Research

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

Ming Tian* and Lei Liu

Author Affiliations

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

For all author emails, please log on.

Fixed Point Theory and Applications 2012, 2012:201  doi:10.1186/1687-1812-2012-201

 Received: 20 March 2012 Accepted: 24 October 2012 Published: 7 November 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 and norm . Let C be a nonempty closed convex subset of H. Let be a nonexpansive mapping, namely , for all . The set of fixed points of T is denoted by .

Let ϕ be a bifunction of into ℝ, where ℝ is the set of real numbers. Consider the equilibrium problem (EP) which is to find such that

(1.1)

We denoted the set of solutions of EP by . Given a mapping , let for all , then if and only if for all , 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]).

On the other hand, consider the constrained convex minimization problem as follows:

(1.2)

where is a real-valued convex function. It is well known that the gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. If g is (Fréchet) differentiable, then the GPA generates a sequence using the following recursive formula:

(1.3)

or more generally,

(1.4)

where in both (1.3) and (1.4) the initial guess is taken from C arbitrarily, and the parameters, λ or , 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 , then the operator

(1.5)

is a contraction; hence the sequence 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 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].

In 2011, Ceng et al.[22] proposed the following explicit iterative scheme:

where and for each . He proved that the sequences converge strongly to a minimizer of the constrained convex minimization problem, which also solves a certain variational inequality.

In 2000, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings, extended in [23]. Let f be a contraction on H, starting with an arbitrary initial , define a sequence recursively by

(1.6)

where is a sequence in . Xu [24] proved that if satisfies certain conditions, the sequence generated by (1.6) converges strongly to the unique solution of the variational inequality

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.

It is widely known that H satisfies Opial’s condition [25]; that is, for any sequence with , the inequality

holds for every with .

In order to solve the equilibrium problem for a bifunction , let us assume that ϕ satisfies the following conditions:

(A1) , for all ;

(A2) ϕ is monotone, that is, for all ;

(A3) for all , ;

(A4) for each fixed , the function is convex and lower semicontinuous.

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

Lemma 2.1[26]

Letϕbe a bifunction fromintosatisfying (A1), (A2), (A3), and (A4), then for anyand, there existssuch that

Further, if

then the following hold:

(1) is single-valued;

(2) is firmly nonexpansive; that is,

(3) ;

(4) is closed and convex.

Definition 2.1 A mapping is said to be firmly nonexpansive if and only if is nonexpansive, or equivalently,

Alternatively, T is firmly nonexpansive if and only if T can be expressed as

where is nonexpansive. Obviously, projections are firmly nonexpansive.

Definition 2.2 A mapping 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,

(2.1)

where and is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.

Clearly, a firmly nonexpansive mapping is a -averaged map.

Proposition 2.1[27]

For given operators:

(i) Iffor someand ifUis 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) Iffor some, Uis 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 mappingsis averaged, then so is the composite. In particular, ifis-averaged, andis-averaged, where, then the compositeisα-averaged, where.

Recall that the metric projection from H onto C is the mapping which assigns, to each point , the unique point satisfying the property

Lemma 2.2For a given:

(a) if and only if, .

(b) if and only if, .

(c) , .

Consequently, is nonexpansive and monotone.

Lemma 2.3The 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])

Letbe a nonexpansive mapping with. Ifis a sequence inCthat converges weakly toxand ifconverges strongly toy, then. In particular, if, then.

Next, we introduce monotonicity of a nonlinear operator.

Definition 2.3 A nonlinear operator G whose domain and range is said to be:

(a) monotone if

(b) β-strongly monotone if there exists such that

(c) ν-inverse strongly monotone (for short, ν-ism) if there exists such that

It can be easily seen that if G is nonexpansive, then is monotone; and the projection map 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]

Letbe an operator from H to itself.

(a) T is nonexpansive if and only if the complementis-ism.

(b) If T isν-ism, then for, γTis-ism.

(c) T is averaged if and only if the complementisν-ism for some. Indeed, for, T isα-averaged if and only ifis-ism.

Lemma 2.5[24]

Letbe a sequence of nonnegative numbers satisfying the condition

where, are sequences of real numbers such that:

(i) and,

(ii) or.

Then.

### 3 Main results

In this paper, we always assume that is a real-valued convex function and ∇g is an L-Lipschitzian mapping with . 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.

