λA)ut-(I-λA)z-tyt‖2+‖zt-z‖212-‖(I-λA)ut-(I-λA)z-(zt-z)-tyt‖2}≤12{‖(I-λA)ut-(I-λA)z‖2+tM+‖zt-z‖212-‖ut-zt-λ(Aut-Az)-tyt‖2}≤12{‖ut-z‖2+tM+‖zt-z‖2-‖ut-zt‖2+2λ〈ut-zt,Aut-Az〉12+2t〈yt,ut-zt〉-‖λA(ut-Az)+tyt‖2}≤12{‖ut-z‖2+tM+‖zt-z‖2-‖ut-zt‖2+2λ‖ut-zt‖‖Aut-Az‖12+2t‖yt‖‖ut-zt‖‖‖2},
that is,
(3.15)‖zt-z‖2≤‖ut-z‖2+tM-‖ut-zt‖2+2λ‖ut-zt‖‖Aut-Az‖+2t‖yt‖‖ut-zt‖≤‖xt-z‖2-‖xt-ut‖2+M‖Bxt-Bz‖+tM-‖ut-zt‖2+2λ‖ut-zt‖‖Aut-Az‖+2t‖yt‖‖ut-zt‖.
Therefore, we have
(3.16)‖xt-z‖2=‖Szt-z‖2≤‖zt-z‖2≤‖xt-z‖2-‖xt-ut‖2+M‖Bxt-Bz‖+tM-‖ut-zt‖2+2λ‖ut-zt‖‖Aut-Az‖+2t‖yt‖‖ut-zt‖.
Hence, we deduce
(3.17)‖xt-ut‖2+‖ut-zt‖2≤M(‖Bxt-Bz‖+t)+2λ‖ut-zt‖‖Aut-Az‖+2t‖yt‖‖ut-zt‖⟶0.
This denotes that
(3.18)lim⁡t→0‖xt-ut‖=lim⁡t→0‖ut-zt‖=0.
Note that
(3.19)‖zt-yt‖=‖PC[(1-t)yt]-PCyt‖≤t‖yt‖⟶0,
thus,
(3.20)‖yt-Syt‖≤‖yt-zt‖+‖zt-ut‖+‖ut-xt‖+‖xt-Syt‖≤2‖yt-zt‖+‖zt-ut‖+‖ut-xt‖⟶0.
Next we show that {xt} is relatively norm compact as t→0. Let {tn}⊂(0,1) be a sequence such that tn→0 as n→∞. Put xn:=xtn, un:=utn and yn:=ytn. From (3.20), we get
(3.21)‖yn-Syn‖⟶0.
By (3.2), we have
(3.22)‖xt-z‖2=‖SPC[(1-t)yt]-SPCz‖2≤‖yt-z-tyt‖2=‖yt-z‖2-2t〈yt,yt-z〉+t2‖yt‖2=‖yt-z‖2-2t〈yt-z,yt-z〉-2t〈z,yt-z〉+t2‖yt‖2≤(1-2t)‖yt-z‖2+2t〈z,z-yt〉+t2‖yt‖2≤(1-2t)‖xt-z‖2+2t〈z,z-yt〉+t2‖yt‖2.
It follows that
(3.23)‖xt-z‖2≤〈z,z-yt〉+tM2.
In particular,
(3.24)‖xn-z‖2≤〈z,z-yn〉+tnM2,z∈Ω.
Since{yn} is bounded, without loss of generality, we may assume that {yn} converges weakly to a point x*∈C. Hence, un and xn also converge weakly to x*. Noticing (3.21) we can use Lemma 2.3 to get x*∈F(S).
Now we showx*∈EP(F,B). Since un=Tμ(xn-μBxn), for any y∈C we have
(3.25)F(un,y)+1μ〈y-un,un-(xn-μBxn)〉≥0.
From the monotonicity of F, we have
(3.26)1μ〈y-un,un-(xn-μBxn)〉≥F(y,un),∀y∈C.
Hence,
(3.27)〈y-uni,uni-xniμ+Bxni〉≥F(y,uni),∀y∈C.
Put vt=ty+(1-t)x* for all t∈(0,1] and y∈C. Then, we have vt∈C. So, from (3.27) we have
(3.28)〈vt-uni,Bvt〉≥〈vt-uni,Bvt〉-〈vt-uni,uni-xniμ+Bxni〉+F(vt,uni)=〈vt-uni,Bvt-Buni〉+〈vt-uni,Buni-Bxni〉-〈vt-uni,uni-xniμ〉+F(vt,uni).
