# General Iterative Methods for System of Equilibrium Problems and Constrained Convex Minimization Problem in Hilbert Spaces

Authors: Peichao Duan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by Tian and Liu (2012) and many others. Furthermore, we give numerical example to demonstrate the effectiveness of our iterative scheme.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈,〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H.Let{Fk} be a countable family of bifunctions from C×C to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems which is to find x∈C such that
(1)Fk(x,y)≥0,∀k∈Γ,∀y∈C,
where Γ is an arbitrary index set. If Γ is a singleton, then problem (1) becomes the following equilibrium problem:
(2)findingx∈CsuchthatF(x,y)≥0,∀y∈C.
The solution set of (2) is denoted by EP(F).Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Many methods have been proposed to solve the equilibrium problem (2); see [2–4] and the references therein. In particular, some methods have been proposed to solve the system of equilibrium problems. See [5–7] and the references therein.On the other hand, we consider the following constrained minimization problem:(3)minimize{f(x):x∈C},
where f:C→ℝ is a real-valued convex function. It is known that the gradient projection algorithm (GPA) is a powerful tool for solving the constrained minimization problems and has extensively been studied; see for instance [8–10]. If f is (Fréchet) differentiable, then the GPA generates a sequence {xn} using the following recursive formula:
(4)xn+1=PC(xn-λ∇f(xn)),∀n≥0,
or more generally,
(5)xn+1=PC(xn-λn∇f(xn)),∀n≥0,
where in both (4) and (5) the initial guess x0 is taken from C arbitrarily, and the parameters, λ or λn, are positive real numbers satisfying certain conditions. The convergence of the algorithms (4) and (5) depends on the behavior of the gradient ∇f. As a matter of fact, it is known that if ∇f is α-strongly monotone and L-Lipschitzian with constants α, L≥0, then the operator
(6)T:=PC(I-λ∇f)
is a contraction; hence, the sequence {xn} defined by the algorithm (4) converges in norm to the unique minimizer of (3). However, if the gradient ∇f fails to be strongly monotone, the operator T by (6) would fail to be contractive; consequently, the sequence {xn} generated by the algorithm (4) may fail to converge strongly [11]. If ∇f is Lipschitzian, then the algorithms (4) and (5) can still converge in the weak topology under certain conditions [10, 12].In 2007, Marino and Xu [3] introduced the general iterative method and proved that the algorithm converged strongly. In 2009, Liu [2] considered two iterative schemes by the general iterative method for equilibrium problems and strict pseudocontractions. In 2011, Xu [11] gave an alternative operator-oriented approach to algorithm (5), namely, an averaged mapping approach. He gave his averaged mapping approach to the GPA (5) and the relaxed GPA. Moreover, he constructed a counter example which shows that the algorithm (4) does not converge in norm in an infinite-dimensional space and also presented two modifications of GPA which are shown to have strong convergence. Recently, Ceng et al. [8] proposed implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and proved that the sequences generated by their schemes converges strongly to a solution of the constrained convex minimization problem. Very recently, Tian and Liu [9] proposed implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium problem and a constrained convex minimization problem; strong convergence theorems are obtained in [9].In this paper, motivated by the above facts, we introduce two iterative schemes by the composite general iterative methods. Further we obtain strong convergence theorems for finding a common element of the set of solutions of a constrained convex minimization problems and the set of solutions of the equilibrium problem(7)Fk(x,y)≥0,∀k∈Γ,∀y∈C,
where Γ={1,2,…,M} is a finite index set.


## 2. Preliminaries

Throughout this paper, we always write⇀ for weak convergence and → for strong convergence. We need some definitions and tools in a real Hilbert space H which are listed as below.A mappingS of C is said to be a nonexpansive mapping such that
(8)∥Sx-Sy∥≤∥x-y∥
for all x,y∈C. The set of fixed points of S is denoted by F(S); that is, F(S)={x∈C:Sx=x}.A mappingT:H→H is said to be an averaged mapping if it can be written as the average of an identity I and a nonexpansive mapping; that is,
(9)T=(1-α)I+αS,
where α is a number in (0,1) and S is nonexpansive. More precisely, we say that T is α-averaged. It is known that the projection is (1/2)-averaged.Lemma 1.
LetH be a real Hilbert space. There hold the following identities:(i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉, ∀x,y∈H,(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2, ∀t∈[0,1], ∀x,y∈H.Lemma 2 (see [10]).
Assume that{αn} is a sequence of nonnegative real numbers such that
(10)αn+1≤(1-γn)αn+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that (i)
∑n=1∞γn=∞;(ii)
limsupn→∞δn/γn≤0or∑n=1∞|δn|<∞.
Then,limn→∞αn=0.Recall that given a nonempty closed convex subsetC of a real Hilbert space H, for any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(11)∥x-PCx∥≤∥x-y∥
for all y∈C. Such a PC is called the metric (or the nearest point) projection of H onto C. As we all know, y=PCx if and only if there holds the relation:
(12)〈x-y,y-z〉≥0,∀z∈C.Lemma 3 (see [13]).
