Proximal point algorithms involving fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces
- Prasit Cholamjiak^{1},
- Afrah AN Abdou^{2}Email author and
- Yeol Je Cho^{2, 3}Email author
https://doi.org/10.1186/s13663-015-0465-4
© Cholamjiak et al. 2015
Received: 10 August 2015
Accepted: 9 November 2015
Published: 10 December 2015
Abstract
In this paper, we introduce a new modified proximal point algorithm involving fixed point iterates of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces and prove that the sequence generated by our iterative process converges to a minimizer of a convex function and a fixed point of mappings.
Keywords
convex minimization problem resolvent identity \(\operatorname{CAT}(0)\) space proximal point algorithm nonexpansive mappingMSC
47H09 47H101 Introduction
It is noted in [1] that (1.3) is independent of (1.2) (and hence (1.1)) and has the convergence rate better than (1.1) and (1.2) for contractions.
Some interesting results for solving a fixed point problem of nonlinear mappings in the framework of \(\operatorname{CAT}(0)\) spaces can also be found, for examples, in [5–14].
We denote \(\operatorname{argmin}_{y\in X}f(y)\) by the set of minimizers of f. A successful and powerful tool for solving this problem is the well-known proximal point algorithm (shortly, the PPA) which was initiated by Martinet [15] in 1970. In 1976, Rockafellar [16] generally studied, by the PPA, the convergence to a solution of the convex minimization problem in the framework of Hilbert spaces.
Recently, many convergence results by the PPA for solving optimization problems have been extended from the classical linear spaces such as Euclidean spaces, Hilbert spaces and Banach spaces to the setting of manifolds [27–30]. The minimizers of the objective convex functionals in the spaces with nonlinearity play a crucial role in the branch of analysis and geometry. Numerous applications in computer vision, machine learning, electronic structure computation, system balancing and robot manipulation can be considered as solving optimization problems on manifolds (see [31–34]).
Can we establish strong convergence of the sequence to minimizers of a convex function and to fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces?
Motivated by the previous works, we propose the modified proximal point algorithm using the S-type iteration process for two nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces and prove some convergence theorems of the proposed processes under some mild conditions.
2 Preliminaries and lemmas
- (1)
Euclidean spaces \(\mathbb{R}^{n}\);
- (2)
Hilbert spaces;
- (3)
simply connected Riemannian manifolds of nonpositive sectional curvature;
- (4)
hyperbolic spaces;
- (5)
trees.
It is well known that, in \(\operatorname{CAT}(0)\) spaces, \(A(\{x_{n}\})\) consists of exactly one point [37].
Definition 2.1
A sequence \(\{x_{n}\}\) in a \(\operatorname{CAT}(0)\) space X is said to Δ-converge to a point \(x\in X\) if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\).
In this case, we write \(\Delta\mbox{-}\!\lim_{n\rightarrow\infty}x_{n}=x\) and call x the Δ-limit of \(\{x_{n}\}\). We denote \(w_{\Delta}(x_{n}):=\bigcup\{A(\{u_{n}\})\}\), where the union is taken over all subsequences \(\{u_{n}\}\) of \(\{x_{n}\}\).
Recall that a bounded sequence \(\{x_{n}\}\) in X is said to be regular if \(r(\{x_{n}\})=r(\{u_{n}\})\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). It is well known that every bounded sequence in X has a Δ-convergent subsequence [38].
Lemma 2.2
[3]
Let C be a closed and convex subset of a complete \(\operatorname{CAT}(0)\) space X and \(T:C\rightarrow C\) be a nonexpansive mapping. Let \(\{x_{n}\}\) be a bounded sequence in C such that \(\lim_{n\rightarrow\infty}d(x_{n},Tx_{n})=0\) and \(\Delta\mbox{-}\!\lim_{n\rightarrow\infty}x_{n}=x\). Then \(x=Tx\).
Lemma 2.3
[3]
If \(\{x_{n}\}\) is a bounded sequence in a complete \(\operatorname{CAT}(0)\) space with \(A(\{x_{n}\})=\{x\}\), \(\{u_{n}\}\) is a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\})=\{u\}\) and the sequence \(\{d(x_{n},u)\}\) converges, then \(x=u\).
Let \(f:X\rightarrow(-\infty,\infty]\) be a proper convex and lower semi-continuous function. It was shown in [25] that the set \(F(J_{\lambda})\) of fixed points of the resolvent associated with f coincides with the set \(\operatorname{argmin}_{y\in X}f(y)\) of minimizers of f.
Lemma 2.4
[39]
Let \((X,d)\) be a complete \(\operatorname{CAT}(0)\) space and \(f:X\rightarrow(-\infty ,\infty]\) be proper convex and lower semi-continuous. For any \(\lambda>0\), the resolvent \(J_{\lambda}\) of f is nonexpansive.
Lemma 2.5
[41]
Proposition 2.6
[39, 40] (The resolvent identity)
For more results in \(\operatorname{CAT}(0)\) spaces, refer to [42].
3 Main results
We are now ready to prove our main results.
Theorem 3.1
- (1)
\(\lim_{n\rightarrow\infty}d(x_{n},q)\) exists for all \(q\in\Omega\);
- (2)
\(\lim_{n\rightarrow\infty}d(x_{n},z_{n})=0\);
- (3)
\(\lim_{n\rightarrow\infty}d(x_{n},T_{1}x_{n})=\lim_{n\rightarrow \infty}d(x_{n},T_{2}x_{n})=0\).
Proof
Next, we prove the Δ-convergence of our iteration.
