A viscosity splitting algorithm for solving inclusion and equilibrium problems
- Buthinah A Bin Dehaish^{1}Email author,
- Abdul Latif^{2},
- Huda O Bakodah^{1} and
- Xiaolong Qin^{3}
https://doi.org/10.1186/s13660-015-0554-2
© Bin Dehaish et al.; licensee Springer. 2015
Received: 9 October 2014
Accepted: 7 January 2015
Published: 11 February 2015
Abstract
In this paper, we present a viscosity splitting algorithm with computational errors for solving common solutions of inclusion and equilibrium problems. Strong convergence theorems are established in the framework of real Hilbert spaces. Applications are also provided to support the main results.
Keywords
1 Introduction
Splitting algorithms have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators; see, for example, [1–4] and the references therein. The central problem is to iteratively find a zero point of the sum of two monotone operators.
One of the classical methods of studying the problem \(0\in Tx\), where T is a maximal monotone operator, is the proximal point algorithm (PPA) which was initiated by Martinet [5] and further developed by Rockafellar [6]. The PPA and its dual version in the context of convex programming, the method of multipliers of Hesteness and Powell, have been extensively studied and are known to yield as special cases decomposition methods such as the method of partial inverses [7], the Douglas-Rachford splitting method, and the alternating direction method of multipliers [8]. In the case of \(T=A+B\), where A and B are monotone mappings, the splitting method \(x_{n+1}=(I+r_{n}B)^{-1}(I-r_{n}A)x_{n}\), \(n=0,1,\ldots\) , where \(r_{n}>0\), was proposed by Lions and Mercier [9] and by Passty [10]. There are many nonlinear problems arising in engineering areas needing more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblems or the solution of one subproblem on the solution set of another subproblem; see [11–18] and the references therein. The viscosity approximation method, which was introduced by Moudafi [19], has been extensively investigated by many authors for solving the problems; see, for example, [20–23] and the references therein. The highlight of the viscosity approximation method is that the desired limit point is not only a solution of a nonlinear problem but a unique solution of a classical monotone variational inequality. The aim of this paper is to introduce and investigate a viscosity splitting algorithm with computational errors for solving common solutions of inclusion and equilibrium problems. The common solution is a unique solution of a monotone variational inequality. Strong convergence is guaranteed without the aid of compactness assumptions or metric projections. The main results mainly improve the corresponding results in [11–15].
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a viscosity splitting algorithm with computational errors is investigated. A strong convergence theorem is established. In Section 4, applications of the main results are discussed.
2 Preliminaries
From now on, we always assume that H is a real Hilbert space with the inner product \(\langle\cdot,\cdot\rangle\) and the norm \(\|\cdot\|\). Let C is a nonempty, closed, and convex subset of H.
- (A1)
\(F(x,x)=0\) for all \(x\in C\);
- (A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\);
- (A3)for each \(x,y,z\in C\),$$\limsup_{t\downarrow0}F\bigl(tz+(1-t)x,y\bigr)\leq F(x,y); $$
- (A4)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and lower semicontinuous.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1
[24]
- (a)
\(F(T_{r})=EP(F)\);
- (b)
\(T_{r}\) is single-valued;
- (c)\(T_{r}\) is firmly nonexpansive, i.e., for any \(x,y\in H\),$$\|T_{r}x-T_{r}y\|^{2}\leq\langle T_{r}x-T_{r}y, x-y\rangle; $$
- (d)
The solution set is closed and convex.
Lemma 2.2
Let C be a nonempty, closed, and convex subset of H. Let \(A:C\rightarrow H\) be a mapping and let \(B:H\rightrightarrows H\) be a maximal monotone operator. Then \(F(J_{r}(I-r A))=(A+B)^{-1}(0)\).
Proof
The following lemma appears implicitly in [16]. For the sake of completeness, we still give the proof.