LetA:H→H be an L-Lipschitzian and η-strongly monotone operator on a Hilbert space H with L>0, η>0, 0<μ<2η/L2 and 0<t<1. Then S=(I-tμA):H→H is a contraction with contractive coefficient 1-tτ and τ=(1/2)μ(2η-μL2).Definition 4.
A nonlinear mappingA whose domain D(A)⊆H and range R(A)⊆H is said to be (i)
monotone if(13)〈Ax-Ay,x-y〉≥0,∀x,y∈D(A);(ii)
β-strongly monotone if there exists β>0 such that
(14)〈Ax-Ay,x-y〉≥β∥x-y∥2,∀x,y∈D(A);(iii)
ν-inverse strongly monotone (for short, ν-ism) if there exists a constant ν>0 such that
(15)〈Ax-Ay,x-y〉≥ν∥Ax-Ay∥2,∀x,y∈D(A).Lemma 5.
LetV:C→H be an l-Lipschitz mapping with coefficient l≥0 and A:C→H a strong positive bounded linear operator with γ¯>0. Then for 0<γ<μγ¯/l,
(16)〈x-y,(μA-γV)x-(μA-γV)y〉≥(μγ¯-γl)∥x-y∥2,x,y∈H;
that is, μA-γV is strongly monotone with coefficient μγ¯-γl.Proof.
SinceA:C→H is a strong positive bounded linear operator with γ¯>0. We have
(17)γ¯∥x-y∥2≤〈x-y,Ax-Ay〉=〈x-y,A(x-y)〉≤∥A∥∥x-y∥2.
HenceA is ∥A∥-Lipschitz and γ¯-strongly monotone:
(18)〈x-y,(μA-γV)x-(μA-γV)y〉=μ〈x-y,Ax-Ay〉-γ〈x-y,Vx-Vy〉≥(μγ¯-γl)∥x-y∥2,x,y∈H.Proposition 6.
For given operatorsS,T,V:H→H.(i)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis averaged andVis nonexpansive, thenTis averaged.(ii)
Tis firmly nonexpansive if and only if the complementI-Tis firmly nonexpansive.(iii)
IfT=(1-α)S+αVfor someα∈(0,1),Sis firmly nonexpansive, andVis nonexpansive, thenTis averaged.(iv)
The composite of finitely many averaged mappings is averaged; that is, if each of the mapping{Ti}i=1Nis averaged, then so is the compositeT1,…,TN. 									In particular, ifT1isα1-averaged andT2isα2-averaged, whereα1,α2∈(0,1), then the compositeT1T2isα-averaged, whereα=α1+α2-α1α2.Proposition 7.
LetT:H→H be an operator from H to itself. (i)
Tis nonexpansive if and only if the complementI-Tis(1/2)-ism.(ii)
IfTisν-ism, then forγ>0,γTis(ν/γ)-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>(1/2). Indeed, forα∈(0,1),Tisα-averaged if and only if the complementI-Tis(1/2α)-ism.For solving the equilibrium problem, let us assume that the bifunctionF satisfies the following conditions.(A1)
F(x,x)=0 for all x∈C.(A2)
F is monotone; that is, F(x,y)+F(y,x)≤0 for any x,y∈C;(A3)
For eachx,y,z∈C, limsupt→0F(tz+(1-t)x,y)≤F(x,y).(A4)
F(x,·) is convex and lower semicontinuous for each x∈C.We recall some lemmas which will be needed in the rest of this paper.Lemma 8 (see [14]).
LetC be a nonempty closed convex subset of H, let F be bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈H. Then there exists z∈C such that
(19)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 9 (see [1]).
Forr>0, x∈H, define a mapping Tr:H→C as follows:
(20)Tr(x)={z∈C∣F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following statements hold: (i)
Tr is single-valued;(ii)
Tr is firmly nonexpansive; that is, for any x,y∈H,
(21)∥Trx-Try∥2≤〈Trx-Try,x-y〉;(iii)
F(Tr)=EP(F);(iv)
EP(F) is closed and convex.Lemma 10 (see [4]).
LetC,H,F, and Trx be as in Lemma 9. Then the following holds:
(22)∥Tsx-Ttx∥2≤s-ts〈Tsx-Ttx,Tsx-x〉
for all s,t>0 and x∈H.Lemma 11 (see [13]).
LetH be a Hilbert space, C a nonempty closed convex subset of H, and T:C→C a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.


