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Affine algorithms for the split variational inequality and equilibrium problems
Fixed Point Theory and Applications volume 2013, Article number: 140 (2013)
Abstract
An affine algorithm for the split variational inequality and equilibrium problems is presented. Strong convergence result is given.
1 Introduction
In the present manuscript, we focus on the following split variational inequality and equilibrium problem: Finding a point such that
where is the solution set of the generalized variational inequality of finding , such that
and is the solution set of the equilibrium problem, which is to find such that
Our main motivations are inspired by the following reasons.
Reason 1 Recently, the split problems have been considered by some authors. Especially, the split feasibility problem which can mathematically be formulated as the problem of finding a point with the property
has received much attention due to its applications in signal processing and image reconstruction with particular progress in intensity modulated radiation therapy [1–13]. Note that the involved operator g is a bounded linear operator. However, in the present paper, the involved mapping ψ in (1.1) is a nonlinear mapping.
Reason 2 The variational inequality problem [14–24] and equilibrium problem [23–27], which include the fixed point problems and optimization problems [28–30], have been studied by many authors. It is an interesting topic associated with the analytical and algorithmic approach to the variational inequality and equilibrium problems.
Motivated and inspired by the results in the literature, we present an affine algorithm for solving the split problem (1.1). Strong convergence theorem is given under some mild assumptions.
2 Preliminaries
Let H be a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H.
2.1 Monotonicity and convexity
An operator is said to be monotone if for all . is said to be strongly monotone if there exists a constant such that for all . is called an inverse-strongly-monotone operator if there exists such that for all . Let be a nonlinear operator. is said to be α-inverse strongly g-monotone iff for all and for some . Let B be a mapping of H into . The effective domain of B is denoted by , that is, . A multi-valued mapping B is said to be a monotone operator on H iff for all , , and . A monotone operator B on H is said to be maximal iff its graph is not strictly contained in the graph of any other monotone operator on H.
A function is said to be convex if for any and for any , .
2.2 Nonexpansivity and continuity
A mapping is said to be nonexpansive [31–38] if for all . We use to denote the set of fixed points of T. is called a firmly nonexpansive mapping if, for all , . It is known that T is firmly nonexpansive if and only if a mapping is nonexpansive, where I is the identity mapping on H. is said to be L-Lipschitz continuous if there exists a constant such that for all . In such a case, T is said to be L-Lipschitz continuous. Given a nonempty, closed convex subset C of H, the mapping that assigns every point to its unique nearest point in C is called a metric projection onto C and denoted by , that is, and . The metric projection is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is for all , .
2.3 Equilibrium problem
In this paper, we consider the split problem (1.1). In the sequel, we assume that the solution set S of (1.1) is nonempty.
Problem 2.1 Assume that
-
(A1)
is an α-inverse strongly ψ-monotone mapping;
-
(A2)
is a weakly continuous and γ-strongly monotone mapping such that ;
-
(A3)
is a bifunction;
-
(A4)
is a β-inverse-strongly monotone mapping.
Our objective is to
where F satisfies the following conditions:
-
(F1)
for all ;
-
(F2)
F is monotone, i.e., for all ;
-
(F3)
for each , ;
-
(F4)
for each , is convex and lower semicontinuous.
In order to solve Problem 2.1, we need the following useful lemmas.
2.4 Useful lemmas
The following three lemmas are important tools for our main results in the next section. Note that these lemmas are used extensively in the literature.
Lemma 2.2 (Combettes and Hirstoaga’s lemma [26])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction which satisfies conditions (F1)-(F4). Let and . Then there exists such that
Further, if , then the following hold:
-
(a)
is single-valued and is firmly nonexpansive;
-
(b)
is closed and convex and .
Lemma 2.3 (Suzuki’s lemma [39])
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose for all and . Then .
Lemma 2.4 (Xu’s lemma [40])
Assume that is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence such that and (or ). Then .
3 Algorithms and convergence analysis
In this section, we first present our algorithm for solving Problem 2.1. Assume that the conditions in Problem 2.1 are all satisfied.
Algorithm 3.1 Let C be a nonempty closed and convex subset of a real Hilbert space H.
Step 0. (Initialization)
Step 1. (Projection step) For , let the sequence be generated iteratively by
where is the metric projection, is a real number sequence, is an L-Lipschitz continuous mapping and is a constant.
Step 2. (Proximal step) Find such that
where is a real number sequence.
Step 3. (Affine step) For the above sequences and , let the th sequence be generated by
where is a real number sequence.
Theorem 3.2 Suppose . Assume that the following restrictions are satisfied:
-
(C1)
, and ;
-
(C2)
and ;
-
(C3)
and ;
-
(C4)
.
Then the sequence generated by Algorithm 3.1 converges strongly to , which solves the following variational inequality:
Remark 3.3 The solution of variational inequality (3.1) is unique. As a matter of fact, if also solves (3.1), we have
Adding up the above two inequalities, we deduce
It follows that
which implies that
Since ψ is γ-strongly monotone, we have
Hence,
This deduces the contraction because of by the assumption. Therefore, . So, the solution of variational inequality (3.1) is unique.
Remark 3.4 Using the characteristic inequality of the projection, we have
Remark 3.5
In fact,
Next, we prove Theorem 3.2.
Proof Let . Hence and . Since , from Remark 3.4 we have for all . Thus,
By Algorithm 3.1, we have for all . Noting that , we deduce for all . It follows that
By induction
Hence, is bounded. Since ψ is γ-strongly monotone, we can get (by a similar technique as that in Remark 3.3) . So, . This implies that is bounded. Next, we show . From Step 2 in Algorithm 3.1, we have
Taking , we get
Similarly, we also have
Adding up the above two inequalities, we get
By the monotonicity of F, we have
So,
Thus,
It follows that
and hence
By Algorithm 3.1, we have
Therefore,
It follows that
Since , , and the sequences , , , and are bounded, we have
By Lemma 2.3, we obtain
Hence,
This together with the γ-strong monotonicity of ψ implies that
By the convexity of the norm, we have
where is some constant. From Remark 3.5, we derive
Thus,
So,
Since , and , we obtain
Set for all n. By using the property of projection, we get
It follows that
From (3.3) and (3.5), we have
Then we obtain
Since , and , we have
Next, we prove , where is the unique solution of (3.1). We take a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to some point . Without loss of generality, we may assume that . This implies that due to the weak continuity of ψ. Now, we show . We firstly show . Since , for any , we have
From the monotonicity of F, we have
Hence,
Put for all and . Then we have . So, from (3.8) we have
Note that . Further, from the monotonicity of A, we have . Letting in (3.9), we have . This together with (F1), (F4) implies that
and hence . Letting , we have . This implies that . Next, we only need to prove . Set
By [41], we know that R is maximal ψ-monotone. Let . Since and , we have . Noting that , we get
It follows that
Then
Since and , we deduce that by taking in (3.10). Thus, by the maximal ψ-monotonicity of R. Hence, . Therefore, . From (3.7), we obtain
Note that
It follows that
Therefore,
where and . It is easily seen that and . We can therefore apply Lemma 2.4 to conclude that and . This completes the proof. □
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Acknowledgements
Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Rudong Chen was supported in part by NSFC 11071279. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
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Yao, Y., Chen, R. & Liou, YC. Affine algorithms for the split variational inequality and equilibrium problems. Fixed Point Theory Appl 2013, 140 (2013). https://doi.org/10.1186/1687-1812-2013-140
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DOI: https://doi.org/10.1186/1687-1812-2013-140