Journal of Applied Mathematics

Volume 2013 (2013), Article ID 957363, 11 pages

http://dx.doi.org/10.1155/2013/957363

Research Article

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

Peichao Duan

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 29 December 2012; Accepted 12 July 2013

Academic Editor: Luigi Muglia

Copyright © 2013 Peichao Duan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of .

Let be a countable family of bifunctions from to , where is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems which is to find such that

where is an arbitrary index set. If is a singleton, then problem (1) becomes the following equilibrium problem:

The solution set of (2) is denoted by .

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:

where 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 is (Fréchet) differentiable, then the GPA generates a sequence using the following recursive formula:

or more generally,

where in both (4) and (5) the initial guess is taken from arbitrarily, and the parameters, or , are positive real numbers satisfying certain conditions. The convergence of the algorithms (4) and (5) depends on the behavior of the gradient . As a matter of fact, it is known that if is -strongly monotone and -Lipschitzian with constants , , then the operator

is a contraction; hence, the sequence defined by the algorithm (4) converges in norm to the unique minimizer of (3). However, if the gradient fails to be strongly monotone, the operator by (6) would fail to be contractive; consequently, the sequence generated by the algorithm (4) may fail to converge strongly [11]. If 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

where 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 which are listed as below.

A mapping of is said to be a nonexpansive mapping such that

for all . The set of fixed points of is denoted by ; that is, .

A mapping is said to be an averaged mapping if it can be written as the average of an identity and a nonexpansive mapping; that is,

where is a number in and is nonexpansive. More precisely, we say that is -averaged. It is known that the projection is -averaged.

Lemma 1. Let be a real Hilbert space. There hold the following identities:

(i), ,(ii), , .

Lemma 2 (see [10]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that

(i); (ii)or

Then, .

Recall that given a nonempty closed convex subset of a real Hilbert space , for any , there exists a unique nearest point in , denoted by , such that

for all . Such a is called the metric (or the nearest point) projection of onto . As we all know, if and only if there holds the relation:

Lemma 3 (see [13]). Let be an -Lipschitzian and -strongly monotone operator on a Hilbert space with , , and . Then is a contraction with contractive coefficient and .

Definition 4. A nonlinear mapping whose domain and range is said to be

(i)monotone if

(ii)-strongly monotone if there exists such that

(iii)-inverse strongly monotone (for short, -ism) if there exists a constant such that

Lemma 5. Let be an -Lipschitz mapping with coefficient and a strong positive bounded linear operator with . Then for ,

that is, is strongly monotone with coefficient .

Proof. Since is a strong positive bounded linear operator with . We have

Hence is -Lipschitz and -strongly monotone:

Proposition 6. For given operators .

(i)If for some and if is averaged and is nonexpansive, then is averaged.(ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive.(iii)If for some , is firmly nonexpansive, and is nonexpansive, then is averaged.(iv)The composite of finitely many averaged mappings is averaged; that is, if each of the mapping is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .

Proposition 7. Let be an operator from to itself.

(i) is nonexpansive if and only if the complement is -ism. (ii)If is -ism, then for , is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed, for , is -averaged if and only if the complement is -ism.

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions.

(A1) for all . (A2)  is monotone; that is, for any ;(A3) For each , .(A4)  is convex and lower semicontinuous for each .

We recall some lemmas which will be needed in the rest of this paper.

Lemma 8 (see [14]). Let be a nonempty closed convex subset of , let be bifunction from to satisfying (A1)–(A4), and let and . Then there exists such that

Lemma 9 (see [1]). For , , define a mapping as follows:

for all . Then, the following statements hold:

(i) is single-valued; (ii) is firmly nonexpansive; that is, for any ,

(iii); (iv) is closed and convex.

Lemma 10 (see [4]). Let , and be as in Lemma 9. Then the following holds:

for all and .

Lemma 11 (see [13]). Let be a Hilbert space, a nonempty closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then .

3. Main Result

Throughout the rest of this paper, we always assume that is an -Lipschitzian mapping with coefficient , and is a strongly positive bounded linear operator with coefficient . Then we obtain that is -Lipschitzian and -strongly monotone. Let be a real-valued convex function and assume that is -ism with , which then implies that is -ism. So by Proposition 7, its complement is -averaged. Since the projection is -averaged, we obtain from Proposition 6 that the composition is -averaged for . Hence we have that, for each , is -averaged. Therefore, we can write

where is nonexpansive.

Suppose that the minimization problem (3) is consistent and let denote its solution set. Assume that and .

Denote for every and for all . Define a mapping . Since both and are nonexpansive, it is easy to get that is also nonexpansive. Consider the following mapping on defined by

where . By Lemmas 3 and 9, we have

Since , it follows that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed pointed such that

For simplicity, we will write for provided no confusion occurs. Next we prove the sequences converges strongly to a point which solves the variational inequality

Equivalently, .

