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
MSC
1 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
References
- Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61-79 (2007) MathSciNetMATHGoogle Scholar
- Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), pp. 195-225. Univ. Sevilla Secr. Publ., Seville (2003) Google Scholar
- Dhompongsa, S, Panyanak, B: On Δ-convergence theorems in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56, 2572-2579 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Khan, SH, Abbas, M: Strong and Δ-convergence of some iterative schemes in \(\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 61, 109-116 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Chang, SS, Wang, L, Lee, HWJ, Chan, CK, Yang, L: Demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Appl. Math. Comput. 219, 2611-2617 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Cho, YJ, Ćirić, L, Wang, S: Convergence theorems for nonexpansive semigroups in \(\operatorname{CAT}(0)\) spaces. Nonlinear Anal. 74, 6050-6059 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Cuntavepanit, A, Panyanak, B: Strong convergence of modified Halpern iterations in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2011, Article ID 869458 (2011) View ArticleMathSciNetGoogle Scholar
- Fukhar-ud-din, H: Strong convergence of an Ishikawa-type algorithm in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2013, 207 (2013) View ArticleGoogle Scholar
- Laokul, T, Panyanak, B: Approximating fixed points of nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Int. J. Math. Anal. 3, 1305-1315 (2009) MathSciNetMATHGoogle Scholar
- Laowang, W, Panyanak, B: Strong and Δ-convergence theorems for multivalued mappings in \(\operatorname{CAT}(0)\) spaces. J. Inequal. Appl. 2009, Article ID 730132 (2009) View ArticleMathSciNetGoogle Scholar
- Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2010, Article ID 268780 (2010) View ArticleMathSciNetGoogle Scholar
- Phuengrattana, W, Suantai, S: Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2012, 230 (2012) View ArticleGoogle Scholar
- Saejung, S: Halpern’s iteration in \(\operatorname{CAT}(0)\) spaces. Fixed Point Theory Appl. 2010, Article ID 471781 (2010) MathSciNetGoogle Scholar
- Shi, LY, Chen, RD, Wu, YJ: Δ-Convergence problems for asymptotically nonexpansive mappings in \(\operatorname{CAT}(0)\) spaces. Abstr. Appl. Anal. 2013, Article ID 251705 (2013) MathSciNetGoogle Scholar
- Martinet, B: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154-158 (1970) MathSciNetMATHGoogle Scholar
- Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877-898 (1976) View ArticleMathSciNetMATHGoogle Scholar
- Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403-419 (1991) View ArticleMathSciNetMATHGoogle Scholar
- Kamimura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226-240 (2000) View ArticleMathSciNetMATHGoogle Scholar
- Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967) View ArticleMATHGoogle Scholar
- Boikanyo, OA, Morosanu, G: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635-641 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Marino, G, Xu, HK: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791-808 (2004) View ArticleMathSciNetMATHGoogle Scholar
- Xu, HK: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115-125 (2006) View ArticleMATHGoogle Scholar
- Yao, Y, Noor, MA: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217, 46-55 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Bačák, M: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689-701 (2013) View ArticleMATHGoogle Scholar
- Ariza-Ruiz, D, Leuştean, L, López, G: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 366, 4299-4322 (2014) View ArticleMATHGoogle Scholar
- Bačák, M: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24, 1542-1566 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Ferreira, OP, Oliveira, PR: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257-270 (2002) View ArticleMathSciNetMATHGoogle Scholar
- Li, C, López, G, Martín-Márquez, V: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663-683 (2009) View ArticleMathSciNetMATHGoogle Scholar
- Papa Quiroz, EA, Oliveira, PR: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16, 49-69 (2009) MathSciNetMATHGoogle Scholar
- Wang, JH, López, G: Modified proximal point algorithms on Hadamard manifolds. Optimization 60, 697-708 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Adler, R, Dedieu, JP, Margulies, JY, Martens, M, Shub, M: Newton’s method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal. 22, 359-390 (2002) View ArticleMathSciNetMATHGoogle Scholar
- Smith, ST: Optimization techniques on Riemannian manifolds. In: Hamiltonian and Gradient Flows, Algorithms and Control. Fields Institute Communications, vol. 3, pp. 113-136. Am. Math. Soc., Providence (1994) Google Scholar
- Udriste, C: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994) View ArticleMATHGoogle Scholar
- Wang, JH, Li, C: Convergence of the family of Euler-Halley type methods on Riemannian manifolds under the γ-condition. Taiwan. J. Math. 13, 585-606 (2009) MATHGoogle Scholar
- Bridson, MR, Haefliger, A: Metric Spaces of Non-positive Curvature. Grundelhren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999) MATHGoogle Scholar
- Bruhat, M, Tits, J: Groupes réductifs sur un corps local: I. Données radicielles valuées. Publ. Math. Inst. Hautes Études Sci. 41, 5-251 (1972) View ArticleMathSciNetMATHGoogle Scholar
- Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762-772 (2006) View ArticleMathSciNetMATHGoogle Scholar
- Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689-3696 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Jost, J: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659-673 (1995) View ArticleMathSciNetMATHGoogle Scholar
- Mayer, UF: Gradient flows on nonpositively curved metric spaces and harmonic maps. Commun. Anal. Geom. 6, 199-253 (1998) MATHGoogle Scholar
- Ambrosio, L, Gigli, N, Savare, G: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2008) MATHGoogle Scholar
- Bačák, M: Convex Analysis and Optimization in Hadamard Spaces. de Gruyter, Berlin (2014) MATHGoogle Scholar