Lemma 2.3
Proof
Lemma 2.4
[28]
Lemma 2.5
[29]
- (i)
\(\sum_{n=1}^{\infty}\gamma_{n}=\infty\);
- (ii)
\(\limsup_{n\rightarrow\infty}\delta_{n}/\gamma_{n}\leq0\) or \(\sum_{n=1}^{\infty}\delta_{n}<\infty\);
- (iii)
\(\sum_{n=1}^{\infty}e_{n}<\infty\).
3 Main results
Theorem 3.1
Let C be a nonempty, closed, and convex subset of H and let F be a bifunction from \(C\times C\) to R which satisfies (A1)-(A4). Let \(f:C\rightarrow C\) be a contraction with the contractive constant \(\kappa\in(0,1)\). Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping and let \(B:H\rightrightarrows H\) be a maximal monotone mapping. Assume \((A+B)^{-1}(0)\cap EP(F)\neq\emptyset\). Let \(\{r_{n}\}\) and \(\{ s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\), \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}\), where \(\{y_{n}\}\) is a sequence in C such that \(F(y_{n},y)+\frac{1}{s_{n}}\langle y-y_{n},y_{n}-J_{r_{n}} (x_{n}-r_{n}Ax_{n}+e_{n} )\rangle\geq0\), \(\forall y\in C\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{ r_{n}\}\), and \(\{s_{n}\}\) satisfy the conditions: \(\lim_{n\rightarrow\infty }\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\), \(0<\liminf_{n\rightarrow\infty}s_{n}\), \(0< r\leq r_{n}\leq r'<2\alpha\), \(\lim_{n\rightarrow\infty}|r_{n}-r_{n+1}|=\lim_{n\rightarrow\infty}|s_{n}-s_{n+1}|=0\), where r and \(r'\) are real constants. Then \(\{x_{n}\}\) converges strongly to \(q=P_{(A+B)^{-1}(0)\cap EP(F)}f(q)\).
Proof
From Theorem 3.1, we have the following result on the equilibrium problem immediately.
Corollary 3.2
Let C be a nonempty, closed, and convex subset of H and let F be a bifunction from \(C\times C\) to R which satisfies (A1)-(A4). Assume \(EP(F)\neq\emptyset\). Let \(f:C\rightarrow C\) be a contraction with the contractive constant \(\kappa\in(0,1)\). Let \(\{s_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\), \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}\), where \(\{y_{n}\}\) is a sequence such that \(F(y_{n},y)+\frac{1}{s_{n}}\langle y-y_{n},y_{n}-x_{n}-e_{n}\rangle\geq0\), \(\forall y\in C\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{ s_{n}\}\) satisfy the conditions: \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty }\beta_{n}<1\), \(0<\liminf_{n\rightarrow\infty}s_{n}\), \(\lim_{n\rightarrow\infty}|s_{n}-s_{n+1}|=0\). Then \(\{x_{n}\}\) converges strongly to \(q=P_{EP(F)}f(q)\).
From Theorem 3.1, we have the following result on the inclusion problem immediately.
Corollary 3.3
Let C be a nonempty, closed, and convex subset of H and let \(f:C\rightarrow C\) be a contraction with the contractive constant \(\kappa\in(0,1)\). Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping and let \(B:H\rightrightarrows H\) be a maximal monotone mapping such that the domain of B is in C. Assume \((A+B)^{-1}(0)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\) and \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}J_{r_{n}} (x_{n}-r_{n}Ax_{n}+e_{n} )\), \(n\geq1\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{n}\}\) satisfy the following conditions: \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\), \(0< r\leq r_{n}\leq r'<2\alpha\), \(\lim_{n\rightarrow\infty}|r_{n}-r_{n+1}|=0\), where r and \(r'\) are real constants. Then \(\{x_{n}\}\) converges strongly to \(q=P_{(A+B)^{-1}(0)}f(q)\).
4 Applications
In this section, we give some results on equilibrium problems, variational inequalities, and convex functions.
