An iterative method for split hierarchical monotone variational inclusions
- Qamrul Hasan Ansari^{1, 2}Email author and
- Aisha Rehan^{1}
https://doi.org/10.1186/s13663-015-0368-4
© Ansari and Rehan 2015
Received: 10 March 2015
Accepted: 26 June 2015
Published: 18 July 2015
Abstract
In this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems, etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result.
Keywords
MSC
1 Introduction
If the sets C and Q are the set of fixed points of the operators \(T : H_{1} \to H_{1}\) and \(S : H_{2} \to H_{2}\), respectively, then the SVIP is called a split hierarchical variational inequality problem (SHVIP). It is introduced and studied by Ansari et al. [6]. Several special cases of a SHVIP, namely, the split convex minimization problem, the split variational inequality problem defined over the solution set of monotone variational inclusion problem, the split variational inequality problem defined over the solution set of equilibrium problem, are also considered in [6].
Theorem 1.1
([7], Theorem 2.1)
Given a bounded linear operator \(A : H_{1} \rightarrow H_{2}\). Let \(f : H_{1}\rightarrow H_{1}\) and \(g : H_{2}\rightarrow H_{2}\) be \(\alpha_{1}\) and \(\alpha_{2}\) inverse strongly monotone operators on \(H_{1}\) and \(H_{2}\), respectively, and \(B_{1}\), \(B_{2}\) be two maximal monotone operators, and set \(\alpha:= \min\{\alpha_{1},\alpha_{2}\}\). Consider the operator \(U:= J_{\lambda}^{B_{1}}(I-\lambda f)\), \(T:= J_{\lambda}^{B_{2}}(I-\lambda g)\) with \(\lambda\in(0,2\alpha)\). Then the sequence \(\{x_{n}\}\) generated by (1.7) converges weakly to an element \(x^{*}\in\Gamma\), provided that \(\Gamma\neq\emptyset\) and \(\gamma \in(0, 1/L)\).
2 Preliminaries
Let H be a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot,\cdot\rangle\) and \(\|\cdot\|\), respectively. We denote by \(x_{n} \rightarrow x\) (respectively, \(x_{n}\rightharpoonup x\)) the strong (respectively, weak) convergence of the sequence \(\{x_{n}\}\) to x. Let \(T : H \rightarrow H\) be an operator whose range is denoted by \(R(T)\). The set of all fixed points of T is denoted by \(\operatorname{Fix}(T)\), that is, \(\operatorname{Fix}(T) = \{ x \in H : x = Tx \}\).
Definition 2.1
- (a)
nonexpansive if \(\|Tx-Ty\| \leq\|x-y\|\), for all \(x, y \in H\);
- (b)
- (c)averaged nonexpansive [11] if it can be written aswhere \(\alpha\in(0,1)\), I is the identity operator on H, and \(S:H\rightarrow H\) is a nonexpansive mapping;$$T = (1-\alpha)I +\alpha S, $$
- (d)
firmly nonexpansive if \(\|Tx-Ty\|^{2}\leq\langle x-y, Tx-Ty\rangle\), for all \(x,y \in H\);
- (e)α-inverse strongly monotone (α-ism) if there exists a constant \(\alpha> 0\) such that$$\langle Tx-Ty, x-y\rangle\geq\alpha\|Tx-Ty\|^{2}, \quad \mbox{for all } x ,y \in H. $$
The following example shows that every nonexpansive operator is not necessarily strongly nonexpansive.
Example 2.1
Let \(T : [-1,1] \to \mathbb {R}\) be defined by \(Tx = -x\), for all \(x \in[-1,1]\). Then T is nonexpansive but not strongly nonexpansive.
The following result will be used in the sequel.
Proposition 2.1
[11]
- (i)
If T is ν-ism, then for \(\gamma> 0\), γT is \(\frac{\nu}{\gamma}\)-ism.
- (ii)
T is averaged if and only if the complement \(I-T\) is ν-ism for some \(\nu> \frac{1}{2}\). Indeed, for \(\alpha\in(0,1)\), T is α-averaged if and only if \(I-T\) is \(\frac{1}{2\alpha}\)-ism.
- (iii)
The composite of finitely many averaged mappings is averaged.