Note that ∥Buni-Bxni∥≤(1/β)∥uni-xni∥→0. Further, from monotonicity of B, we have 〈vt-uni,Bvt-Buni〉≥0. Letting i→∞ in (3.28), we have
(3.29)〈vt-x*,Bvt〉≥F(vt,x*).
From (H1), (H4), and (3.29), we also have
(3.30)0=F(vt,vt)≤tF(vt,y)+(1-t)F(vt,x*)≤tF(vt,y)+(1-t)〈vt-x*,Bvt〉=tF(vt,y)+(1-t)t〈y-x*,Bvt〉,
and hence
(3.31)0≤F(vt,y)+(1-t)〈Bvt,y-x*〉.
Letting t→0 in (3.31), we have, for each y∈C,
(3.32)0≤F(x*,y)+〈y-x*,Bx*〉.
This implies that x*∈EP(F,B). By the same argument as that of [13], we have x*∈VI(C,A). Therefore, x*∈Ω.
We substitutex* for z in (3.24) to get
(3.33)‖xn-x*‖2≤〈x*,x*-yn〉+tn2M.
Hence, the weak convergence of {yn} to x* implies that xn→x* strongly. This has proved the relative norm compactness of the net {xt} as t→0.
Now we return to (3.24) and take the limit as n→∞ to get
(3.34)‖x*-z‖2≤〈z,z-x*〉,z∈Ω.
To show that the entire net {xt} converges to x*, assume xsn→x~∈Ω, where sn→0. In (3.34), we take z=x~ to get
(3.35)‖x*-x~‖2≤〈x~,x~-x*〉.
Interchange x* and x~ to obtain
(3.36)‖x~-x*‖2≤〈x*,x*-x~〉.
Adding up (3.35) and (3.36) yields
(3.37)2‖x*-x~‖2≤‖x*-x~‖2,
which implies that x~=x*.
We note that (3.34) is equivalent to
(3.38)‖x*‖2≤〈x*,z〉,z∈Ω.
This clearly implies that
(3.39)‖x*‖≤‖z‖,z∈Ω.
Therefore, x* solves the minimization problem (3.1). This completes the proof.Next we introduce an explicit algorithm for finding a solution of the minimization problem (3.1). This scheme is obtained by discretizing the implicit scheme (3.2). We will show the strong convergence of this algorithm.Theorem 3.2.
Suppose thatΩ≠∅. For given x0∈C arbitrarily, let the sequence {xn} be generated iteratively by
(3.40)xn+1=βnxn+(1-βn)SPC[(1-αn)(I-λA)Tμ(I-μB)xn],n≥0,
where {αn} and {βn} are two sequences in (0,1) satisfying the following conditions:(a)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(b)
0<liminfn→∞βn≤limsupn→∞βn<1. Then the sequence {xn} converges strongly to a solution of the minimization problem (3.1).Proof.
Takez∈Ω. First we need use the following fact:
z
=
S
z
=
T
μ
(
I
-
μ
B
)
z
=
P
C
(
I
-
λ
A
)
z
=
S
P
C
[
(
I
-
λ
A
)
T
μ
(
I
-
μ
B
)
z
] for all λ>0,μ>0. In particular,
(3.41)z=PC[z-(1-αn)λAz]=PC[αnz+(1-αn)(I-λA)z]=SPC[αnz+(1-αn)(I-λA)Tμ(I-μB)z],∀n≥0.
Setun=Tμ(xn-μBxn), yn=(I-λA)un and zn=PC[(1-αn)yn] for all n≥0. From (3.40), we get
(3.42)‖xn+1-z‖=‖βn(xn-z)+(1-βn)(SPC[(1-αn)(I-λA)Tμ(I-μB)xn]-SPC[αnz+(1-αn)(I-λA)Tμ(I-μB)z])‖≤βn‖xn-z‖+(1-βn)‖(1-αn)[(I-λA)Tμ(I-μB)xn-(I-λA)Tμ(I-μB)z]-αnz‖≤βn‖xn-z‖+(1-βn)[(1-αn)‖xn-z‖+αn‖z‖]=[1-(1-βn)αn]‖xn-z‖+αn(1-βn)‖z‖≤max⁡{‖xn-z‖,‖z‖}.
By induction, we obtain, for all n≥0,
(3.43)‖xn-z‖≤max⁡{‖x0-z‖,‖z‖}.
Hence, {xn} is bounded. Consequently, we deduce that {un}, {yn}, and {zn} are all bounded. We will use M>0 to denote some possible constant appearing in the following.
