A Predictor-Corrector Method for Solving Equilibrium Problems

and Applied Analysis 3 Then x∗ is a solution of EP if and only if it is a solution of the auxiliary equilibrium problem (AEP): finding x ∗ ∈ C, s.t. εf (x ∗ ,y) + H (x ∗ ,y) ≥ 0, ∀y ∈ C. (AEP) Proof. It is easy to know that if x∗ is a solution of EP, then it is also a solution of AEP. Vice versa, let x∗ be a solution of AEP. Then x∗ is a minimum point of the problem min x∈K [εf (x ∗ , y) + H (x ∗ , y)] . (6) Because K is convex then x∗ is an optimal solution for (6) if and only if


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][2][3][4].Equilibrium problems include variational inequalities as special cases.In recent years, several numerical techniques [5][6][7][8][9][10][11][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 (, ) = 0 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: finding  * ∈  such that  ( * , ) ≥ 0, ∀ ∈ .(EP) 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 (EP), 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 (EP).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] A function ℎ :  →  is said to be strongly convex on  with modulus  ( ≥ 0), if and only if If ℎ is differentiable, then ℎ is strongly convex on  with modulus  ( ≥ 0), if and only if A function ℎ :  →  is said to be Lipschitz continuous on  with modulus  ( > 0), 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 (EP) 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 (VIP) of (EP) mentioned above and we apply our results in these frameworks and the predictor-corrector algorithm is applied to (VIP).The paper ends with some concluding remarks.

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 (, ) = 0, for all  ∈ .Then the following statements are equivalent: ( We can define the following general iterative algorithm framework.
Algorithm 3. Consider the following.
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  > 0. Let (, ) :  ×  →  be a nonnegative, differentiable function on the convex set  with respect to  and such that (i) (, ) = 0, for all  ∈ ; (ii)    (, ) = 0, for all  ∈ .
Applying Algorithm 3 to the AEP, we obtain the following iterative method.Algorithm 6.Consider the following.
Remark 8.According to the above, we extend the coefficient of approximate function from  ∈ (0, 1] in [20] to  ∈  \ {0}, which is a more generic case.Now, we describe the framework of predictor-corrector algorithm as follows.
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  +1 helps to correct the new prediction.

Convergence Analysis
In this section, we will give some convergence results about the algorithm.
Next we give the convergence result under perfect foresight, which has been stated in [20].
At the same time, respective to convergence under imperfect foresight, we first give some denotations and results.
Proof.Let  * be a solution of (EP) and consider for each  ∈  the Lyapunov function Γ  :  ×  →  defined for all ,  ∈ : Since ℎ is strongly convex on  with modulus , we can easily obtain that, for all   ∈ , Consider the following relation: where For  1 , we can easily get the following from the strong convexity of ℎ: For  2 , 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): , we derive the following from (13): In particular, let  =  + ; we have That is, Hence, ( Similarly to the proof of (1), we omit the process and get the conclusion.
We finish the proof.
For practical implementation, it is necessary to give a stopping criterion.
Next, we give the predictor-corrector algorithm about the (EP) with stopping criterion.

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][22][23][24][25][26][27] and the references therein.
Similarly to Assumption 14, we have the following.
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.; then the sequence {  } ∈ generated by the predictor-corrector methods is bounded and lim  → ∞ ‖  −  +1 ‖ = 0. Theorem 26.Assume that   ≥  > 0 for all  ∈ .If the sequence {  } ∈ generated by the predictor-corrector algorithm is bounded and lim  → ∞ ‖  −  +1 ‖ = 0, then every limit point of {  } ∈ is a solution of (VIP).
Similar to Theorems 15 and 16, we can prove Theorems 25 and 26.Here, we will omit their details.
Theorem 29.Assume that   ≥  > 0 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.

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 Δstationary 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 (EP), 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.