Abstract and Applied Analysis
Volume 2014 (2014), Article ID 313217, 11 pages
http://dx.doi.org/10.1155/2014/313217

Research Article

A Predictor-Corrector Method for Solving Equilibrium Problems

Zong-Ke Bao,1,2 Ming Huang,2,3 and Xi-Qiang Xia²

¹School of Accounting, Zhejiang University of Finance and Economics, Hangzhou 310018, China
²Dalian University of Technology, Dalian 116024, China
³Dongbei University of Finance and Economics, Dalian 116025, China

Received 28 April 2014; Accepted 30 June 2014; Published 15 October 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 Zong-Ke Bao et al. 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 suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise to -stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.

1. Introduction

Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization. This theory has witnessed an explosive growth in theoretical advances and applications across all disciplines of pure and applied sciences. As a result of this interaction, we have a variety of techniques to study existence results for equilibrium problems; see [1–4]. Equilibrium problems include variational inequalities as special cases. In recent years, several numerical techniques [5–12] including projection, resolvent, and auxiliary principle have been developed and analyzed for solving equilibrium problems.

Let be a nonempty closed convex subset of , and let be a continuous function satisfying for all , is convex on for all , and is lower semicontinuous (l.s.c.) on for all . The equilibrium problems (for short EP) proposed by Blum-Oettli [1] are as follows:

Recently, much attention has been given to reformulate the equilibrium problem as an optimization problem. This problem is very general in the sense that it includes, as particular cases, the optimization problem, the variational inequality problem, the Nash equilibrium problem in noncooperative games, the fixed-point problem, the nonlinear complementarity problem, and the vector optimization problem (see, e.g., [1, 13] and the references quoted therein). Multiobjective optimization problems can also be obtained by , as shown by Iusem and Sosa [13]. The above particular cases are useful models of many practical problems arising in game theory, physics, economics, and so forth. The interest of this problem is that it unifies all these particular problems in a convenient way. For example, the work of Brezis et al. extended results concerning variational inequalities, corresponding to the case where and is a monotone operator (see [14], pages 296-297). Moreover, many methods devoted to solving one of these problems can be extended, with suitable modifications, to solve the general equilibrium problem. In this paper we suppose that there exists at least one solution to problem . In particular, it is true when is compact. Other existence results for this problem can be found, for instance, in [1, 15].

In this paper, one uses usually the auxiliary principle technique. This technique deals with finding a suitable auxiliary problem and proving that the solution of the auxiliary problem is the solution of the original problem by using the fixed-point approach. Glowinski et al. [6] used this technique to study the existence of a solution of mixed variational inequalities. Noor [8] has used this technique to suggest and analyze a number of iterative methods for solving various classes of variational inequalities. It has been shown that a substantial number of numerical methods can be obtained as special cases from this technique. In this paper, we use again the auxiliary principle technique to suggest and analyze some predictor-corrector methods for solving equilibrium problems. In this respect, our results represent an improvement of previously known results. Noor [16] and Noor et al. [17] have introduced inertial proximal methods for variational inequalities using the auxiliary principle technique and proved that the convergence criteria of inertial proximal methods require only pseudomonotonicity. Inertial proximal methods include proximal methods as a special case. For recent development and applications of the proximal methods, see [5, 11, 18]. Our results can be considered as novel and important applications of the auxiliary principle technique. This paper is an extension over the related work of [19, 20]; the main contributions can be summarized as follows. First of all, we extend the coefficient of approximate function from to , which is a better conclusion. Secondly, approximate function does not need to satisfy the conditions in [20]; that is to say, our condition is weaker than therein. Moreover, we present a new algorithm, predictor-corrector methods for solving , and give a stopping criterion. In this sense, our result represents an improvement and refinement of the known results.

We recall the main notations and definitions that will be used in the sequel.

A function is said to be strongly monotone on with modulus , if and only if

A function is said to be strongly convex on with modulus (), if and only if

If is differentiable, then is strongly convex on with modulus (), if and only if

A function is said to be Lipschitz continuous on with modulus (), if and only if

Usually, we need there to be at least one solution for equilibrium problems. In particular, it is true when is compact.

Proposition 1 (existence of equilibrium (see [19])). Suppose is nonempty compact convex and is jointly lower semicontinuous, separately continuous in , and convex in . Then admits at least one solution.

This paper is organized as follows. In Section 2, we introduce some algorithms. In particular, we will give a predictor-corrector algorithmic frame. We present some convergence analysis under perfect and imperfect foresight in Section 3. Section 4 is devoted to an application: we focus on the particular case variational inequalities problem of mentioned above and we apply our results in these frameworks and the predictor-corrector algorithm is applied to . The paper ends with some concluding remarks.

