Halpern-type proximal point algorithm in CAT(0) spaces
DOI:
https://doi.org/10.12775/TMNA.2022.038Keywords
Monotone inclusion problem, fixed point problem, nonexpansive mapping, resolvent operators, CAT(0) spacesAbstract
A method which is a combination of the Halpern method and proximal point method (PPA) is introduced in this paper. It is proved that the sequence of iterates generated by our method converges strongly to a point which is a common solution to some monotone inclusion problem and fixed point problem in CAT$(0)$ spaces under some appropriate conditions.References
B. Ahmadi Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73 (2010), no. 10, 3450–3455.
M. Bacák, The proximal point algorithm in metric spaces, Israel J. Math. 194 (2013), no. 2, 689–701.
I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), 195–218.
O.A. Boikanyo and G. Morosanu, Four parameter proximal point algorithms, Nonlinear Anal. 74 (2011), 544–555.
O.A. Boikanyo and G. Morosanu, Inexact Halpern-type proximal point algorithm, J. Global Optim. 51 (2011), 11–26.
M.R. Bridson and A. Haefilger, Metric Spaces of Non-Positive Curvature, Fundamental Principle of Mathematical Sciences, vol. 319, Springer, Berlin, Germany, 1999.
F. Bruhat and J. Tits, Groupes Réductifs sur un Corp Local, I. Donneéadicielles Valuées, vol. 41, Institut des Hautes Études Scientifiques, 1972.
S.S. Chang, J.C. Yao, C.F. Wen, L. Yang and L.J. Qin, Common zero for a finite family of monotone mappings in Hadamard spaces with applications, Mediterr. J. Math. 15 (2018), article 160.
P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006), no. 2, 983–987.
C.E. Chidume, A.U. Bello and P. Ndambomve, Strong and ∆-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal. 2014 (2014).
P. Cholamjiak, The modified proximal point algorithm in CAT(0) spaces, Optim. Lett. 9 (2015), 1401–1410.
J.X. DaCruz Neto, O.P. Ferreira, L.R. Lucambio Pérez and S.Z. Németh, Convexand monotone-transformable mathematical programming problems and a proximal-like point method, J. Global Optim. 35 (2006), 53–69.
H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces, arXiv: 1410.1137VI [math.FA], (2014).
S. Dhompongsa, W.A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007), 35–45.
S. Dhompongsa abd B. Panyanak, On ∆-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), 2572–2579.
Q. Dong, D. Jiang, P. Cholamjiak and Y. Shehu, A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions, J. Fixed Point Theory Appl. 19 (2017), 3097–3118.
O.P. Ferreira and P.R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization 51 (2002), 257–270.
Goebel K, Reich S, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, (1984).
M. Gromov and S.M. Bates, Metric Structures for Riemannian and Non-Riemannian Spaces, with Appendices by M. Katz, P. Pansu and S. Semmes (S.M. Lafontaine, P. Pansu, eds.), Progr. Math., vol. 152, Birkhäuser, 1999.
S. Jain and L.B. Jain, On Banach contraction principle in a cone metric space, J. Nonlinear Sci. Appl. 5 (2012), 252–258.
J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math., ETH Zürich, Birkhäuser, Basel, 199).
B.A. Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73 (2010), 3450–3455.
S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000), 226–240.
K.R. Kazmi and S.H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett. 8 (2014), no. 3, 1113–1124.
H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math. Soc. 103 (2017), no. 1, 70–90.
H. Khatibzadeh and S. Ranjbar, A variational inequality in complete CAT(0) spaces, J. Fixed Point Theory Appl. 17 (2015), 557–574.
K.S. Kim, Some convergence theorems for contractive type mappings in CAT(0) spaces, Abstr. Appl. Anal. (2013), Art. ID 381715, 9 pp.
W.A. Kirk, Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta, vol. 64, Iniversidad de Sevilla, Sevilla, Spain, 2003, pp. 195–225.
W.A. Kirk, Geometry and fixed point theory II, International Conference on Fixed Point Theory and Applications, Yokohama Publishers, Yokohama, 2004, pp. 113–142.
W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689–3696.
L. Leustean, Nonexpansive iterations uniformly cover W -hyperbolic spaces, Nonlinear Analysis and Optimization 1: Nonlinear Analysis, Contemporary Math. Am. Math. Soc., Providence 513 (2010), 193–209.
C. Li, G. López and V. Martı́n-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc. 79 (2009), 663–683.
T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182.
P.E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899–912.
G. Marino and H.K. Xu, Convergence of generalized proximal point algorithm, Comm. Pure Appl. Anal. 3 (2004), 791–808.
J. Merryfield, Generalization of the Banach contraction principle, J. Math. Anal. Appl. 273 (2002), 112–120.
F.U. Oguisis and O.T. Mewomo, Iterative solution of split variational inclusion problem in real Banach space, Afr. Math. 28 (2017), 295–309.
C.C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces, Optim. Methods Softw. 34 (2018), no. 6, 1168–1183.
R.S. Palais, A simple proof of the Banach contraction principle, J. Fixed Point Theory Appl. 2 (2007), 221–223.
E.A. Papa Quiroz and P.R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds, J. Convex Anal. 16 (2009), 49–69.
S. Ranjbar and H. Khatibzadeh, Strong and ∆-convergence to a zero of a monotone operator in CAT(0) spaces, Mediterr. J. Math. 14 (2017), article 56.
S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537–558.
S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal. 75 (2012), 742–750.
Y. Shehu, Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces, Results Math. 74, 138 (2019).
S. Suantai, Y. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problem in Banach spaces, Optim. Methods Softw. 34 (2019), 853–874.
R. Suparatulatorn, P. Cholamjiak and R. Suparatulatorn, On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods Softw. 32 (2017), no. 1, 182–192.
R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877–898.
J. Tang, Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces, Abstr. Appl. Anal. (2014), Art. ID 389804, 9 pp.
F. Wang and H. Cui, On the contraction-proximal point algorithms with multi-parameters, J. Global Optim. 54 (2012), 485–491.
J.H. Wang and G. López, Modified proximal point algorithms on Hadamard manifolds, Optimization 60 (2011), 697–708.
L. Wang, C.F. Wen and J.Q. Zhang, The modified proximal point algorithm in Hadamard spaces, J. Inequal. Appl. 2018 (2018), article no. 124.
H.K. Xu, Iterative algorithms for nonlinear operators, J. London. Math. Soc. 66 (2002), no. 1, 240–256.
H.K. Xu, A regularization method for the proximal point algorithm, J. Global Optim. 36 (2006), 115–125.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Chibueze Christian Okeke
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0