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# Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem

- Ming Tian
^{1}Email author and - Lei Liu
^{1}

**2012**:201

https://doi.org/10.1186/1687-1812-2012-201

© Tian and Liu; licensee Springer 2012

**Received: **20 March 2012

**Accepted: **24 October 2012

**Published: **7 November 2012

## Abstract

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this paper. Under suitable conditions, strong convergence theorems are obtained.

**MSC:**46N10, 47J20, 74G60.

## Keywords

- iterative algorithm
- equilibrium problem
- constrained convex minimization
- variational inequality

## 1 Introduction

Let *H* be a real Hilbert space with inner product $\u3008\cdot ,\cdot \u3009$ and norm $\parallel \cdot \parallel $. Let *C* be a nonempty closed convex subset of *H*. Let $T:C\to C$ be a nonexpansive mapping, namely $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $, for all $x,y\in C$. The set of fixed points of *T* is denoted by $F(T)$.

*ϕ*be a bifunction of $C\times C$ into ℝ, where ℝ is the set of real numbers. Consider the equilibrium problem (EP) which is to find $z\in C$ such that

We denoted the set of solutions of EP by $EP(\varphi )$. Given a mapping $F:C\to H$, let $\varphi (x,y)=\u3008Fx,y-x\u3009$ for all $x,y\in C$, then $z\in EP(\varphi )$ if and only if $\u3008Fz,y-z\u3009\ge 0$ for all $y\in C$, that is, *z* is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1–3] and the references therein.

Composite iterative algorithms were proposed by many authors for finding a common solution of an equilibrium problem and a fixed point problem (see [4–18]).

*g*is (

*Fréchet*) differentiable, then the GPA generates a sequence $\{{x}_{n}\}$ using the following recursive formula:

*C*arbitrarily, and the parameters,

*λ*or ${\lambda}_{n}$, are positive real numbers satisfying certain conditions. The convergence of the algorithms (1.3) and (1.4) depends on the behavior of the gradient ∇

*g*. As a matter of fact, it is known that if ∇

*g*is

*α*-strongly monotone and

*L*-Lipschitzian with constants $\alpha ,L\ge 0$, then the operator

is a contraction; hence the sequence $\{{x}_{n}\}$ defined by the algorithm (1.3) converges in norm to the unique minimizer of (1.2). However, if the gradient ∇*g* fails to be strongly monotone, the operator *W* defined by (1.5) would fail to be contractive; consequently, the sequence $\{{x}_{n}\}$ generated by the algorithm (1.3) may fail to converge strongly (see [19]). If ∇*g* is Lipschitzian, then the algorithms (1.3) and (1.4) can still converge in the weak topology under certain conditions.

Recently, Xu [19] proposed an explicit operator-oriented approach to the algorithm (1.4); that is, an averaged mapping approach. He gave his averaged mapping approach to the GPA (1.4) and the relaxed gradient-projection algorithm. Moreover, he constructed a counterexample which shows that the algorithm (1.3) does not converge in norm in an infinite-dimensional space and also presented two modifications of GPA which are shown to have strong convergence [20, 21].

*et al.*[22] proposed the following explicit iterative scheme:

where ${s}_{n}=\frac{2-{\lambda}_{n}L}{4}$ and ${P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1-{s}_{n}){T}_{n}$ for each $n\ge 0$. He proved that the sequences $\{{x}_{n}\}$ converge strongly to a minimizer of the constrained convex minimization problem, which also solves a certain variational inequality.

*f*be a contraction on

*H*, starting with an arbitrary initial ${x}_{0}\in H$, define a sequence $\{{x}_{n}\}$ recursively by

The purpose of the paper is to study the iterative method for finding the common solution of an equilibrium problem and a constrained convex minimization problem. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative method for finding the common element of the set of solutions of an equilibrium problem and the solution set of a constrained convex minimization problem. We also prove some strong convergence theorems.

## 2 Preliminaries

Throughout this paper, we always assume that *C* is a nonempty closed convex subset of a Hilbert space *H*. We use ‘⇀’ for weak convergence and ‘→’ for strong convergence.

*H*satisfies Opial’s condition [25]; that is, for any sequence $\{{x}_{n}\}$ with ${x}_{n}\rightharpoonup x$, the inequality

holds for every $y\in H$ with $y\ne x$.

