Proximal Point Algorithm Based on AP Iterative Technique for Nonexpansive Mappings in CAT(0) Spaces
DOI:
https://doi.org/10.26713/cma.v14i1.1831Keywords:
Proximal point algorithm, Nonexpansive mapping, CAT(0) space, Convex minimization problemAbstract
In this paper, we introduce the modified proximal point algorithm for solving minimization problems in CAT(0) spaces. We then show that the sequence converges to a common fixed point of nonexpansive mapping and a minimizer of a convex function. Finally, we present a numerical illustration for supporting our main result. The findings in this paper are a generalization of certain corresponding results given by some authors.
Downloads
References
R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton’s method on Riemannian manifolds and a geometric model for human spine, IMA Journal of Numerical Analysis 22(3) (2002), 359-390, DOI: 10.1093/imanum/22.3.359.
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel (2008), DOI: 10.1007/978-3-7643-8722-8.
D. Ariza-Ruiz, L. Leu¸stean and G. López-Acedo, Firmly nonexpansive mappings in classes of geodesic spaces, Transactions of the American Mathematical Society 366 (2014), 4299 – 4322, DOI: 10.1090/S0002-9947-2014-05968-0.
M. Bacák, Computing medians and means in Hadamard spaces, SIAM Journal on Optimization 24(3) (2014), 1542 – 1566, DOI: 10.1137/140953393.
M. Bacák, The proximal point algorithm in metric spaces, Israel Journal of Mathematics 194 (2013), 689 – 701, DOI: 10.1007/s11856-012-0091-3.
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften Book series, Vol. 319, Springer, Berlin (1999), URL: https://link.springer.com/book/10.1007/978-3-662-12494-9.
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 41 (1972), 5 – 251, DOI: 10.1007/BF02715544.
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, Rhode Island (2001), DOI: 10.1090/gsm/033.
P. Cholamjiak, A. A. N. Abdou and Y. J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Applications 2015 (2015), 227, 1 – 13 DOI: 10.1186/s13663-015-0465-4.
S. Dhompongsa and B. Panyanak, On ∆-convergence theorems in CAT(0) spaces, Computers & Mathematics with Applications 56(10) (2008), 2572 – 2579, DOI: 10.1016/j.camwa.2008.05.036.
S. Dhompongsa, W. A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, Journal of Nonlinear and Convex Analysis 8 (2007), 35 – 45.
S. Dhompongsa, W. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Analysis: Theory, Methods & Applications 65(4) (2006), 762 – 772, DOI: 10.1016/j.na.2005.09.044.
O. P. Ferreira and P. R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization 51(2) (2002), 257 – 270, DOI: 10.1080/02331930290019413.
H. Fukhar-ud-din, Strong convergence of an Ishikawa-type algorithm in CAT(0) spaces, Fixed Point Theory and Applications 2013 (2013), Article number: 207, DOI: 10.1186/1687-1812-2013-207.
M. Gromov, Hyperbolic groups, In: Essays in Group Theory, S. M. Gersten (eds), Mathematical Sciences Research Institute Publications, Vol. 8, Springer, New York, (1987), DOI: 10.1007/978-1-4613-9586-7_3.
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization 29(2) (1991), 403 – 419, DOI: 10.1137/0329022.
B. Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society 73 (1967), 957 – 961, DOI: 10.1090/S0002-9904-1967-11864-0.
J. Jost, Convex functionals and generalized harmonic maps into spaces of non positive curvature, Commentarii Mathematici Helvetici 70 (1995), 659 – 673, DOI: 10.1007/BF02566027.
S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, Journal of Approximation Theory 106(2) (2000), 226 – 240, DOI: 10.1006/jath.2000.3493.
S. H. Khan and M. Abbas, Strong and ∆-convergence of some iterative schemes in CAT(0) spaces, Computers & Mathematics with Applications 61(1) (2011), 109 – 116, DOI: 10.1016/j.camwa.2010.10.037.
W. A. Kirk, Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta. University Seville Secretary of Publications, Seville, Spain 64 (2003), 195 – 225.
W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods & Applications 68(13) (2008), 3689 – 3696, DOI: 10.1016/j.na.2007.04.011.
P. Lamba and A. Panwar, On different results for new three step iteration process in CAT(0) space, Journal of Interdisciplinary Mathematics 24(4) (2021), 897 – 909, DOI: 10.1080/09720502.2020.1860285.
T. Laokul and B. Panyanak, Approximating fixed points of nonexpansive mappings in CAT(0) spaces, International Journal of Mathematical Analysis 3(25-28) (2009), 1305 – 1315, URL: http://cmuir.cmu.ac.th/jspui/handle/6653943832/59721.
W. Laowang and B. Panyanak, Strong and ∆-convergence theorems for multivalued mappings in CAT(0) spaces, Journal of Inequalities and Applications 2009 (2009), Article ID 730132, DOI: 10.1155/2009/730132.
C. Li, G. López and V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, Journal of the London Mathematical Society 79(3) (2009), 663 – 683, DOI: 10.1112/jlms/jdn087.
T. C. Lim, Remarks on some fixed point theorems, Proceedings of the American Mathematical Society 60 (1976), 179 – 182, DOI: 10.1090/S0002-9939-1976-0423139-X.
B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Revue française d’informatique et de recherche opérationnelle 4(R-3) (1970), 154 – 158, URL: http://www.numdam.org/article/M2AN_1970__4_3_154_0.pdf.
U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Communications in Analysis and Geometry 6(2) (1998), 199 – 253, DOI: 10.4310/CAG.1998.v6.n2.a1.
E. A. P. Quiroz and P. R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on hadamard manifolds, Journal of Convex Analysis 16(1) (2009), 49 – 69, URL: https://www.heldermann-verlag.de/jca/jca16/jca0604_b.pdf.
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization 14(5) (1976), 877 – 898, DOI: 10.1137/0314056.
S. T. Smith, Optimization techniques on Riemannian manifolds, Hamiltonian and gradient flows, algorithms and control, Fields Institute Communications 3 (1994), 113 – 136.
C. Udri¸ste, Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications book series (MAIA, Vol. 297), Springer Science & Business Media (1994), DOI: 10.1007/978-94-015-8390-9.
J. H. Wang and G. López, Modified proximal point algorithms on Hadamard manifolds, Optimization 60(6) (2011), 697 – 708, DOI: 10.1080/02331934.2010.505962.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