Theorem 12. Let be a nonempty closed convex subset of a real Hilbert space with , and let be bifunctions from to satisfying (A1)–(A4). Let be a real-value convex function and -ism with . Assume the set . Let be an -Lipschitzian mapping with and a strongly positive bounded linear operator with coefficient , and . Let {} and {} be sequences generated by the following algorithm:

where , , and ; if and satisfy the following conditions:

(i), ; (ii), for ,

then, as , the sequence converges strongly to a point , which solves the variational inequality (27).

Proof. The proof is divided into several steps.

Step  1. It shows first that is bounded.

First, since , we can assume that . By Lemma 3, we have .

Take any , since for each , is nonexpansive, , and ; we have

for all .

Thus, by (28) and Lemma 3, we derive that

It follows that .

Hence, is bounded and so . It follows from the Lipschitz continuity of , , and that , , and are also bounded. From the nonexpansivity of , it follows that is also bounded.

Step  2. It shows that

Next we will show that

Indeed, for , it follows from the firmly nonexpansivity of that for each , we have

Thus we get

which implies that for each ,

Thus, from Lemma 1 and (35), we get

It follows that

Since , (32) holds, then we have

Step  3. It shows that

Observe that

Since and , it is easy to get (39).

Thus,

We obtain .

Notice that

where . Hence we have

From the boundedness of , and , we conclude that

Since is bounded, there exists a subsequence which converges weakly to .

Step  4. It shows that .

Since is closed and convex, is weakly closed. So we have . By Lemma 11 and (44), we have .

Next we will show that .

Indeed, by Lemma 9, we have that for each ,

From (A2), we get

Hence,

From (32), we obtain that as for each (especially, ). Together with (32), condition (ii), and (A4) we have, for each , that

For any, and , let . Since and , we obtain that , and hence . So, we have

Dividing by , we get, for each , that

Letting and from (A3), we get

For all and for each ; that is, . Hence .

Step  5. It shows that , where

Hence, we obtain

It follows that

This implies that

In particular,

Since , it follows from (56) that as . Next, we show that solves the variational inequality (27).

By the iterative algorithm (28), we have

Therefore we have

that is,

Due to the nonexpansivity of , we have that is monotone; that is, , for  all . Hence for any ,

Now replacing in (60) with and letting , we obtain

that is, is a solution of the variational inequality (27).

Further, by the uniqueness of the solution of the variational inequality (27), we conclude that as . We rewrite (27) as

This is equivalent to the fixed point equation

Theorem 13. Let be a nonempty closed convex subset of a real Hilbert space with , and let be bifunctions from to which satisfies conditions (A1)–(A4). Let be a real-value convex function and a -ism mapping with . Assume the set . Let is an -Lipschitzian mapping with and is a strongly positive bounded linear operator with coefficient , and . Given , let {} and {} be sequences generated by the following algorithm:

where , , and ; if , , and satisfy the following conditions:

(i), and ; (ii), and for ; (iii), and ,

then, the sequence converges strongly to a point , which solves the variational inequality (27).

Proof. The proof is divided into several steps.

Step  1. It shows first that is bounded.

Take any , we have

Thus, by (64), we derive that

By induction, we obtain , , . Hence, is bounded and so . It follows from the Lipschitz continuity of , , and that , , and are also bounded. From the nonexpansivity of , it follows that is also bounded.

Step  2. It shows that

By (68), we have

Next we estimate .

Observe that

where .

Substitute (69) into (68), we get

for some appropriate positive constant such that

Observe that

By Lemma 10, we get

Combing (70) and (73), we have

By Lemma 2, It follows from conditions (i)–(iii) that (67) holds. Further from (73), we have

Step  3. It shows that

For any , as the same proof of Theorem 12, we have

Then from (64) and (77), we derive that

From and (67), we have

Further we have

Next,

It follows from (67) and (80) that (76) holds. Further we have .

Step  4. It shows that

where is a unique solution of the variational inequality (27). Indeed, take a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume . By the same argument as in the proof of Theorem 12, we have . Since , it follows that

Step  5. It shows that

Consider

It follows from (67) and (82) that

This implies that

where , . It is easily to see that . Hence by Lemma 2, the sequence converges strongly to .

Remark 14. If , , 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 Theorem 13.

Let be the two dimensional Euclidean space with usual inner product , and induced norm . Next, we consider the following simple example.

Example 15. Consider the following constrained convex minimization problem:

In Theorem 13, let , , and , for  all , . Give , , for all , with Lipschitz coefficient .

Compute the gradient of , we have is -ism and is positively definite; thus is convex. Give the parameters , for every , fix , and . Then is the sequence generated by

where

As , we have , .

Take the initial guess , we obtain the numerical results as shown in Table 1.

Table 1: (initial guess).

n  (iterative number)

errors (n)

2262

(1.9986, 1.9979)

−1.9986

22627

(1.9999, 1.9998)

−1.9999

226274

(2.0000, 2.0000)

−2.0000

Acknowledgments

The author would like to thank the referee for valuable suggestions to improve the paper and the Fundamental Research Funds for the Central Universities (Grant: 3122013k004).

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