## 3. Main Result

Throughout the rest of this paper, we always assume thatV is an l-Lipschitzian mapping with coefficient l≥0, and A is a strongly positive bounded linear operator with coefficient γ¯. Then we obtain that A is ∥A∥-Lipschitzian and γ¯-strongly monotone. Let f:C→R be a real-valued convex function and assume that ∇f is ν-ism with ν>0, which then implies that λ∇f is ν/λ-ism. So by Proposition 7, its complement I-λ∇f is λ/(2ν)-averaged. Since the projection PC is 1/2-averaged, we obtain from Proposition 6 that the composition PC(I-λ∇f) is (2ν+λ)/(4ν)-averaged for 0<λ<2/L. Hence we have that, for each n, PC(I-λn∇f) is (2ν+λn)/(4ν)-averaged. Therefore, we can write
(23)PC(I-λn∇f)=2ν-λn4νI+2ν+λn4νSn,
where Sn is nonexpansive.Suppose that the minimization problem (3) is consistent and let U denote its solution set. Assume that 0<μ<2γ¯/∥A∥2 and 0<γ<μ(γ¯-(μ∥A∥2/2))/l=τ/l.DenoteΘnk=Trk,nFk⋯Tr2,nF2Tr1,nF1 for every k∈{1,2,…,M} and Θn0=I for all n∈ℕ. Define a mapping Wn=SnΘnM. Since both Sn and Trk,nFk,k∈{1,2,…,M} are nonexpansive, it is easy to get that Wn is also nonexpansive. Consider the following mapping Gn on H defined by
(24)Gnx=αnγVΘnMx+(I-αnμA)Wnx,∀x∈H,n∈N,
where αn∈(0,1). By Lemmas 3 and 9, we have
(25)∥Gnx-Gny∥≤αnγ∥Vx-Vy∥+(1-αnτ)∥Wnx-Wny∥≤αnγl∥x-y∥+(1-αnτ)∥x-y∥=(1-αn(τ-γl))∥x-y∥.
Since 0<1-αn(τ-γl)<1, it follows that Gn is a contraction. Therefore, by the Banach contraction principle, Gn has a unique fixed pointed xnV∈H such that
(26)xnV=αnγVΘnM(xnV)+(I-αnμA)WnxnV.For simplicity, we will writexn for xnV provided no confusion occurs. Next we prove the sequences {xn} converges strongly to a point x*∈Ω=U∩∩k=1MEP(Fk) which solves the variational inequality
(27)〈(γV-μA)x*,p-x*〉≤0,∀p∈Ω.
Equivalently, x*=PΩ(I-μA+γV)x*.Theorem 12.
LetC be a nonempty closed convex subset of a real Hilbert space H with C±C⊆C, and let Fk,k∈{1,2,…M} be bifunctions from C×C to ℝ satisfying (A1)–(A4). Let f:C→ℝ be a real-value convex function and ∇fν-ism with ν>0. Assume the set Ω=U∩∩k=1MEP(Fk)≠∅. Let V:C→C be an l-Lipschitzian mapping with l≥0 and A a strongly positive bounded linear operator with coefficient γ¯≥0, 0<μ<2γ¯/∥A∥2 and 0<γ<μ(γ¯-(μ∥A∥2/2))/l=τ/l. Let {xn} and {un} be sequences generated by the following algorithm:
(28)un=TrM,nFMTrM-1,nFM-1⋯Tr2,nF2Tr1,nF1xn,xn=αnγVun+(I-μαnA)Snun,
where PC(I-λn∇f)=βnI+(1-βn)Sn, βn=((2ν-λn)/4ν), and λn∈(0,2ν); if {αn} and {rk,n} satisfy the following conditions: (i)
{αn}⊂(0,1), limn→∞αn=0;(ii)
{rk,n}⊂(0,∞), liminfn→∞rk,n>0 for k∈{1,2,…,M}, then, as βn→0(λn→2ν), the sequence {xn} converges strongly to a point x*∈Ω, which solves the variational inequality (27).Proof.
The proof is divided into several steps.
Step  1. It shows first that {xn} is bounded.
First, sincelimn→∞αn=0, we can assume that αn∈(0,∥A∥-1). By Lemma 3, we have ∥I-αnμA∥≤1-αnτ.
Take anyp∈Ω, since for each k∈{1,2,…,M}, Trk,nFk is nonexpansive, p=Trk,nFkp, and un=ΘnMxn; we have
(29)∥un-p∥=∥ΘnMxn-ΘnMp∥≤∥xn-p∥
for all n∈N.
Thus, by (28) and Lemma 3, we derive that
(30)∥xn-p∥=∥αnγVun+(I-μαnA)Snun-p∥=∥(I-μαnA)Snun-(I-μαnA)Snp+αnγVun-αnγVp+αnγVp-αnμAp∥≤(1-αnτ)∥un-p∥+αnγl∥un-p∥+αn∥(γV-μA)p∥≤(1-αn(τ-γl))∥xn-p∥+αn∥(γV-μA)p∥.
It follows that ∥xn-p∥≤(∥(γV-μA)p∥/(τ-γl)).
Hence,{xn} is bounded and so {un}. It follows from the Lipschitz continuity of A, ∇f, and V that {Axn}, {Aun}, {∇f(un)} and {Vun} are also bounded. From the nonexpansivity of Sn, it follows that {Snun} is also bounded.
Step  2. It shows that
(31)limn→∞∥un-xn∥=0.
