Proximal algorithms for a class of mixed equilibrium problems
© Song and Zhang; licensee Springer 2012
Received: 21 February 2012
Accepted: 17 September 2012
Published: 2 October 2012
We present two proximal algorithms for solving the mixed equilibrium problems. Under some simpler framework, the strong and weak convergence of the sequences defined by two general algorithms is respectively obtained. In particular, we deal with several iterative schemes in a united way and apply our algorithms for solving the classical equilibrium problem, the minimization problem, the classical variational inequality problem and the generalized variational inequality problem. Our results properly include some corresponding results in this field as a special case.
MSC:47H06, 47J05, 47J25, 47H10, 90C33, 90C25, 49M45, 65C25, 49J40, 65J15, 47H09.
The set of solutions of EP (1.2) is denoted by .
and is denoted by .
The mixed equilibrium problem (MEP) is very interesting because it covers mathematical programs and optimization problems over equilibrium constraints, hierarchical minimization problems, variational inequality, complementarity problems, monotone inclusion problems, saddle point problems, Nash equilibria in noncooperative games as well as certain fixed point problems. In other words, the mixed equilibrium problem unifies several problems arising from engineering, physics, statistics, computer science, optimization theory, operation research, economics and others. The interest of this problem is that it unites all these particular problems in a convenient way. Moreover, many methods devoted to solving one of these problems can be extended, with suitable modifications, to solving the mixed equilibrium problem (MEP).
In the last 20 years or so, many mathematical works have been devoted to studying the method for finding an approximate solution of with various types of additional conditions. Moudafi  extended the proximal method to monotone equilibrium problems and Konnov  used the proximal method to solve equilibrium problems with weakly monotone bifunctions. In 2005, Combettes and Hirstoaga  introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and they also proved a strong convergence theorem under the condition . Song and Zheng  and Song, Kang and Cho  considered the convergence of a Halpern-type iteration for . The bundle methods and extragradient methods were extended to equilibrium problems in  and . In 2007, Takahashi and Takahashi  introduced the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T under the conditions and . Subsequently, the above viscosity approximation method was studied by different mathematicians using varying environmental conditions; see references [9–13] for details. Recently, Moudafi  showed weak convergence of proximal methods for a class of bilevel monotone equilibrium problems. For other solution methods regarding the equilibrium problems see [2, 15–22].
In this paper, our main objective is to show the mixed equilibrium problem (1.1) can be solved by two very simple proximal methods under simpler conditions, where a bifunction F and a function φ satisfy the following standard assumptions.
Condition 1.1 The function is a proper lower semicontinuous convex function and the bifunction satisfies the following:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
To this end, we introduce two proximal algorithms.
Some other proximal methods and the related strong and weak convergence results can be derived from our main theorem by particularizing the bifunction F, the function H and the parameters , . Our results treat several iterative schemes in a united way and properly include some results of [3–5, 8, 20, 21, 23, 24] as a special case.
The rest of the paper is organized as follows. In Section 2, we introduce some necessary lemmas and results, and show our algorithms are well defined. In Section 3, we study the strong convergence of a sequence iteratively given by Algorithm 1. In Section 4, we deal with the weak convergence of an iterative scheme defined by Algorithm 2. Some other proximal methods and the related results derived from our main theorems and some concluding remarks can be found in Section 5.
2 Preliminaries and basic results
For a bifunction , we have the following lemmas which were also given in .
Lemma 2.1 (Blum-Oettli [, Corollary 1])
Using a similar proof technique of Combettes-Hirstoaga [, Lemma 2.12], also see Ceng and Yao [, Lemma 3.1], Peng, Liou and Yao [, Lemma 2.2] obtained the following lemma which guarantees that our algorithms are well defined.
Lemma 2.2 ([, Lemma 2.2])
the domain of is H.
- (2)is single-valued and firmly nonexpansive, i.e., for any ,
is closed and convex.
where and satisfy the restrictions and . Then converges to zero as .
3 Strong convergence of Algorithm 1
In this section, we deal with an iterative scheme given by Algorithm 1 for finding an element of the set of solutions of the MEP (1.1) in a Hilbert space.
then converge strongly to some element of .
Thus, is bounded, and hence so is .
where . Following the proof technique in Mainge [, Lemma 3.2, Theorem 3.1], the proof may be divided into two cases.
So, an application of Lemma 2.3 onto (3.5) yields .
Now, it follows from (3.6) that . The proof is completed. □
4 Weak convergence of Algorithm 2
In this section, we show a weak convergence theorem which solves the MEP (1.1) in a Hilbert space.
If, in addition, satisfies (C3) , then and converge weakly to some element x of .
So, and are bounded. Moreover, the limit exists for each .
Therefore, the limit exists for each since .
By properties of Hilbert spaces, the boundedness of means that the sequence is weak compact in H, and hence there exists a subsequence of such that . Then using the same argument as in the proof of Theorem 3.1 by means of (4.3) and the condition (C3) together with the properties of the bifunction F and the function φ, we must have .
Adding the above equations, we must have , and so . This implies . By (4.3), it is obvious that also converges weakly to x. The desired results are reached. □
5 Deduced results and some remarks
F satisfies (A1)-(A4) in Condition 1.1 and ;
- (ii)for , and ,
Using Theorems 3.1 and 4.1 along with Lemma 5.1, we have the following theorems which solve GVIP (1.5) and VIP (1.4) under the simpler framework.
- (1)let and be sequences generated iteratively by(5.1)
- (2)let and be sequences generated iteratively by(5.2)
where and satisfy (C3) and (C4) . Then converges weakly to some .
Following Theorems 3.1 and 4.1 (), the desired results are proved. □
- (1)let and be sequences generated iteratively by(5.3)
- (2)let and be sequences generated iteratively by(5.4)
where and satisfy (C3) and (C4). Then converges weakly to some .
Proof Let for all . It follows from Lemma 5.1, Theorems 3.1 and 4.1, the desired results are obtained. □
For the classical equilibrium problem (1.2), the following is obvious ( in Theorems 3.1 and 4.1).
- (1)let and be sequences generated iteratively by(5.5)
- (2)let and be sequences generated iteratively by(5.6)
where and satisfy (C3) and (C4). Then converges weakly to some .
When in Theorems 3.1 and 4.1, we also have better approximated algorithm about solving the minimization problem (1.3) of a function φ.
- (1)let and be sequences generated iteratively by(5.7)
- (2)let and be sequences generated iteratively by(5.8)
where and satisfy (C3) and (C4). Then converges weakly to some .
We also observe the condition (C4) that includes the case that as a special case. Therefore, our iteration scheme contains several proximal point algorithms that compute the solution of some regularized problem.
, given by (5.9) converge weakly to some element x of ;
, given by (5.10) converge weakly to some element x of .
Proof It follows from Lemma 2.2 or Lemma 5.1 together with (5.9) or (5.10) that . Then by Theorem 4.1, the desired results are obtained. □
All strong convergence theorems of this paper remain true if one replaces the anchor point u by a contraction f (that is, so-called viscosity approximation methods with a contraction f) since Song  has showed their equivalency. So our main results could recover or develop some viscosity approximation results such as ones in references [8, 10, 11, 13] as well as others not referenced here.
The framework of holding our conclusions is more general and our results treat several iterative schemes and corresponding results in a united way. Consequently, our main results could be considered as recovering, developing and improving some known related convergence results in this field.
The authors would like to thank the editor and the anonymous referee for useful comments and valuable suggestions on the language and structure of our manuscript. This work is supported by the National Natural Science Foundation of P.R. China (11071279, 11171094, 11271112).
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