In order to solve the equilibrium problem for a bifunction $\varphi :C\times C\to \mathbb{R}$, let us assume that *ϕ* satisfies the following conditions:

(A1) $\varphi (x,x)=0$, for all $x\in C$;

(A2) *ϕ* is monotone, that is, $\varphi (x,y)+\varphi (y,x)\le 0$ for all $x,y\in C$;

(A3) for all $x,y,z\in C$, ${lim}_{t\downarrow 0}\varphi (tz+(1-t)x,y)\le \varphi (x,y)$;

(A4) for each fixed $x\in C$, the function $y\mapsto \varphi (x,y)$ is convex and lower semicontinuous.

Let us recall the following lemmas which will be useful for our paper.

**Lemma 2.1** [26]

*Let*

*ϕ*

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying*(A1), (A2), (A3),

*and*(A4),

*then for any*$r>0$

*and*$x\in H$,

*there exists*$z\in C$

*such that*

*Further*,

*if*

*then the following hold*:

- (1)
${Q}_{r}$

*is single*-*valued*; - (2)${Q}_{r}$
*is firmly nonexpansive*;*that is*,${\parallel {Q}_{r}x-{Q}_{r}y\parallel}^{2}\le \u3008{Q}_{r}x-{Q}_{r}y,x-y\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in H;$ - (3)
$F({Q}_{r})=EP(\varphi )$;

- (4)
$EP(\varphi )$

*is closed and convex*.

**Definition 2.1**A mapping $T:H\to H$ is said to be firmly nonexpansive if and only if $2T-I$ is nonexpansive, or equivalently,

*T*is firmly nonexpansive if and only if

*T*can be expressed as

where $S:H\to H$ is nonexpansive. Obviously, projections are firmly nonexpansive.

**Definition 2.2**A mapping $T:H\to H$ is said to be an

*averaged mapping*if it can be written as the average of the identity

*I*and a nonexpansive mapping; that is,

where $\alpha \in (0,1)$ and $S:H\to H$ is nonexpansive. More precisely, when (2.1) holds, we say that *T* is *α*-averaged.

Clearly, a firmly nonexpansive mapping is a $\frac{1}{2}$-averaged map.

**Proposition 2.1** [27]

*For given operators*$S,T,V:H\to H$:

- (i)
*If*$T=(1-\alpha )S+\alpha V$*for some*$\alpha \in (0,1)$*and if**U**is averaged and V is nonexpansive*,*then T is averaged*. - (ii)
*T is firmly nonexpansive if and only if the complement I*-*T is firmly nonexpansive*. - (iii)
*If*$T=(1-\alpha )S+\alpha V$*for some*$\alpha \in (0,1)$,*U**is firmly nonexpansive and V is nonexpansive*,*then T is averaged*. - (iv)
*The composite of finitely many averaged mappings is averaged*.*That is*,*if each of the mappings*${\{{T}_{i}\}}_{i=1}^{N}$*is averaged*,*then so is the composite*${T}_{1}\cdots {T}_{N}$.*In particular*,*if*${T}_{1}$*is*${\alpha}_{1}$-*averaged*,*and*${T}_{2}$*is*${\alpha}_{2}$-*averaged*,*where*${\alpha}_{1},{\alpha}_{2}\in (0,1)$,*then the composite*${T}_{1}{T}_{2}$*is**α*-*averaged*,*where*$\alpha ={\alpha}_{1}+{\alpha}_{2}-{\alpha}_{1}{\alpha}_{2}$.

*H*onto

*C*is the mapping ${P}_{C}:H\to C$ which assigns, to each point $x\in H$, the unique point ${P}_{C}x\in C$ satisfying the property

**Lemma 2.2**

*For a given*$x\in H$:

- (a)
$z={P}_{C}x$

*if and only if*$\u3008x-z,y-z\u3009\le 0$, $\mathrm{\forall}y\in C$. - (b)
$z={P}_{C}x$

*if and only if*${\parallel x-z\parallel}^{2}\le {\parallel x-y\parallel}^{2}-{\parallel y-z\parallel}^{2}$, $\mathrm{\forall}y\in C$. - (c)
$\u3008{P}_{C}x-{P}_{C}y,x-y\u3009\ge {\parallel {P}_{C}x-{P}_{C}y\parallel}^{2}$, $\mathrm{\forall}x,y\in H$.