Lemma 4.1
[12]
Theorem 4.2
Let C be a nonempty, closed, and convex subset of H and let F and G be two bifunctions from \(C\times C\) to R which satisfies (A1)-(A4). Assume that \(EP(G)\cap EP(F)\neq\emptyset\). Let \(f:C\rightarrow C\) be a contraction with the contractive constant \(\kappa\in(0,1)\). Let \(\{r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\), \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}\), \(n\geq1\), where \(F(y_{n},y)+\frac{1}{s_{n}}\langle y-y_{n},y_{n}-(I+r_{n}W)^{-1} (x_{n}+e_{n} )\rangle\geq0\), \(\forall y\in C\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{ r_{n}\}\), and \(\{s_{n}\}\) satisfy the following conditions: \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty }\beta_{n}<1\), \(0<\liminf_{n\rightarrow\infty}s_{n}\), \(0< r\leq r_{n}\), \(\lim_{n\rightarrow\infty}|r_{n}-r_{n+1}|=\lim_{n\rightarrow\infty }|s_{n}-s_{n+1}|=0\), where r is a real constant. Then \(\{x_{n}\}\) converges strongly to \(q=P_{EP(G)\cap EP(F)}f(q)\).
Theorem 4.3
Let C be a nonempty, closed, and convex subset of H and let \(f:C\rightarrow C\) be a contraction with the contractive constant \(\kappa\in(0,1)\). Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping such that \(VI(C,A)\neq \emptyset\). Let \(\{r_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\), \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}\), \(n\geq 1\), \(y_{n}=\operatorname{Proj}_{C} (x_{n}-r_{n}Ax_{n}+e_{n} )\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{n}\}\) satisfy the conditions: \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta _{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\), \(0< r\leq r_{n}\leq r'<2\alpha\), \(\lim_{n\rightarrow\infty}|r_{n}-r_{n+1}|=0\), where r and \(r'\) are real constants. Then \(\{x_{n}\}\) converges strongly to \(q=P_{VI(C,A)}f(q)\).
Proof
Put \(F(x,y)=0\) for any \(x,y\in C\) and \(s_{n}=1\). From Theorem 3.1, we draw the desired conclusion immediately. □
Now, we are in a position to consider the problem of finding minimizers of proper lower semicontinuous convex functions. For a proper lower semicontinuous convex function \(g:H\rightarrow(-\infty,\infty]\), the subdifferential mapping ∂g of g is defined by \(\partial g(x)=\{x^{*}\in H:g(x)+\langle y-x,x^{*}\rangle\leq g(y),\forall y\in H\}\), \(\forall x\in H\). Rockafellar [30] proved that ∂g is a maximal monotone operator. It is easy to verify that \(0\in\partial g(v)\) iff \(g(v)=\min_{x\in H} g(x)\).
Theorem 4.4
Let \(g:H\rightarrow(-\infty,+\infty]\) be a proper convex lower semicontinuous function such that \((\partial g)^{-1}(0)\) is not empty. Let f be a contraction on H with the contractive constant \(\kappa\in(0,1)\). Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{ e_{n}\}\) be a sequence in H such that \(\sum_{n=1}^{\infty}\|e_{n}\|<\infty\). Let \(\{x_{n}\}\) be a sequence generated in the following process: \(x_{1}\in C\), \(x_{n+1}=\alpha_{n} f(x_{n})+\beta_{n}x_{n}+\gamma_{n}y_{n}\), \(n\geq 1\), \(y_{n}=\arg\min_{z\in H}\{g(z)+\frac{\|z-x_{n}+e_{n}\|^{2}}{2r_{n}}\}\). Assume that the control sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{ r_{n}\}\) satisfy the conditions: \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\), \(0<\liminf_{n\rightarrow\infty}\beta _{n}\leq\limsup_{n\rightarrow\infty}\beta_{n}<1\), \(0< r\leq r_{n}<\infty\) \(\lim_{n\rightarrow\infty}|r_{n}-r_{n+1}|=0\), where r is a real constant. Then \(\{x_{n}\}\) converges strongly to \(q=P_{(\partial g)^{-1}(0)}f(q)\).
Proof
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant No. (58-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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