Note that ∇g is L-Lipschitzian. This implies that ∇g is ()-ism, which then implies that is ()-ism. So, by Proposition 2.2, is ()-averaged. Now since the projection is (1/2)-averaged, we see from Proposition 2.1 that the composite is ()-averaged for . Hence, we have that for each n, is ()-averaged. Therefore, we can write

where is nonexpansive and .

Let be a contraction with the constant . Suppose that the minimization problem (1.2) is consistent, and let U denote its solution set. Let be a sequence of mappings defined as in Lemma 2.1. Consider the following mapping on C defined by

where . By Lemma 2.1, we have

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

For simplicity, we will write for provided no confusion occurs. Next, we prove the convergence of , while we claim the existence of the , which solves the variational inequality

(3.1)

Equivalently, .

Theorem 3.1LetCbe a nonempty closed convex subset of a real Hilbert spaceHandϕbe a bifunction fromintosatisfying (A1), (A2), (A3), and (A4). Letbe a real-valued convex function, and assume thatgis anL-Lipschitzian mapping withandis a contraction with the constant. Assume that. Letbe a sequence generated by

where, , and. Letandsatisfy the following conditions:

(i) , ;

(ii) , .

Thenconverges strongly, as (), to a pointwhich solves the variational inequality (3.1).

Proof First, we claim that is bounded. Indeed, pick any , since and , then we know that for any ,

(3.2)

Thus, we derive that (noting and is nonexpansive)

Then we have

and hence is bounded. From (3.2), we also derive that is bounded.

Next, we claim that . Indeed, for any , by Lemma 2.1, we have

This implies that

(3.3)

Then from (3.3), we derive that

Since , it follows that

Then we show that . Indeed,

Since and , we obtain that

Thus,

and

we have

Observe that

where . Hence, we have

From the boundedness of , () and , we conclude that

Since ∇g is L-Lipschitzian, ∇g is -ism. Consequently, is a nonexpansive self-mapping on C. As a matter of fact, we have for each

Consider a subsequence of . Since is bounded, there exists a subsequence of which converges weakly to q. Next, we show that . Without loss of generality, we can assume that . Then, by Lemma 2.4, we obtain

This shows that .

Next, we show that . Since , for any , we obtain

From (A2), we have

Replacing n by , we have

Since and , it follows from (A4) that for all . Let

then we have and hence . Thus, from (A1) and (A4), we have

and hence . From (A3), we have for all and hence . Therefore, .

On the other hand, we note that

Hence, we obtain

It follows that

In particular,

(3.4)

Since , it follows from (3.4) that as .

Next, we show that q solves the variational inequality (3.1). Observe that

Hence, we conclude that

Since is nonexpansive, we have that is monotone. Note that for any given ,

Now, replacing n with in the above inequality, and letting , we have

From the arbitrariness of , it follows that is a solution of the variational inequality (3.1). Further, by the uniqueness of solution of the variational inequality (3.1), we conclude that as . The variational inequality (3.1) can be written as

So, in terms of Lemma 2.2, it is equivalent to the following equality:

This completes the proof. □

Theorem 3.2LetCbe a nonempty closed convex subset of a real Hilbert spaceHandϕbe a bifunction fromintosatisfying (A1), (A2), (A3), and (A4). Letbe a real-valued convex function, and assume thatgis anL-Lipschitzian mapping withandis a contraction with the constant. Assume that. Letbe a sequence generated byand

where, , and. Let, andsatisfy the following conditions:

(i) , , ;

(ii) , , , ;

(iii) , (), .

Thenconverges strongly to a pointwhich solves the variational inequality (3.1).

Proof First, we show that is bounded. Indeed, pick any , since and , then we know that for any ,

(3.5)

Thus, we derive that (noting and is nonexpansive)

By induction, we have

and hence is bounded. From (3.5), we also derive that is bounded.

Next, we show that . Indeed, since ∇g is -ism, is nonexpansive. It follows that for any given ,

This together with the boundedness of implies that is bounded.

Also, observe that

for some appropriate constant such that

Thus, we get

(3.6)

for some appropriate constant such that

From and , we note that

(3.7)

and

(3.8)

Putting in (3.7) and in (3.8), we have

and

So, from (A2), we have

and hence

Since , without loss of generality, let us assume that there exists a real number a such that for all . Thus, we have

thus,

(3.9)

where .