Next we will show that(32)limn→∞∥Θnkxn-Θnk-1xn∥=0,k=1,2,…,M.
Indeed, forp∈Ω, it follows from the firmly nonexpansivity of Trk,nFk that for each k∈{1,2,…,M}, we have
(33)∥Θnkxn-p∥2=∥Trk,nFkΘnk-1xn-Trk,nFkp∥2≤〈Θnkxn-p,Θnk-1xn-p〉=12(∥Θnkxn-p∥2+∥Θnk-1xn-p∥2-∥Θnkxn-Θnk-1xn∥2).
Thus we get
(34)∥Θnkxn-p∥2≤∥Θnk-1xn-p∥2-∥Θnkxn-Θnk-1xn∥2,k=1,2,…,M,
which implies that for each k∈{1,2,…,M},
(35)∥Θnkxn-p∥2≤∥Θn0xn-p∥2-∥Θnkxn-Θnk-1xn∥2-∥Θnk-1xn-Θnk-2xn∥2⋯-∥Θn1xn-Θn0xn∥2≤∥xn-p∥2-∥Θnkxn-Θnk-1xn∥2.
Thus, from Lemma 1 and (35), we get
(36)∥xn-p∥2=∥αnγVun+(I-μαnA)Snun-p∥2=∥(I-μαnA)Snun-(I-μαnA)Snp+αnγVun-αnμAp∥2≤(1-αnτ)2∥un-p∥2+2αn〈γVun-μAp,xn-p〉≤(1-αnτ)2∥un-p∥2+2αn〈γVun-γVp+γVp-μAp,xn-p〉≤(1-αnτ)2(∥xn-p∥2-∥Θnkxn-Θnk-1xn∥2)+2αnγl∥xn-p∥2+2αn∥γVp-μAp∥∥xn-p∥≤∥xn-p∥2+(αnτ)2∥xn-p∥2-(1-αnτ)2∥Θnkxn-Θnk-1xn∥2+2αn∥γVp-μAp∥∥xn-p∥.
It follows that
(37)(1-αnτ)2∥Θnkxn-Θnk-1xn∥2≤(αnτ)2∥xn-p∥2+2αn∥γVp-μAp∥∥xn-p∥.
Since αn→0, (32) holds, then we have
(38)∥un-xn∥≤∥un-ΘnM-1xn∥+∥ΘnM-1xn-ΘnM-2xn∥+⋯+∥Θn1xn-xn∥⟶0.
Step  3. It shows that
(39)limn→∞∥xn-Snxn∥=0.
Observe that(40)∥xn-Snxn∥=∥xn-Snun+Snun-Snxn∥≤∥xn-Snun∥+∥Snun-Snxn∥≤αn∥γVun-μASnun∥+∥un-xn∥.
Since αn→0 and ∥un-xn∥→0, it is easy to get (39).
Thus,(41)∥un-Snun∥=∥un-xn+xn-Snxn+Snxn-Snun∥≤2∥xn-un∥+∥xn-Snxn∥.
We obtain ∥un-Snun∥→0.
Notice that(42)∥PC(I-λn∇f)un-un∥=∥βnun+(1-βn)Snun-un∥=(1-βn)∥un-Snun∥≤∥un-Snun∥,
where βn∈(0,1/2). Hence we have
(43)∥PC(I-2ν∇f)un-un∥≤∥PC(I-2ν∇f)un-PC(I-λn∇f)un∥+∥PC(I-λn∇f)un-un∥≤∥(I-2ν∇f)un-(I-λn∇f)un∥+∥PC(I-λn∇f)un-un∥≤(2ν-λn)∥∇f(un)∥+∥Snun-un∥.
From the boundedness of{∇f(un)}, βn→0 and ∥Snun-un∥→0, we conclude that
(44)limn→∞∥PC(I-2ν∇f)un-un∥=0.
Since{un} is bounded, there exists a subsequence {unj} which converges weakly to x*.
Step  4. It shows that x*∈Ω.
SinceC is closed and convex, C is weakly closed. So we have x*∈C. By Lemma 11 and (44), we have x*∈U.
Next we will show thatx*∈∩i=1MEP(Fk).
Indeed, by Lemma9, we have that for each k=1,2,…,M,
(45)Fk(Θnkxn,y)+1rk,n〈y-Θnkxn,Θnkxn-Θnk-1xn〉≥0,∀y∈C.
From (A2), we get
(46)1rk,n〈y-Θnkxn,Θnkxn-Θnk-1xn〉≥Fk(y,Θnkxn),∀y∈C.
Hence,
(47)〈y-Θnjkxnj,Θnjkxnj-Θnjk-1xnjrk,nj〉≥Fk(y,Θnjkxnj),∀y∈C.
From (32), we obtain that Θnjkxnj⇀x* as j→∞ for each k=1,2,…,M (especially, unj=ΘnjMxnj). Together with (32), condition (ii), and (A4) we have, for each k=1,2,…,M, that
(48)0≥Fk(y,x*),∀y∈C.