*Consequently*, ${P}_{C}$ *is nonexpansive and monotone*.

**Lemma 2.3**

*The following inequality holds in an inner product space X*:

The so-called demiclosedness principle for nonexpansive mappings will be used.

**Lemma 2.4** (Demiclosedness principle [28])

*Let* $T:C\to C$ *be a nonexpansive mapping with* $Fix(T)\ne \mathrm{\varnothing}$. *If* $\{{x}_{n}\}$ *is a sequence in* *C* *that converges weakly to* *x* *and if* $\{(I-T){x}_{n}\}$ *converges strongly to* *y*, *then* $(I-T)x=y$. *In particular*, *if* $y=0$, *then* $x\in Fix(T)$.

Next, we introduce monotonicity of a nonlinear operator.

**Definition 2.3**A nonlinear operator

*G*whose domain $D(G)\subseteq H$ and range $R(G)\subseteq H$ is said to be:

- (a)monotone if$\u3008x-y,Gx-Gy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G),$
- (b)
*β*-strongly monotone if there exists $\beta >0$ such that$\u3008x-y,Gx-Gy\u3009\ge \beta {\parallel x-y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G),$ - (c)
*ν*-inverse strongly monotone (for short,*ν*-ism) if there exists $\nu >0$ such that$\u3008x-y,Gx-Gy\u3009\ge \nu {\parallel Gx-Gy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in D(G).$

It can be easily seen that if *G* is nonexpansive, then $I-G$ is monotone; and the projection map ${P}_{C}$ is a 1-ism.

The inverse strongly monotone (also referred to as co-coercive) operators have been widely used to solve practical problems in various fields, for instance, in traffic assignment problems; see, for example, [29, 30] and reference therein.

The following proposition summarizes some results on the relationship between averaged mappings and inverse strongly monotone operators.

**Proposition 2.2** [27]

*Let*$T:H\to H$

*be an operator from H to itself*.

- (a)
*T is nonexpansive if and only if the complement*$I-T$*is*$\frac{1}{2}$-*ism*. - (b)
*If T is**ν*-*ism*,*then for*$\gamma >0$,*γT**is*$\frac{\nu}{\gamma}$-*ism*. - (c)
*T is averaged if and only if the complement*$I-T$*is**ν*-*ism for some*$\nu >\frac{1}{2}$.*Indeed*,*for*$\alpha \in (0,1)$,*T is**α*-*averaged if and only if*$I-T$*is*$\frac{1}{2\alpha}$-*ism*.

**Lemma 2.5** [24]

*Let*$\{{a}_{n}\}$

*be a sequence of nonnegative numbers satisfying the condition*

*where*$\{{\gamma}_{n}\}$, $\{{\delta}_{n}\}$

*are sequences of real numbers such that*:

- (i)
$\{{\gamma}_{n}\}\subset (0,1)$

*and*${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}$, - (ii)
${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\delta \le 0$

*or*${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}|{\delta}_{n}|<\mathrm{\infty}$.

*Then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.

## 3 Main results

In this paper, we always assume that $g:C\to \mathbb{R}$ is a real-valued convex function and ∇*g* is an *L*-Lipschitzian mapping with $L\ge 0$. Since the Lipschitz continuity of ∇*g* implies that it is indeed inverse strongly monotone, its complement can be an averaged mapping. Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that an averaged mapping plays an important role in the gradient-projection algorithm.

*g*is

*L*-Lipschitzian. This implies that ∇

*g*is ($1/L$)-ism, which then implies that $\lambda \mathrm{\nabla}g$ is ($1/\lambda L$)-ism. So, by Proposition 2.2, $I-\lambda \mathrm{\nabla}g$ is ($\lambda L/2$)-averaged. Now since the projection ${P}_{C}$ is (1/2)-averaged, we see from Proposition 2.1 that the composite ${P}_{C}(I-\lambda \mathrm{\nabla}g)$ is ($(2+\lambda L)/4$)-averaged for $0<\lambda <2/L$. Hence, we have that for each

*n*, ${P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g)$ is ($(2+{\lambda}_{n}L)/4$)-averaged. Therefore, we can write

where ${T}_{n}$ is nonexpansive and ${s}_{n}=\frac{2-{\lambda}_{n}L}{4}$.