2. Main Algorithm

Most of the algorithms developed for solving EP can be derived from equivalent formulations of the equilibrium problem. We will focus our attention on fixed-point formulations of EP: we will show that such formulations lead to a generalization of the methods developed by Cohen for variational inequalities and optimization problems.

Let us recall the following preliminary result which states the above mentioned equivalent formulation of EP.

Lemma 2. Suppose that , for all . Then the following statements are equivalent:

(a)there exists , s.t. , for all ;(b) is a solution of the problem

We can define the following general iterative algorithm framework.

Algorithm 3. Consider the following.
Step 1. Set , .
Step   2. Denote by the solution of the problem: .
Step 3. If , for some fixed , then stop; otherwise let and go to Step 2.

Unfortunately, in most of the cases, it is not possible to apply the previous algorithm directly to the equilibrium problems, for the previous algorithm may cause instabilities in the iterate process. So it is necessary to introduce an auxiliary equilibrium problem, which is equivalent to the equilibrium problem.

Proposition 4. Let be a convex differentiable function with respect to at and . Let be a nonnegative, differentiable function on the convex set with respect to and such that

(i), for all ;(ii), for all .

Then is a solution of EP if and only if it is a solution of the auxiliary equilibrium problem :

Proof. It is easy to know that if is a solution of EP, then it is also a solution of AEP.
Vice versa, let be a solution of AEP. Then is a minimum point of the problem

Because is convex then is an optimal solution for (6) if and only if

so that

Dividing by , we obtain that (8) implies, by the convexity of , that

Remark 5. Suppose is a strongly convex differentiable function; denote , for all . We have

That is, satisfies Proposition 4.

Applying Algorithm 3 to the AEP, we obtain the following iterative method.

Algorithm 6. Consider the following.
Step 1. Set , .
Step   2. Denote by the solution of the problem: .
Step 3. If , for some fixed , then stop; otherwise let and go to Step 2.
Most papers about EP only study the existence of EP’s solution. In this paper, we will give a predictor-corrector method to solve the equilibrium problems.

Definition 7. Let and . A convex function is a -approximation of at , if on and , where .

Remark 8. According to the above, we extend the coefficient of approximate function from in [20] to , which is a more generic case.

Now, we describe the framework of predictor-corrector algorithm as follows.

Algorithm 9. Let , , for all .
Step 1. Let , .
Step   2. Find -approximation of at , by predictor-corrector method. Let

Step 3. If , for some fixed , then stop; otherwise let and go to Step 2.

Remark 10. In Algorithm 9, each stage of computation requires two proximal steps. In Step 2, is served to predict the next point; the other helps to correct the new prediction.

3. Convergence Analysis

In this section, we will give some convergence results about the algorithm.

Definition 11. In Algorithm 9, if , is called a perfect foresight point of ; otherwise is an imperfect foresight point of .

Next we give the convergence result under perfect foresight, which has been stated in [20].

Proposition 12 (see [20]). Assume that there exist numbers and a nonnegative function such that, for all ,

(i);(ii).

If the sequence is nonincreasing and for all and if , then the sequence generated by the predictor-corrector algorithm is bounded and .

Proposition 13 (see [20]). Assume that for all . If the sequence generated by the predictor-corrector algorithm is bounded and , then every limit point of is a solution of problem .

At the same time, respective to convergence under imperfect foresight, we first give some denotations and results.

By the previous introduction, we have

Using (12) and (13), we get

Arranging (15), we have

Let in (14) and (16); then, adding them, we can get

Assumption 14. Assume that there exist and , for all . Consider the following:

(i);(ii).

We denote

It is convenient to prove the following theorem.

Theorem 15. Assume that the function satisfies Assumption 14 and , ; then the sequence generated by the predictor-corrector methods is bounded and .

Proof. Let be a solution of and consider for each the Lyapunov function defined for all :

Since is strongly convex on with modulus , we can easily obtain that, for all ,

Consider the following relation:

where

For , we can easily get the following from the strong convexity of :

For , we derive the following from (17):

Then

For the last term on the right of the above equality, we have

We can obtain the following from assumption (ii):

Similarly,

Because of , we derive the following from (13):

In particular, let ; we have

That is,

Hence,

Finally, we obtain

So

Arrange the previous inequality relation; we can get

Under the condition of , in order to obtain , we only need to prove the following result:

That is,

When ,

Then

We discuss in two cases.
If ,

So

We know that

Finally, we get ; it follows that is a nonincreasing sequence. By (21), we know that is bounded below by 0. Hence, converges in and is bounded. Passing to the limit in (42), then ().
If ,

Similarly to , we can obtain the result.
When ,

Likewise, we also discuss in two cases.
When ,

Similarly to the proof of , we omit the process and get the conclusion.
When .
Similar to the proof of , we omit the process and get the conclusion.

Theorem 16. Assume that for all . If the sequence generated by the predictor-corrector algorithm is bounded and , then every limit point of is a solution of equilibrium problem.