Definexn+1=βnxn+(1-βn)vn for all n≥0. It follows that
(3.44)‖vn+1-vn‖=‖SPC[(1-αn+1)yn+1]-SPC[(1-αn)yn]‖≤‖(1-αn+1)yn+1-(1-αn)yn‖≤‖yn+1-yn‖+αn+1‖yn+1‖+αn‖yn‖≤‖(I-λA)un+1-(I-λA)un‖+M(αn+1+αn)≤‖un+1-un‖+M(αn+1+αn)=‖Tμ(I-μB)xn+1-Tμ(I-μB)xn‖+M(αn+1+αn)≤‖xn+1-xn‖+M(αn+1+αn).
This together with (a) implies that
(3.45)limsup⁡n→∞(‖vn+1-vn‖-‖xn+1-xn‖)≤0.
Hence by Lemma 2.2, we get
(3.46)lim⁡n→∞‖vn-xn‖=0.
Therefore,
(3.47)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖vn-xn‖=0.
By the convexity of the norm ∥·∥, we have
(3.48)‖xn+1-z‖2=‖βn(xn-z)+(1-βn)(vn-z)‖2≤βn‖xn-z‖2+(1-βn)‖PC[(1-αn)(I-λA)Tμ(I-μB)xn]-PC[αnz+(1-αn)(I-λA)Tμ(I-μB)z]‖2≤βn‖xn-z‖2+(1-βn)‖(1-αn)[(I-λA)Tμ(I-μB)xn-(I-λA)Tμ(I-μB)z]-αnz‖2=βn‖xn-z‖2+(1-βn)[(1-αn)2‖(I-λA)un-(I-λA)z‖2(1-αn)2-2αn(1-αn)〈(I-λA)un-(I-λA)z,z〉+αn2‖z‖2]≤βn‖xn-z‖2+(1-βn)[‖un-z‖2+λ(λ-2α)‖Aun-Az‖2+αnM]=βn‖xn-z‖2+(1-βn)[‖Tμ(I-μB)xn-Tμ(I-μB)z‖2+λ(λ-2α)‖Aun-Az‖2+αnM]≤βn‖xn-z‖2+(1-βn)[‖xn-z‖2+μ(μ-2β)‖Bxn-Bz‖2+λ(λ-2α)‖Aun-Az‖2+αnM].
It follows that
(3.49)(1-βn)λ(2α-λ)‖Aun-Az‖2+(1-βn)μ(2β-μ)‖Bxn-Bz‖2≤‖xn-z‖2-‖xn+1-z‖2+αnM≤(‖xn-z‖+‖xn+1-z‖)‖xn-xn+1‖+αnM.
Since liminfn→∞(1-βn)λ(2α-λ)>0, liminfn→∞(1-βn)μ(2β-μ)>0, ∥xn-xn+1∥→0 and αn→0, we derive
(3.50)lim⁡n→∞‖Aun-Az‖=lim⁡n→∞‖Bxn-Bz‖=0.
From Lemma 2.1 and (2.2), we obtain
(3.51)‖un-z‖2=‖Tμ(xn-μBxn)-Tμ(z-μBz)‖2≤〈(xn-μBxn)-(z-μBz),un-z〉=12(‖(xn-μBxn)-(z-μBz)‖2+‖un-z‖2-‖(xn-un)-μ(Bxn-Bz)‖2)≤12(‖xn-z‖2+‖un-z‖2-‖(xn-un)-μ(Bxn-Bz)‖2)=12(‖xn-z‖2+‖un-z‖2-‖xn-un‖2+2μ〈xn-un,Bxn-Bz〉-μ2‖Bxn-Bz‖2).
It follows that
(3.52)‖un-z‖2≤‖xn-z‖2-‖xn-un‖2+2μ〈xn-un,Bxn-Bz〉-μ2‖Bxn-Bz‖2≤‖xn-z‖2-‖xn-un‖2+2μ‖xn-un‖‖Bxn-Bz‖≤‖xn-z‖2-‖xn-un‖2+M‖Bxn-Bz‖.