For any,0<t≤1 and y∈C, let yt=ty+(1-t)x*. Since y∈C and x*∈C, we obtain that yt∈C, and hence Fk(yt,x*)≤0. So, we have
(49)0=Fk(yt,yt)≤tFk(yt,y)+(1-t)Fk(yt,x*)≤tFk(yt,y).
Dividing by t, we get, for each k=1,2,…,M, that
(50)Fk(yt,y)≥0,∀y∈C.
Letting t→0 and from (A3), we get
(51)Fk(x*,y)≥0.
For all y∈C and x*∈EP(Fk) for each k=1,2,…,M; that is, x*∈∩k=1MEP(Fk). Hence x*∈Ω.
Step  5. It shows that xn→x*, where x*=PΩ(I-μA+γV)x*(52)xn-x*=αn(γVun-μAx*)+(I-μαnA)Snun-(I-μαnA)x*.
Hence, we obtain
(53)∥xn-x*∥2=αn〈γVun-μAx*,xn-x*〉+〈(I-μαnA)Snun-(I-μαnA)x*,xn-x*〉≤αn〈γVun-μAx*,xn-x*〉+(1-αnτ)∥xn-x*∥2.
It follows that
(54)∥xn-x*∥2≤1τ〈γVun-μAx*,xn-x*〉=1τ(γ〈Vun-Vx*,xn-x*〉+〈γVx*-μAx*,xn-x*〉)≤1τ(γl∥xn-x*∥2+〈γVx*-μAx*,xn-x*〉).
This implies that
(55)∥xn-x*∥2≤〈γVx*-μAx*,xn-x*〉τ-γl.
In particular,
(56)∥xnj-x*∥2≤〈γVx*-μAx*,xnj-x*〉τ-γl.
Sincexnj⇀x*, it follows from (56) that xnj→x* as j→∞. Next, we show that x* solves the variational inequality (27).
By the iterative algorithm (28), we have
(57)xn=αnγVun+(I-μαnA)Snun.
Therefore we have
(58)αn(μA-γV)xn=αn(μA-γV)(I-Wn)xn+αn(μA-γV)Snun+αnγVun+(I-μαnA)Snun-xn=αn(μA-γV)(I-Wn)xn+αnγ(Vun-VSnun)-(I-SnΘnM)xn;
that is,
(59)(μA-γV)xn=(μA-γV)(I-Wn)xn+γ(Vun-VSnun)-1αn(I-Wn)xn.
Due to the nonexpansivity of Wn, we have that I-Wn is monotone; that is, 〈x-y,(I-Wn)x-(I-Wn)y〉≥0, for  all x,y∈H. Hence for any p∈Ω,
(60)〈(μA-γV)xn,xn-p〉=-1αn〈(I-Wn)xn-(I-Wn)p,xn-p〉+〈(μA-γV)(I-Wn)xn,xn-p〉+γ〈Vun-VSnun,xn-p〉≤〈(μA-γV)(I-Wn)xn,xn-p〉+γ∥Vun-VSnun∥∥xn-p∥≤〈(μA-γV)(I-Wn)xn,xn-p〉+γl∥un-Snun∥∥xn-p∥.
Now replacingn in (60) with nj and letting j→∞, we obtain
(61)〈(μA-γV)x*,x*-p〉=limj→∞〈(μA-γV)xnj,xnj-p〉≤limj→∞〈(μA-γV)(xnj-Snjunj),xnj-p〉=0;
that is, x*∈Ω is a solution of the variational inequality (27).
Further, by the uniqueness of the solution of the variational inequality (27), we conclude that xn→x* as n→∞. We rewrite (27) as
(62)〈(I-μA+γV)x*-x*,x*-p〉≥0,∀p∈Ω.
This is equivalent to the fixed point equation
(63)PΩ(I-μA+γV)x*=x*.Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H with C±C⊆C, and let Fk,k∈{1,2,…M} be bifunctions from C×C to ℝ which satisfies conditions (A1)–(A4). Let f:C→ℝ be a real-value convex function and ∇f a ν-ism mapping with ν>0. Assume the set Ω=U∩∩i=1MEP(Fk)≠∅. Let V:C→C is an l-Lipschitzian mapping with l≥0 and A is a strongly positive bounded linear operator with coefficient γ¯≥0, 0<μ<2γ¯/∥A∥2 and 0<γ<μ(γ¯-(μ∥A∥2/2))/l=τ/l. Given x1∈H, let {xn} and {un} be sequences generated by the following algorithm:
(64)un=TrM,nFMTrM-1,nFM-1⋯Tr2,nF2Tr1,nF1xn,xn+1=αnγVun+(I-μαnA)Snun,
where PC(I-λn∇f)=βnI+(1-βn)Sn, βn=((2ν-λn)/4ν), and λn∈(0,2ν); if {αn}, {βn}, and {rk,n} satisfy the following conditions:(i)
{αn}⊂(0,1), limn→∞αn=0 and ∑n=1∞‍|αn+1-αn|<∞;(ii)
{rk,n}⊂(0,∞), liminfn→∞rk,n>0 and ∑n=1∞‍|rk,n+1-rk,n|<∞ for k∈{1,2,…M};(iii)
{βn}⊂(0,(1/2)), limn→∞βn=0 and ∑n=1∞‍|βn+1-βn|<∞, then, the sequence {xn} converges strongly to a point x*∈Ω, which solves the variational inequality (27).Proof.