*U*denote its solution set. Let $\{{Q}_{{\beta}_{n}}\}$ be a sequence of mappings defined as in Lemma 2.1. Consider the following mapping ${G}_{n}$ on

*C*defined by

Equivalently, $q={P}_{U\cap EP(\varphi )}f(q)$.

**Theorem 3.1**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and*

*ϕ*

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying*(A1), (A2), (A3),

*and*(A4).

*Let*$g:C\to \mathbb{R}$

*be a real*-

*valued convex function*,

*and assume that*∇

*g*

*is an*

*L*-

*Lipschitzian mapping with*$L\ge 0$

*and*$f:C\to C$

*is a contraction with the constant*$\rho \in (0,1)$.

*Assume that*$U\cap EP(\varphi )\ne \mathrm{\varnothing}$.

*Let*$\{{x}_{n}\}$

*be a sequence generated by*

*where*${u}_{n}={Q}_{{\beta}_{n}}{x}_{n}$, ${P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1-{s}_{n}){T}_{n}$, ${s}_{n}=\frac{2-{\lambda}_{n}L}{4}$

*and*$\{{\lambda}_{n}\}\subset (0,\frac{2}{L})$.

*Let*$\{{\beta}_{n}\}$

*and*$\{{\alpha}_{n}\}$

*satisfy the following conditions*:

- (i)
$\{{\beta}_{n}\}\subset (0,\mathrm{\infty})$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}>0$;

- (ii)
$\{{\alpha}_{n}\}\subset (0,1)$, ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$.

*Then* $\{{x}_{n}\}$ *converges strongly*, *as* ${s}_{n}\to 0$ ($\iff {\lambda}_{n}\to \frac{2}{L}$), *to a point* $q\in U\cap EP(\varphi )$ *which solves the variational inequality* (3.1).

*Proof*First, we claim that $\{{x}_{n}\}$ is bounded. Indeed, pick any $p\in U\cap EP(\varphi )$, since ${u}_{n}={Q}_{{\beta}_{n}}{x}_{n}$ and $p={Q}_{{\beta}_{n}}p$, then we know that for any $n\in \mathbb{N}$,

and hence $\{{x}_{n}\}$ is bounded. From (3.2), we also derive that $\{{u}_{n}\}$ is bounded.

*g*is

*L*-Lipschitzian, ∇

*g*is $\frac{1}{L}$-ism. Consequently, ${P}_{C}(I-\frac{2}{L}\mathrm{\nabla}g)$ is a nonexpansive self-mapping on

*C*. As a matter of fact, we have for each $x,y\in C$

*q*. Next, we show that $q\in U\cap EP(\varphi )$. Without loss of generality, we can assume that ${u}_{{n}_{i}}\rightharpoonup q$. Then, by Lemma 2.4, we obtain

This shows that $q\in U$.

*n*by ${n}_{i}$, we have

and hence $0\le \varphi ({z}_{t},y)$. From (A3), we have $0\le \varphi (q,y)$ for all $y\in C$ and hence $q\in EP(\varphi )$. Therefore, $q\in EP(\varphi )\cap U$.

Since ${x}_{{n}_{i}}\rightharpoonup q$, it follows from (3.4) that ${x}_{{n}_{i}}\to q$ as $i\to \mathrm{\infty}$.

*q*solves the variational inequality (3.1). Observe that

*n*with ${n}_{i}$ in the above inequality, and letting $i\to \mathrm{\infty}$, we have

This completes the proof. □

**Theorem 3.2**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and*

*ϕ*

*be a bifunction from*$C\times C$

*into*ℝ

*satisfying*(A1), (A2), (A3),

*and*(A4).

*Let*$g:C\to \mathbb{R}$

*be a real*-

*valued convex function*,

*and assume that*∇

*g*

*is an*

*L*-

*Lipschitzian mapping with*$L\ge 0$

*and*$f:C\to C$

*is a contraction with the constant*$\rho \in (0,1)$.

*Assume that*$U\cap EP(\varphi )\ne \mathrm{\varnothing}$.