Theorem 3.2
Proof
Next, we show that \(w_{\Delta}(x_{n})\subset\Omega\). Let \(u\in w_{\Delta}(x_{n})\). Then there exists a subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\) such that \(A(\{u_{n}\})=\{u\}\). From Lemma 2.2, there exists a subsequence \(\{v_{n}\}\) of \(\{u_{n}\}\) such that \(\Delta\mbox{-}\!\lim_{n\rightarrow\infty}v_{n}=v\) for some \(v\in\Omega\). So, \(u=v\) by Lemma 2.3. This shows that \(w_{\Delta}(x_{n})\subset \Omega\).
Finally, we show that the sequence \(\{x_{n}\}\) Δ-converges to a point in Ω. To this end, it suffices to show that \(w_{\Delta}(x_{n})\) consists of exactly one point. Let \(\{u_{n}\}\) be a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\})=\{u\}\) and let \(A(\{x_{n}\})=\{x\}\). Since \(u\in w_{\Delta}(x_{n})\subset\Omega\) and \(\{d(x_{n},u)\}\) converges, by Lemma 2.3, we have \(x=u\). Hence \(w_{\Delta}(x_{n})=\{x\}\). This completes the proof. □
If \(T_{1}=T_{2}=T\) in Theorem 3.2, then we obtain the following result.
Corollary 3.3
Since every Hilbert space is a complete \(\operatorname{CAT}(0)\) space, we obtain directly the following result.
Corollary 3.4
Next, we establish the strong convergence theorems of our iteration.
Theorem 3.5
Let X be a complete \(\operatorname{CAT}(0)\) space and \(f:X\rightarrow(-\infty,\infty]\) be a proper convex and lower semi-continuous function. Let \(T_{1}\) and \(T_{2}\) be nonexpansive mappings on X such that \(\Omega=F(T_{1})\cap F(T_{2})\cap \operatorname{argmin}_{y\in X} f(y)\) is nonempty. Assume that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences such that \(0< a\leq\alpha_{n}\), \(\beta_{n}\leq b<1\) for all \(n\in\mathbb{N}\) and for some a, b, and \(\{\lambda_{n}\}\) is a sequence such that \(\lambda_{n}\geq\lambda>0\) for all \(n\in\mathbb{N}\) and for some λ. Then the sequence \(\{x_{n}\}\) generated by (3.12) strongly converges to a common element of Ω if and only if \(\liminf_{n\rightarrow\infty}d(x_{n},\Omega)=0\), where \(d(x,\Omega)=\inf\{d(x,q):q\in\Omega\}\).
Proof
A family \(\{A, B, C\}\) of mappings is said to satisfy the condition \((\Omega)\) if there exists a nondecreasing function \(f:[0,\infty)\rightarrow[0,\infty)\) with \(f(0)=0\), \(f(r)>0\) for all \(r\in(0,\infty)\) such that \(d(x,Ax)\geq f(d(x,F))\) or \(d(x,Bx)\geq f(d(x,F))\) or \(d(x,Cx)\geq f(d(x,F))\) for all \(x\in X\). Here \(F=F(A)\cap F(B)\cap F(C)\).
Theorem 3.6
Let X be a complete \(\operatorname{CAT}(0)\) space and \(f:X\rightarrow(-\infty,\infty]\) be a proper convex and lower semi-continuous function. Let \(T_{1}\) and \(T_{2}\) be nonexpansive mappings on X such that \(\Omega=F(T_{1})\cap F(T_{2})\cap \operatorname{argmin}_{y\in X} f(y)\) is nonempty. Assume that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences such that \(0< a\leq\alpha_{n}\), \(\beta_{n}\leq b<1\) for all \(n\in\mathbb{N}\) and for some a, b, and \(\{\lambda_{n}\}\) is a sequence such that \(\lambda_{n}\geq\lambda>0\) for all \(n\in\mathbb{N}\) and for some λ. If \(\{J_{\lambda}, T_{1}, T_{2}\}\) satisfies the condition \((\Omega)\), then the sequence \(\{x_{n}\}\) generated by (3.12) strongly converges to a common element of Ω.
Proof
A mapping \(T:C\rightarrow C\) is said to be semi-compact if any sequence \(\{x_{n}\}\) in C satisfying \(d(x_{n},Tx_{n})\rightarrow0\) has a convergent subsequence.
Theorem 3.7
Let X be a complete \(\operatorname{CAT}(0)\) space and \(f:X\rightarrow(-\infty,\infty]\) be a proper convex and lower semi-continuous function. Let \(T_{1}\) and \(T_{2}\) be nonexpansive mappings on X such that \(\Omega=F(T_{1})\cap F(T_{2})\cap \operatorname{argmin}_{y\in X} f(y)\) is nonempty. Assume that \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences such that \(0< a\leq\alpha_{n}\), \(\beta_{n}\leq b<1\) for all \(n\in\mathbb{N}\) and for some a, b, and \(\{\lambda_{n}\}\) is a sequence such that \(\lambda_{n}\geq\lambda>0\) for all \(n\in\mathbb{N}\) and for some λ. If \(T_{1}\) or \(T_{2}\), or \(J_{\lambda}\) is semi-compact, then the sequence \(\{x_{n}\}\) generated by (3.12) strongly converges to a common element of Ω.
Proof
Remark 3.8
- (1)
Our main results generalize Theorem 1, Theorem 2 and Theorem 3 of Khan-Abbas [4] from one nonexpansive mapping to two nonexpansive mappings involving the convex and lower semi-continuous function in \(\operatorname{CAT}(0)\) spaces.
- (2)
Theorem 3.1 extends that of Bačák [24] in \(\operatorname{CAT}(0)\) spaces. In fact, we present a new modified proximal point algorithm for solving the convex minimization problem as well as the fixed point problem of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces.
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Prasit Cholamjiak thanks University of Phayao. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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