The proof is divided into several steps.
Step  1. It shows first that {xn} is bounded.
Take anyp∈Ω, we have
(65)∥un-p∥=∥ΘnMxn-ΘnMp∥≤∥xn-p∥.
Thus, by (64), we derive that
(66)∥xn+1-p∥=∥αnγVun+(I-μαnA)Snun-p∥=∥(I-μαnA)Snun-(I-μαnA)Snp+αnγVun-αnγVp+αnγVp-αnμAp∥≤(1-αnτ)∥un-p∥+αnγl∥un-p∥+αn∥γVp-μAp∥≤(1-αn(τ-γl))∥xn-p∥+αn∥γVp-μAp∥=(1-αn(τ-γl))∥xn-p∥+αn(τ-γl)∥γVp-μAp∥τ-γl≤max{∥xn-p∥,∥γVp-μAp∥τ-γl}.
By induction, we obtain ∥xn-p∥≤max{∥x1-p∥, (∥γVp-μAp∥/(τ-γl))}, n≥1. Hence, {xn} is bounded and so {un}. It follows from the Lipschitz continuity of A, ∇f, and V that {Axn}, {Aun}{∇f(un)}, and {Vun} are also bounded. From the nonexpansivity of Sn, it follows that {Snun} is also bounded.
Step  2. It shows that
(67)∥xn+1-xn∥⟶0.
By (68), we have
(68)∥xn+1-xn∥=∥αnγVun+(I-μαnA)Snun-αn-1γVun-1-(I-μαn-1A)Sn-1un-1∥≤αnγ∥Vun-Vun-1∥+|αn-αn-1|γ∥Vun-1∥+∥(I-μαnA)Snun-(I-μαnA)Snun-1∥+∥(I-μαnA)Snun-1-(I-μαnA)Sn-1un-1∥+∥(I-μαnA)Sn-1un-1-(I-μαn-1A)Sn-1un-1∥≤αnγl∥un-un-1∥+|αn-αn-1|×(∥γVun-1∥+∥μASn-1un-1∥)+(1-αnτ)(∥un-un-1∥+∥Snun-1-Sn-1un-1∥)≤(1-αn(τ-γl))∥un-un-1∥+|αn-αn-1|(∥γVun-1∥+∥μASn-1un-1∥)+(1-αnτ)∥Snun-1-Sn-1un-1∥.
Next we estimate ∥Snun-1-Sn-1un-1∥.
Observe that(69)∥Snun-1-Sn-1un-1∥=∥PC(I-λn∇f)-βnI1-βnun-1-PC(I-λn-1∇f)-βn-1I1-βn-1un-1∥=∥4νPC(I-λn∇f)-(2ν-λn)I2ν+λnun-1-4νPC(I-λn-1∇f)-(2ν-λn-1)I2ν+λn-1un-1∥≤∥4νPC(I-λn∇f)2ν+λnun-1-4νPC(I-λn-1∇f)2ν+λn-1un-1∥+∥2ν-λn-12ν+λn-1un-1-2ν-λn2ν+λnun-1∥=∥(4ν(2ν+λn-1)PC(I-λn∇f)un-1-4ν(2ν+λn)PC(I-λn-1∇f)un-1)×((2ν+λn)(2ν+λn-1))-1∥+4ν|λn-λn-1|(2ν+λn)(2ν+λn-1)∥un-1∥≤∥(4ν(λn-1-λn)PC(I-λn∇f)un-1+4ν(2ν+λn)×(PC(I-λn∇f)un-1-PC(I-λn-1∇f)un-1))×((2ν+λn)(2ν+λn-1))-1∥+4ν|λn-λn-1|(2ν+λn)(2ν+λn-1)∥un-1∥≤4ν|λn-1-λn|∥PC(I-λn∇f)un-1∥(2ν+λn)(2ν+λn-1)+4ν∥PC(I-λn∇f)un-1-PC(I-λn-1∇f)un-1∥2ν+λn-1+4ν|λn-λn-1|(2ν+λn)(2ν+λn-1)∥un-1∥≤|λn-λn-1|(1ν∥PC(I-λn∇f)un-1∥+2∥∇f(un-1)∥+1ν∥un-1∥)=4ν|βn-βn-1|(1ν∥PC(I-λn∇f)un-1∥+2∥∇f(un-1)∥+1ν∥un-1∥)≤M1|βn-βn-1|,
where M1=supn{4∥PC(I-λn∇f)un-1∥+8ν∥∇f(un-1)∥+4∥un-1∥}.