*Let*$\{{x}_{n}\}$

*be a sequence generated by*${x}_{1}\in C$

*and*

*where*${u}_{n}={Q}_{{\beta}_{n}}{x}_{n}$, ${P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g)={s}_{n}I+(1-{s}_{n}){T}_{n}$, ${s}_{n}=\frac{2-{\lambda}_{n}L}{4}$

*and*$\{{\lambda}_{n}\}\subset (0,\frac{2}{L})$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$

*and*$\{{s}_{n}\}$

*satisfy the following conditions*:

- (i)
$\{{\beta}_{n}\}\subset (0,\mathrm{\infty})$, $lim\hspace{0.17em}inf{\beta}_{n}>0$, ${\sum}_{n=1}^{\mathrm{\infty}}|{\beta}_{n+1}-{\beta}_{n}|<\mathrm{\infty}$;

- (ii)
$\{{\alpha}_{n}\}\subset (0,1)$, ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, ${\sum}_{n=1}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$;

- (iii)
$\{{s}_{n}\}\subset (0,\frac{1}{2})$, ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$ ($\iff {lim}_{n\to \mathrm{\infty}}{\lambda}_{n}=\frac{2}{L}$), ${\sum}_{n=1}^{\mathrm{\infty}}|{s}_{n+1}-{s}_{n}|<\mathrm{\infty}$.

*Then* $\{{x}_{n}\}$ *converges strongly to a point* $q\in U\cap EP(\varphi )$ *which solves the variational inequality* (3.1).

*Proof*First, we show that $\{{x}_{n}\}$ is bounded. Indeed, pick any $p\in U\cap EP(\varphi )$, since ${u}_{n}={Q}_{{\beta}_{n}}{x}_{n}$ and $p={Q}_{{\beta}_{n}}p$, then we know that for any $n\in \mathbb{N}$,

and hence $\{{x}_{n}\}$ is bounded. From (3.5), we also derive that $\{{u}_{n}\}$ is bounded.

*g*is $\frac{1}{L}$-ism, ${P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g)$ is nonexpansive. It follows that for any given $p\in S$,

This together with the boundedness of $\{{u}_{n}\}$ implies that $\{{P}_{C}(I-{\lambda}_{n}\mathrm{\nabla}g){u}_{n-1}\}$ is bounded.

*a*such that ${\beta}_{n}>a>0$ for all $n\in \mathbb{N}$. Thus, we have

where ${M}_{3}=sup\{\parallel {u}_{n}-{x}_{n}\parallel :n\in \mathbb{N}\}$.

Then, $\parallel {x}_{n}-{T}_{n}{x}_{n}\parallel \to 0$, it follows that $\parallel {u}_{n}-{T}_{n}{u}_{n}\parallel \to 0$.

Since $\{{x}_{n}\}$ is bounded, without loss of generality, we may assume that ${x}_{{n}_{k}}\rightharpoonup \tilde{x}$. By the same argument as in the proof of Theorem 3.1, we have $\tilde{x}\in U\cap EP(\varphi )$.

where ${M}^{\ast}=sup\{{\parallel {x}_{n}-q\parallel}^{2}:n\in \mathbb{N}\}$, and ${\delta}_{n}=\frac{{\alpha}_{n}}{2(1-\rho )(1-{\alpha}_{n}\rho )}{M}^{\ast}+\frac{1}{(1-\rho )(1-{\alpha}_{n}\rho )}\u3008-(I-f)q,{x}_{n+1}-q\u3009$.

It is easy to see that ${lim}_{n\to \mathrm{\infty}}2(1-\rho ){\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}2(1-\rho ){\alpha}_{n}=\mathrm{\infty}$, and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0$ by (3.12). Hence, by Lemma 2.5, the sequence $\{{x}_{n}\}$ converges strongly to *q*. This completes the proof. □

## 4 Conclusions

Methods for solving the equilibrium problem and the constrained convex minimization problem have extensively been studied respectively in a Hilbert space. But to the best of our knowledge, it would probably be the first time in the literature that we introduce implicit and explicit algorithms for finding the common element of the set of solutions of an equilibrium problem and the set of solutions of a constrained convex minimization problem, which also solves a certain variational inequality.

## Declarations

### Acknowledgements

The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012K001), and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).

## Authors’ Affiliations

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