Affine algorithms for the split variational inequality and equilibrium problems
© Yao et al.; licensee Springer. 2013
Received: 11 October 2012
Accepted: 14 May 2013
Published: 29 May 2013
An affine algorithm for the split variational inequality and equilibrium problems is presented. Strong convergence result is given.
Our main motivations are inspired by the following reasons.
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.
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.
is an α-inverse strongly ψ-monotone mapping;
is a weakly continuous and γ-strongly monotone mapping such that ;
is a bifunction;
is a β-inverse-strongly monotone mapping.
for all ;
F is monotone, i.e., for all ;
for each , ;
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 )
is single-valued and is firmly nonexpansive;
is closed and convex and .
Lemma 2.3 (Suzuki’s lemma )
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 )
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.
where is the metric projection, is a real number sequence, is an L-Lipschitz continuous mapping and is a constant.
where is a real number sequence.
where is a real number sequence.
, and ;
This deduces the contraction because of by the assumption. Therefore, . So, the solution of variational inequality (3.1) is unique.
Next, we prove Theorem 3.2.
where and . It is easily seen that and . We can therefore apply Lemma 2.4 to conclude that and . This completes the proof. □
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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