Let \(\varphi: H \rightarrow H\) be a given single-valued α-inverse strongly monotone operator and \(\lambda\in(0, 2\alpha)\). Then \((I-\lambda\varphi)\) is averaged. Indeed, since φ is α-inverse strongly monotone, λφ is \(\frac{\alpha}{\lambda}\)-ism. Thus, \(I-\lambda\varphi\) is averaged as \(\frac{\alpha}{\lambda} > \frac{1}{2}\).
Lemma 2.1
(Demiclosedness principle) [12], Lemma 2
Let C be a nonempty, closed, and convex subset of a real Hilbert space H and \(T : C \rightarrow C \) be a nonexpansive operator with \(\operatorname{Fix}(T) \neq\emptyset\). If the sequence \(\{x_{n}\} \subseteq C\) converges weakly to x and the sequence \(\{(I-T)x_{n}\}\) converges strongly to y, then \((I-T)x = y\). In particular, if \(y = 0\), then \(x \in\operatorname{Fix}(T)\).
Definition 2.2
- (a)monotone if$$\langle u-v, x-y\rangle\geq0, \quad\mbox{for all } u \in Bx, v \in By; $$
- (b)maximal monotone if it is monotone and the graphof B is not properly contained in the graph of any other monotone operator.$$G(B) =\bigl\{ (x,u) \in H \times H : u \in Bx \bigr\} $$
3 Algorithm and convergence result
It is well known that when the set-valued mapping \(B : H \rightrightarrows H\) is maximal monotone, then for each \(x \in H\) and \(\lambda> 0\), there is a unique \(z \in H \) such that \(x \in(I+\lambda B)z\) [13, 14]. In this case, the operator \(J_{\lambda}^{B} := (I+\lambda B)^{-1}\) is called resolvent of B with parameter λ. It is well known that \(J_{\lambda}^{B}\) is a single-valued and firmly nonexpansive mapping.
Now we propose the following algorithm to compute the approximate solutions of the SHMVIP.
Algorithm 3.1
Initialization: Let \(\lambda> 0\) and take arbitrary \(x_{0} \in H_{1}\).
Last step: Update \(n := n+1\).
Next we prove the weak convergence of the sequence generated by Algorithm 3.1.
Theorem 3.1
Let \(A : H_{1}\rightarrow H_{2}\) be a bounded linear operator, \(f : H_{1}\rightarrow H_{1}\) be an \(\alpha_{1}\)-inverse strongly monotone operator, \(T : H_{1}\rightarrow H_{1}\) be a strongly nonexpansive operator such that \(\operatorname{Fix}(T) \neq\emptyset\), \(g : H_{2}\rightarrow H_{2}\) be an \(\alpha_{2}\)-inverse strongly monotone operator, \(S : H_{2}\rightarrow H_{2}\) be a nonexpansive operator such that \(\operatorname{Fix}(S)\neq\emptyset\), and \(\alpha:= \min\{\alpha_{1},\alpha_{2}\}\). Consider the operator \(U := J_{\lambda}^{B_{1}}(I-\lambda f)\) and \(V := J_{\lambda}^{B_{2}}(I-\lambda g)\) with \(\lambda\in(0,2\alpha)\), and \(B_{1} : H_{1} \rightrightarrows H_{1}\) and \(B_{2} : H_{2} \rightrightarrows H_{2}\) are two maximal monotone set-valued mappings with nonempty values. Then the sequence \(\{x_{n}\}\) generated by Algorithm 3.1 converges weakly to an element \(x^{*}\in\Psi\), provided \(\Psi\neq\emptyset\).
Proof
Now, we illustrate Algorithm 3.1 and Theorem 3.1 by the following example.
Example 3.1
The values of \(\pmb{\{x_{n}\}}\) with initial guess \(\pmb{x_{1}=10}\) , \(\pmb{x_{1}=15}\) , and \(\pmb{x_{1}=20}\)
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
\(x_{n}\) | 10 | 0.7037 | 0.0495 | 0.0035 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
\(x_{n}\) | 15 | 1.0556 | 0.0743 | 0.0052 | 0.0004 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
\(x_{n}\) | 20 | 1.4074 | 0.0990 | 0.0070 | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Table 1 shows that the sequence \(\{x_{n}\}\) converges to 0, which is the required solution.
Declarations
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Authors’ Affiliations
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