Substitute (69) into (68), we get
(70)∥xn+1-xn∥≤(1-αn(τ-γl))∥un-un-1∥+|αn-αn-1|(∥γVun-1∥+∥μASn-1un-1∥)+M1|βn-βn-1|≤(1-αn(τ-γl))∥un-un-1∥+(|αn-αn-1|+|βn-βn-1|)×(∥γVun-1∥+∥μASn-1un-1∥+M1)≤(1-αn(τ-γl))∥un-un-1∥+M2(|αn-αn-1|+|βn-βn-1|),
for some appropriate positive constant M2 such that
(71)M2=supn{∥γVun-1∥+∥μASn-1un-1∥+M1}.
Observe that(72)∥un+1-un∥=∥Θn+1Mxn+1-ΘnMxn∥≤∥Θn+1Mxn+1-Θn+1Mxn∥+∥Θn+1Mxn-ΘnMxn∥≤∥xn+1-xn∥+∥Θn+1Mxn-ΘnMxn∥≤∥xn+1-xn∥+∥TrM,n+1FMΘn+1M-1xn-TrM,nFMΘn+1M-1xn∥+∥TrM,nFMΘn+1M-1xn-TrM,nFMTrM-1,nFM-1Θn+1M-2xn∥+⋯+∥TrM,nFMTrM-1,nFM-1⋯Θn+12xn-TrM,nFM⋯Tr2,nF2Tr1,n+1F1xn∥+∥TrM,nFM⋯Tr2,nF2Tr1,n+1F1xn-un∥≤∥xn+1-xn∥+∥TrM,n+1FMΘn+1M-1xn-TrM,nFMΘn+1M-1xn∥+∥Θn+1M-1xn-TrM-1,nFM-1Θn+1M-2xn∥+⋯+∥Tr2,n+1F2Tr1,n+1F1xn-Tr2,nF2Tr1,n+1F1xn∥+∥Tr1,n+1F1xn-Tr1,nF1xn∥.
By Lemma 10, we get
(73)∥un+1-un∥≤∥xn+1-xn∥+|rM,n+1-rM,n|rM,n+1∥TrM,n+1FMΘn+1M-1xn-Θn+1M-1xn∥+|rM-1,n+1-rM-1,n|rM-1,n+1∥TrM-1,nFM-1Θn+1M-2xn-Θn+1M-2xn∥+⋯+|r2,n+1-r2,n|r2,n+1∥Tr2,n+1F2Tr1,n+1F1xn-Tr1,n+1F1xn∥+|r1,n+1-r1,n|r1,n+1∥Tr1,n+1F1xn-xn∥.
Combing (70) and (73), we have
(74)∥xn+1-xn∥≤(1-αn(τ-γl))∥xn-xn-1∥+|rM,n-rM,n-1|rM,n∥TrM,nFMΘnM-1xn-1-ΘnM-1xn-1∥+|rM-1,n-rM-1,n-1|rM-1,n×∥TrM-1,n-1FM-1ΘnM-2xn-1-ΘnM-2xn-1∥+⋯+|r2,n-r2,n-1|r2,n∥Tr2,nF2Tr1,nF1xn-1-Tr1,nF1xn-1∥+|r1,n-r1,n-1|r1,n∥Tr1,nF1xn-1-xn-1∥+M2(|αn-αn-1|+|βn-βn-1|).
By Lemma 2, It follows from conditions (i)–(iii) that (67) holds. Further from (73), we have
(75)∥un+1-un∥⟶0.
Step  3. It shows that
(76)∥xn-Snxn∥⟶0.
For anyp∈Ω, as the same proof of Theorem 12, we have
(77)∥un-p∥2≤∥xn-p∥2-∥Θnkxn-Θnk-1xn∥2.
Then from (64) and (77), we derive that
(78)∥xn+1-p∥2=∥αnγVun+(I-μαnA)Snun-p∥2=∥(I-μαnA)Snun-(I-μαnA)Snp+αnγVun-αnμAp∥2≤(1-αnτ)2∥un-p∥2+2(1-αnτ)αn∥un-p∥∥γVun-μAp∥+αn2∥γVun-μAp∥2≤∥xn-p∥2-∥Θnkxn-Θnk-1xn∥2+2(1-αnτ)αn∥un-p∥∥γVun-μAp∥+αn2∥γVun-μAp∥2.
From αn→0 and (67), we have
(79)∥Θnkxn-Θnk-1xn∥2⟶0.
Further we have
(80)∥xn-un∥⟶0.
Next,
(81)∥xn-Snxn∥=∥xn-xn+1+xn+1-Snun+Snun-Snxn∥≤∥xn-xn+1∥+∥xn+1-Snun∥+∥Snun-Snxn∥≤∥xn-xn+1∥+αn∥γVun-μASnun∥+∥un-xn∥.
It follows from (67) and (80) that (76) holds. Further we have ∥un-Snun∥→0.