Proof. Let be the limiting point of and denote by some subsequence converging to . According to

we obtain

In particular, we set ; then

That is,

Passing to the limit in (50) as , then

At the same time, , so .
From , we have .
Moreover, ; we get .
Due to at and , then

For all , when , we have

In addition, is continuous; we have

Hence

Since , we have

That is,

where denotes the indicate function of the set . Using the definition of subdifferential, we get

Applying the Cauchy-Schwarz inequality and the properties and that is Lipschitz continuous on with constant , we have, for all ,

Take the limit about ; we deduce

Because is continuous, when ,

We finish the proof.

For practical implementation, it is necessary to give a stopping criterion.

Definition 17. Let . A point is called a -stationary point of problem if and

Proposition 18 (see [20]). Let ; ; .
Then and .

Theorem 19. Assume that for all and that the assumptions of Theorem 15 hold. Let be generated by the predictor-corrector algorithm, then the sequences and converge to zero.

Proof. Here we still discuss in two cases.
Under perfect foresight.
Under perfect foresight, it is easy to get .
Since is infinite, it follows from Theorem 16 that the sequence converges to some solution of problem .
On the other hand, for all , we have

Because is Lipschitz-continuous with constant , .
Since , we obtain that the sequence converges to zero.
Moreover,

Finally, by continuity of , so that when ,

(); that is, ().
Under imperfect foresight.
We derive that in the process of proving Theorem 16.
Hence, the sequence converges to zero.
Moreover,

because is continuous, so, when , .
At the same time,

Hence, ; that is, ().

Next, we give the predictor-corrector algorithm about the with stopping criterion.

Algorithm 20 (the predictor-corrector algorithms for ). Let , , for all .
Step 1. Let , , and .
Step   2. Finding a -approximation of at by predictor-corrector method, let

Step 3. If , then stop; otherwise put and go to Step 2.

4. Application to Variational Inequality Problems

Variational inequalities theory, which was introduced by Stampacchia [21], provides us with a simple, direct, natural, general, efficient, and unified framework to study a wide class of problems arising in pure and applied sciences. It has been extended and generalized in several directions using innovative and novel techniques for studying a wide class of equilibrium problems arising in financial, economics, transportation, elasticity, and optimization. During the last three decades, there has been considerable activity in the development for solving variational inequalities. For the applications, physical formulation, numerical methods, and other aspects of variational inequalities, see [21–27] and the references therein.

Let be a given mapping; variational inequality problems are as follows:

We denote ; then the problem is equivalent to the problem .

Similarly to Assumption 14, we have the following.

Assumption 21. Suppose that there exist and , for all :

(i);(ii).

In the same way, we consider the following two cases: perfect foresight and unperfect foresight cases.

First case is under perfect foresight.

Similar to Propositions 12 and 13, we have the following.

Proposition 22. Assume that there exist and a nonnegative function such that, for all ,

(i);(ii).

If the sequence is nonincreasing and the for all and if , then the sequence generated by the predictor-corrector algorithm is bounded and .

Proposition 23. Assume that for all . If the sequence generated by the predictor-corrector algorithm is bounded and , then every limit point of is a solution of .

Second case is under imperfect foresight.

Assumption 24. Assume that there exist and , for all :

(i);(ii).

We denote

Theorem 25. Suppose that satisfies Assumption 24 and , ; then the sequence generated by the predictor-corrector methods is bounded and .

Theorem 26. Assume that for all . If the sequence generated by the predictor-corrector algorithm is bounded and , then every limit point of is a solution of .

Similar to Theorems 15 and 16, we can prove Theorems 25 and 26. Here, we will omit their details.

Moreover, we can also give a stopping criterion.

Definition 27. Let . A point is called a -stationary point of problem if and

Proposition 28 (see [20]). Let ; ; .
Then and .

Theorem 29. Assume that for all and that the assumptions of Theorem 25 hold. Let be generated by the predictor-corrector algorithm, then the sequences and converge to zero.

Likewise, we omit the proof.

Finally, we have the predictor-corrector algorithm for variational inequalities problems as follows.

Algorithm 30 (the predictor-corrector algorithms for ). Let , , for all .
Step 1. Let , , and .
Step   2. Find a -approximation of at by predictor-corrector method. Let

Step 3. If , then stop; otherwise put and go to Step 2.

5. Conclusions

In this paper, we mainly present a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. This method can operate well in practice. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise to -stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.

For further work, the need can be anticipated: here we only give the conceptual algorithmic framework to solve this class of , we will continue to study its rapidly convergent executable algorithm, and we will consider how to use bundle techniques to approximate proximal points and other related quantities. Moreover, we will strive to extend the nonsmooth convex equilibrium problems to nonconvex cases, where its related theory will be researched in later papers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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