Again, by Lemma 2.1 and (2.2), we have
(3.53)‖zn-z‖2=‖PC[(1-αn)(I-λA)un]-PC[(I-λA)z]‖2≤〈(1-αn)(I-λA)un-(I-λA)z,zn-z〉=12{‖(I-λA)un-(I-λA)z-αnyn‖2+‖zn-z‖2-‖(I-λA)un-(I-λA)z-(zn-z)-αnyn‖2}≤12{‖(I-λA)un-(I-λA)z‖2+αnM+‖zn-z‖2-‖un-zn-λ(Aun-Az)-αnyn‖2}≤12{‖un-z‖2+αnM+‖zn-z‖2-‖un-zn‖2+2λ〈un-zn,Aun-Az〉+2αn〈yn,un-zn〉-‖λ(Aun-Az)+αnyn‖2}≤12{‖un-z‖2+αnM+‖zn-z‖2-‖un-zn‖2+2λ‖un-zn‖‖Aun-Az‖+2αn‖yn‖‖un-zn‖(1-αn)2},
that is,
(3.54)‖zn-z‖2≤‖un-z‖2+αnM-‖un-zn‖2+2λ‖un-zn‖‖Aun-Az‖+2αn‖yn‖‖un-zn‖≤‖xn-z‖2-‖xn-un‖2+M‖Bxn-Bz‖+αnM-‖un-zn‖2+2λ‖un-zn‖‖Aun-Az‖+2αn‖yn‖‖un-zn‖.
Hence,
(3.55)‖xn+1-z‖2≤βn‖xn-z‖2+(1-βn)‖zn-z‖2≤βn‖xn-z‖2+(1-βn)[‖xn-z‖2-‖xn-un‖2+M‖Bxn-Bz‖+αnM-‖un-zn‖2+2λ‖un-zn‖‖Aun-Az‖‖xn-z‖2+2αn‖yn‖‖un-zn‖‖‖2]≤‖xn-z‖2-(1-βn)‖xn-un‖2+(‖Bxn-Bz‖+αn)M-(1-βn)‖un-zn‖2+2λ‖un-zn‖‖Aun-Az‖+2αn‖yn‖‖un-zn‖.
It follows that
(3.56)(1-βn)(‖xn-un‖2+‖un-zn‖2)≤‖xn-xn+1‖(‖xn-z‖+‖xn+1-z‖)+(‖Bxn-Bz‖+αn)M+2λ‖un-zn‖‖Aun-Az‖+2αn‖yn‖‖un-zn‖.
Since liminfn→∞(1-βn)>0, αn→0, ∥xn+1-xn∥→0, ∥Aun-Az∥→0 and ∥Bxn-Bz∥→0, we derive that
(3.57)lim⁡n→∞‖xn-un‖=lim⁡n→∞‖un-zn‖=0.
Note that
(3.58)‖zn-yn‖=‖PC[(1-αn)yn]-PCyn‖≤αn‖yn‖⟶0,
therefore,
(3.59)‖Syn-yn‖≤‖Syn-Szn‖+‖Szn-xn‖+‖xn-un‖+‖un-zn‖+‖zn-yn‖≤2‖yn-zn‖+‖Szn-xn‖+‖xn-un‖+‖un-zn‖⟶0.
Next we prove
(3.60)limsup⁡n→∞〈x*,x*-yn〉≤0,
where x*=PΩ(0) is a solution of the minimization problem (3.1).
Indeed, we can choose a subsequence{yni} of {yn} such that
(3.61)limsup⁡n→∞〈x*,x*-yn〉=lim⁡i→∞〈x*,x*-yni〉.
Without loss of generality, we may further assume that yni→x~ weakly. By the same argument as that of Theorem 3.1, we can deduce that x~∈Ω. Therefore,
(3.62)limsup⁡n→∞〈x*,x*-yn〉=〈x*,x*-x~〉≤0.
Finally, we prove xn→x*. As a matter of fact, we have
(3.63)‖xn+1-x*‖2=‖βn(xn-x*)+(1-βn)(SPC[(1-αn)yn]-SPC[x*])‖2≤βn‖xn-x*‖2+(1-βn)‖(1-αn)(yn-x*)-αnx*‖2=βn‖xn-x*‖2+(1-βn)[(1-αn)2‖yn-x*‖2-2αn(1-αn)〈x*,yn-x*〉+αn2‖x*‖2]≤βn‖xn-x*‖2+(1-βn)[(1-αn)‖xn-x*‖2-2αn(1-αn)〈x*,yn-x*〉+αn2‖x*‖2]≤[1-2(1-βn)αn]‖xn-x*‖2+2αn(1-αn)(1-βn)〈x*,x*-yn〉+(1-βn)αn2M=(1-γn)‖xn-x*‖2+δnγn,
where γn=2(1-βn)αn and δn=(1-αn)〈x*,x*-yn〉+(αnM/2). It is clear that ∑n=0∞‍γn=∞ and limsupn→∞δ≤0. Hence, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that xn→x* strongly. This completes the proof.



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*Source: 310801-2012-07-08.xml*