From (3.6) and (3.9), we obtain

where . Hence, by Lemma 2.5, we have

(3.10)

Then, from (3.9) and (3.10), and , we have

For any , as in the proof of Theorem 3.1, we have

(3.11)

Then from (3.11), we derive that

Since and , we have

Next, we have

Then, , it follows that .

Now, we show that

where is a unique solution of the variational inequality (3.1). Indeed, take a subsequence of such that

Since is bounded, without loss of generality, we may assume that . By the same argument as in the proof of Theorem 3.1, we have .

Since , it follows that

(3.12)

From

we have

This implies that

Then, we have

where , and .

It is easy to see that , , and by (3.12). Hence, by Lemma 2.5, the sequence 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.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors read and approved the final manuscript.

### 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).

### References

1. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc.. 4, 506–510 (1953). Publisher Full Text

2. Moudafi, A: Viscosity approximation method for fixed-points problems. J. Math. Anal. Appl.. 241, 46–55 (2000). Publisher Full Text

3. Moudafi, A, Théra, M: Proximal and dynamical approaches to equilibrium problems. Ill-Posed Variational Problems and Regularization Techniques, pp. 187–201. Springer, Berlin (1999)

4. Ceng, LC, Al-Homidan, S, Ansari, QH, Yao, J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math.. 223, 967–974 (2009). Publisher Full Text

5. Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl.. 331, 506–515 (2007). Publisher Full Text

6. Tian, M: An application of hybrid steepest descent methods for equilibrium problems and strict pseudocontractions in Hilbert spaces. J. Inequal. Appl.. 2011, Article ID 173430. doi:10.1155/2011/173430 (2011)

7. Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Haifa. (2001)

8. Liu, Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal.. 71, 4852–4861 (2009). Publisher Full Text

9. Plubtieng, S, Punpaeng, R: A general iterative method for equilibrium problems and fixed point problems in Hilbert space. J. Math. Anal. Appl.. 336, 455–469 (2007). Publisher Full Text

10. Jung, JS: Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces. Appl. Math. Comput.. 215, 3746–3753 (2010). Publisher Full Text

11. Jung, JS: Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl. Math. Comput.. 213, 498–505 (2009). Publisher Full Text

12. Kumam, P: A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping. J. Appl. Math. Comput.. 29, 263–280 (2009). Publisher Full Text

13. Kumam, P: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping. Nonlinear Anal. Hybrid Syst.. 2(4), 1245–1255 (2008). Publisher Full Text

14. Saewan, S, Kumam, P: The shrinking projection method for solving generalized equilibrium problem and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings. Fixed Point Theory Appl. doi:10.1186/1687-1812-2011-9 (2011)

15. Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl.. 133, 359–370 (2007). Publisher Full Text

16. Yao, Y, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl.. 2007, Article ID 64363 (2007)

17. Su, YF, Shang, MJ, Qin, XL: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal., Theory Methods Appl.. 69, 2709–2719 (2008). Publisher Full Text

18. Plubtieng, S, Punpaeng, R: A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings. Appl. Math. Comput.. 179, 548–558 (2008)

19. Xu, HK: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl.. 150, 360–378 (2011). Publisher Full Text

20. Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl.. 116, 659–678 (2003). Publisher Full Text

21. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.. 66, 240–256 (2002). Publisher Full Text

22. Ceng, LC, Ansari, QH, Yao, J-C: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal.. 74, 5286–5302 (2011). Publisher Full Text

23. Marino, G, Xu, HK: A general method for nonexpansive mappings in Hilbert space. J. Math. Anal. Appl.. 318, 43–52 (2006). Publisher Full Text

24. Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl.. 298, 279–291 (2004). Publisher Full Text

25. Bkun, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.. 63, 123–145 (1994)

26. Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal.. 6, 117–136 (2005)

27. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl.. 20, 103–120 (2004). Publisher Full Text

28. Geobel, K, Kirk, WA: Topics on Metric Fixed Points Theory, Cambridge University Press, Cambridge (1990)

29. Bretarkas, DP, Gafin, EM: Projection methods for variational inequalities with applications to the traffic assignment problem. Math. Program. Stud.. 17, 139–159 (1982). Publisher Full Text

30. Han, D, Lo, HK: Solving non additive traffic assignment problems: a descent method for cocoercive variational inequalities. Eur. J. Oper. Res.. 159, 529–544 (2004). Publisher Full Text