Weak and strong convergence of hybrid subgradient method for pseudomonotone equilibrium problem and multivalued nonexpansive mappings
© Wen; licensee Springer. 2014
Received: 23 April 2014
Accepted: 30 October 2014
Published: 17 November 2014
In this paper, we introduce a hybrid subgradient method for finding a common element of the set of solutions of a class of pseudomonotone equilibrium problems and the set of fixed points of a finite family of multivalued nonexpansive mappings in Hilbert space. The proposed method involves only one projection rather than two as in the existing extragradient method and the inexact subgradient method for an equilibrium problem. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. Moreover, a numerical example is given to illustrate our algorithm and our results.
MSC:47H05, 47H09, 47H10.
The set of solution of equilibrium problem is denoted by . It is well known that some important problems such as convex programs, variational inequalities, fixed point problems, minimax problems, and Nash equilibrium problem in noncooperative games and others can be reduced to finding a solution of the equilibrium problem (1.1); see [1–3] and the references therein.
Recently, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of nonlinear mappings has become an attractive subject, and various methods have been extensively investigated by many authors. It is worth mentioning that almost all the existing algorithms for this problem are based on the proximal point method applied to the equilibrium problem combining with a Mann iteration to fixed point problems of nonexpansive mappings, of which the convergence analysis has been considered if the bifunction F is monotone. This is because the proximal point method is not valid when the underlying operator F is pseudomonotone. Another basic idea for solving equilibrium problems is the projection method. However, Facchinei and Pang  show that the projection method is not convergent for monotone inequality, which is a special case of monotone equilibrium problems. In order to obtain convergence of the projection method for equilibrium problems, Tran et al.  introduced an extragradient method for pseudomonotone equilibrium problems, which is computationally expensive because of the two projections defined onto the constrained set. Efforts for deducing the computational costs in computing the projection have been made by using penalty function methods or relaxing the constrained convex set by polyhedral convex ones; see, e.g., [6–15].
In 2011, Santos and Scheimberg  further proposed an inexact subgradient algorithm for solving a wide class of equilibrium problems that requires only one projection rather than two as in the extragradient method, and of which computational results show the efficiency of this algorithm in finite dimensional Euclidean spaces. On the other hand, iterative schemes for multivalued nonexpansive mappings are far less developed than those for nonexpansive mappings though they have more powerful applications in solving optimization problems; see, e.g., [16–23] and the references therein.
where , , for all and , are finite families of nonspreading mappings and multivalued nonexpansive mappings for , respectively. Moreover, he further proved the weak and strong convergence theorems of the iterative sequences under the condition of monotone defined on a bifunction F.
where , , and , , , and are nonnegative real sequences.
Our purpose is not only to modify the proximal point iterative schemes (1.4) for the equilibrium problem to a hybrid subgradient method for a class of pseudomonotone equilibrium problems and a finite family of multivalued nonexpansive mappings, but also to establish weak and strong convergence theorems involving only one projection rather than two as in the extragradient method  and the inexact subgradient method  for the equilibrium problem. Our theorems presented in this paper improve and extend the corresponding results of [5, 15, 18, 21].
- (i)r-strongly monotone if there exists a number such that
- (ii)monotone on K if
- (iii)pseudomonotone on K with respect to if(2.3)
It is clear that (i) ⇒ (ii) ⇒ (iii), for every . Moreover, F is said to be pseudomonotone on K with respect to , if it is pseudomonotone on K with respect to every . When , F is called pseudomonotone on K.
Clearly, . Since for every , this bifunction is pseudomonotone on K with respect to the solution . However, F is not pseudomonotone on K. In fact, both and .
To study the equilibrium problem (1.1), we may assume that Δ is an open convex set containing K and the bifunction satisfy the following assumptions:
(C1) for each and is convex and lower semicontinuous on K;
(C2) is weakly upper semicontinuous for each on the open set Δ;
(C3) F is pseudomonotone on K with respect to and satisfies the strict paramonotonicity property, i.e., for and implies ;
(C4) if is bounded and as , then the sequence with is bounded, where stands for the ϵ-subdifferential of the convex function at x.
Throughout this paper, weak and strong convergence of a sequence in H to x are denoted by and , respectively. In order to prove our main results, we need the following lemmas.
Lemma 2.1 
Lemma 2.2 
Lemma 2.3 
where . Then the sequence is convergent.
Lemma 2.4 
Let K be a nonempty closed convex subset of a real Hilbert space H. Let be a multivalued nonexpansive mapping. If and , then .
3 Weak convergence
, , and .
Then the sequence generated by (1.5) converges weakly to .
Applying Lemma 2.3 to (3.6), by condition (ii), we obtain the existence of .
Combining with and , we can deduced that as desired.
Applying Lemma 2.4 to (3.21), we can deduce that for and hence .
which is a contradiction. Therefore, . This shows that is a single point set, i.e., . This completes the proof. □
, , and .
Then the sequence converges weakly to .
Proof Putting , then , a single multivalued nonexpansive mapping, and the conclusion follows immediately from Theorem 3.1. This completes the proof. □
4 Strong convergence
To obtain strong convergence results, we either add the control condition , or we remove the condition for all and adjust the nonempty compact subset to a proximal bounded subset of K as follows.
, , and .
Then the sequence generated by (4.1) converges strongly to .
which implies that . Then, from (3.21), (4.4), and (4.8), we can conclude that . This completes the proof. □
, , and .
Then the sequence converges strongly to .
which implies that . Since converges strongly to and exists (as in the proof of Theorem 3.1), we find that converges strongly to . This completes the proof. □
In addition, we supply an example and numerical results to illustrate our method and the main results of this paper.
Clearly, F is pseudomonotone on K. Note that is convex for and by taking for all .
Numerical results for an initial point
Numerical results for an initial point
The computations are performed by Matlab R2008a running on a PC Desktop Intel(R) Core(TM) i3-2330M, CPU @2.20 GHz, 790 MHz, 1.83 GB, 2 GB RAM.
Remark 4.1 Our hybrid subgradient method improves the extragradient method of Tran et al.  and the inexact subgradient algorithm of Santos and Scheimberg  for an equilibrium problem in deducing the computational costs of an iterative process.
Remark 4.2 Our results generalize the results of Eslamian , a proximal point method for an equilibrium problem, to a hybrid subgradient method for a pseudomonotone equilibrium problem.
The author is grateful to the anonymous referees for valuable remarks suggestions which helped him very much in improving this manuscript. This work was supported by the National Science Foundation of China (11471059, 11271388), Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1400618).
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