Step  4. It shows that
(82)limsupn→∞〈(γV-μA)x*,xn-x*〉≤0,
where x*=PΩ(I-μA+γV)x* is a unique solution of the variational inequality (27). Indeed, take a subsequence {xnj} of {xn} such that
(83)limsupn→∞〈(γV-μA)x*,xn-x*〉=limj→∞〈(γV-μA)x*,xnj-x*〉.
Since{xnj} is bounded, there exists a subsequence {xnjk} of {xnj} which converges weakly to q. Without loss of generality, we can assume xnj⇀q. By the same argument as in the proof of Theorem 12, we have q∈Ω. Since x*=PΩ(I-μA+γV)x*, it follows that
(84)limsupn→∞〈(γV-μA)x*,xn-x*〉=limj→∞〈(γV-μA)x*,xnj-x*〉=〈(γV-μA)x*,q-x*〉≤0.
Step  5. It shows that
(85)xn⟶x*.
Consider
(86)〈(γV-μA)x*,xn+1-x*〉=〈(γV-μA)x*,xn+1-xn〉+〈(γV-μA)x*,xn-x*〉≤∥(γV-μA)x*∥∥xn+1-xn∥+〈(γV-μA)x*,xn-x*〉.
It follows from (67) and (82) that
(87)limsupn→∞〈(γV-μA)x*,xn+1-x*〉≤0,∥xn+1-x*∥2=∥αnγVun+(I-μαnA)Snun-x*∥2=∥(I-μαnA)Snun-(I-μαnA)x*+αn(γVun-μAx*)∥2≤∥(I-μαnA)Snun-(I-μαnA)x*∥2+2αn〈γVun-μAx*,xn+1-x*〉≤(1-αnτ)2∥un-x*∥2+2αn〈γVun-γVx*,xn+1-x*〉+2αn〈(γV-μA)x*,xn+1-x*〉≤(1-αnτ)2∥xn-x*∥2+αnγl(∥xn-x*∥2+∥xn+1-x*∥2)+2αn〈(γV-μA)x*,xn+1-x*〉.
This implies that
(88)∥xn+1-x*∥2≤(1-αnτ)2+αnγl1-αnγl∥xn-x*∥2+2αn1-αnγl〈(γV-μA)x*,xn+1-x*〉≤(1-2αn(τ-γl)1-αnαγ)∥xn-x*∥2+2αn1-αnγl〈(γV-μA)x*,xn+1-x*〉+(αnτ)21-αnγlM3,
where M3=supn∥xn-x*∥2, n≥1. It is easily to see that γn=((2αn(τ-γl))/(1-αnγl)). Hence by Lemma 2, the sequence {xn} converges strongly to x*.Remark 14.
IfM=1, μ=1, then Theorem 13 reduces to Theorem  3.2 of Tian and Liu [9].


## 4. Numerical Result

In this section, we consider the following simple example to demonstrate the effectiveness, realization and convergence of the algorithm in Theorem13.LetR2 be the two dimensional Euclidean space with usual inner product 〈x(1),x(2)〉=x1(1)x1(2)+x2(1)x2(2), (∀x(1)=(x1(1),x2(1))⊤,x(2)=(x1(2),x2(2))⊤∈R2) and induced norm ∥x∥=x12+x22(∀x=(x1,x2)⊤∈R2). Next, we consider the following simple example.Example 15.
Consider the following constrained convex minimization problem:(89)minimizef(x)=x12+12x22-x1x2-2x1,x∈R2,s.t.x1≥0,x2≥0.
In Theorem 13, let H=R2, C=[0,∞)×[0,∞), and Fk≡0, for  all x,y∈C, k∈{1,2,…,M}. Give A=I, Vx=2x, for all x∈H, with Lipschitz coefficient l=2.
Compute the gradient off, we have ∇f(x)=(2x1-x2-2,x2-x1)⊤ is (1/3)-ism and ∇f2(x)=(2-1-11) is positively definite; thus f is convex. Give the parameters αn=(1/n), βn=(1/4n) for every n≥1, fix μ=1, and γ=(1/10). Then {x(n)} is the sequence generated by
(90)x(n+1)=15nx(n)+n-1nSnx(n),
where
(91)Snx(n)=PC(I-λn∇f)x(n)-βnx(n)1-βn.
As n→∞, we have {x(n)}→x*=(2,2)⊤, f(x*)=-2.Take the initial guessx(1)=(0,0)⊤, we obtain the numerical results as shown in Table 1.Table 1
x
(
1
)
=
(
0,0
)
⊤ (initial guess).
n(iterative number)
x
(
n
)
errors (n)
f
(
n
)
2262
(1.9986, 1.9979)
2.5
×
1
0
-
3
−1.9986
22627
(1.9999, 1.9998)
2.479
×
1
0
-
4
−1.9999
226274
(2.0000, 2.0000)
2.478
×
1
0
-
5
−2.0000



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*Source: 957363-2013-09-23.xml*