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A Korpelevich-like algorithm for variational inequalities
Journal of Inequalities and Applications volume 2013, Article number: 76 (2013)
A Korpelevich-like algorithm has been introduced for solving a generalized variational inequality. It is shown that the presented algorithm converges strongly to a special solution of the generalized variational inequality.
Now it is well-known that the variational inequality of finding such that
where C is a nonempty closed convex subset of a real Hilbert space H and is a given mapping, is a fundamental problem in variational analysis and, in particular, in optimization theory. For related works, please see [1–20] and the references contained therein. Especially, Yao, Marino and Muglia  presented the following modified Korpelevich method for solving (1.1):
Recently, Aoyama, Iiduka and Takahashi  extended the variational inequality (1.1) to Banach spaces as follows:
where C is a nonempty closed convex subset of a real Banach space E. We use to denote the solution set of (1.3). The generalized variational inequality (1.3) is connected with the fixed point problem for nonlinear mappings. For solving the above generalized variational inequality (1.3), Aoyama, Iiduka and Takahashi  introduced the iterative algorithm
where is a sunny nonexpansive retraction from E onto C and , are two real number sequences. Motivated by (1.4), Yao and Maruster  presented a modification of (1.4) as follows:
Motivated and inspired by the above algorithms (1.2), (1.4) and (1.5), in this paper, we suggest an extragradient-type method via the sunny nonexpansive retraction for solving the variational inequalities (1.3) in Banach spaces. It is shown that the presented algorithm converges strongly to a special solution of the variational inequality (1.3).
Let C be a nonempty closed convex subset of a real Banach space E. Recall that a mapping A of C into E is said to be accretive if there exists such that
for all . A mapping A of C into E is said to be α-strongly accretive if for ,
for all . A mapping A of C into E is said to be α-inverse-strongly accretive if for ,
for all .
Let . A Banach space E is said to uniformly convex if for each , there exists such that for any ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit
exists for all . It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for . The norm of E is said to be Frechet differentiable if for each , the limit (2.1) is attained uniformly for . And we define a function called the modulus of smoothness of E as follows:
It is known that E is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space E is said to be q-uniformly smooth if there exists a constant such that for all .
We need the following lemmas for the proof of our main results.
Lemma 2.1 
Let q be a given real number with and let E be a q-uniformly smooth Banach space. Then
for all , where K is the q-uniformly smoothness constant of E and is the generalized duality mapping from E into defined by
Let D be a subset of C and let Q be a mapping of C into D. Then Q is said to be sunny if
whenever for and . A mapping Q of C into itself is called a retraction if . If a mapping Q of C into itself is a retraction, then for every , where is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. We know the following lemma concerning sunny nonexpansive retraction.
Lemma 2.2 
Let C be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C and Q be a retraction from C onto D. Then Q is sunny and nonexpansive if and only if
for all and .
Lemma 2.3 
Let C be a nonempty closed convex subset of a smooth Banach space X. Let be a sunny nonexpansive retraction from X onto C and let A be an accretive operator of C into X. Then for all ,
Lemma 2.4 
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mapping be α-inverse-strongly accretive. Then we have
In particular, if , then is nonexpansive.
Proof Indeed, for all , from Lemma 2.1, we have
It is clear that if , then is nonexpansive. □
Lemma 2.5 
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be a nonexpansive mapping of C into itself. If is a sequence of C such that weakly and strongly, then x is a fixed point of T.
Lemma 2.6 
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in R such that
3 Main results
In this section, we present our Korpelevich-like algorithm and consequently we will show its strong convergence.
3.1 Conditions assumptions
(A1) E is a uniformly convex and 2-uniformly smooth Banach space with a weakly sequentially continuous duality mapping;
(A2) C is a nonempty closed convex subset of E;
(A3) is an α-strongly accretive and L-Lipschitz continuous mapping with ;
(A4) is a sunny nonexpansive retraction from E onto C.
3.2 Parameters restrictions
(P1) λ, μ and γ are three positive constants satisfying:
, for some a, b with ;
where K is the smooth constant of E.
(P2) is a sequence in such that and .
Algorithm 3.1 For given , define a sequence iteratively by
Theorem 3.2 The sequence generated by (3.1) converges strongly to , where is a sunny nonexpansive retraction of E onto .
Proof Let . First, from Lemma 2.2, we have for all . In particular, for all .
Since is α-strongly accretive and L-Lipschitzian, it must be -inverse-strongly accretive mapping. Thus, by Lemma 2.4, we have
Since and , we get for enough large n. Without loss of generality, we may assume that for all , , i.e., . Hence, is nonexpansive.
From (3.1), we have
By (3.1) and (3.2), we have
Hence, is bounded.
Set . From (3.1), we have for all . Then we have
It follows that
This together with Lemma 2.6 implies that
From (3.2), we have
From (3.1), (3.3) and (3.4), we obtain
Therefore, we have
Since and , we obtain
It follows that
Since A is α-strongly accretive, we deduce
which implies that
It follows that
Next, we show that
To show (3.6), since is bounded, we can choose a sequence of converging weakly to z such that
We first prove . It follows that
By Lemma 2.5 and (3.8), we have , it follows from Lemma 2.3 that .
Now, from (3.7) and Lemma 2.2, we have
Noticing that , we deduce that
Since and for all , we can deduce from Lemma 2.2 that
Therefore, we have
which implies that
Finally, we will prove that the sequence . As a matter of fact, from (3.1) and (3.9), we have
Applying Lemma 2.6 to the last inequality, we conclude that converges strongly to . This completes the proof. □
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Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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Wu, Z., Yao, Y., Liou, YC. et al. A Korpelevich-like algorithm for variational inequalities. J Inequal Appl 2013, 76 (2013). https://doi.org/10.1186/1029-242X-2013-76
- Korpelevich-like algorithm
- sunny nonexpansive retraction
- generalized variational inequalities
- α-inverse-strongly accretive mappings
- Banach spaces