# W-convergence of the proximal point algorithm in complete CAT(0) metric spaces

Document Type : Research Paper

Author

Department of Mathematics‎, ‎College of Sciences‎, ‎Higher Education Center of Eghlid‎, ‎Eghlid‎, ‎Iran.

Abstract

‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎X\rightarrow‎ ‎]-\infty‎, +‎\infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal point algorithm $w$-converges to a zero of the subdifferential of $f$‎, ‎i.e., a minimizer of $f$‎. ‎Some strong convergence results (convergence in metric) are also presented with additional assumptions on the monotone operator and‎ ‎the convex function $f$.

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Main Subjects

#### References

B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141 (2013), no. 3, 1029--1039.
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. Bacak, 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 Aleksandrov spaces, Geom. Dedicata 133 (2008) 195--218.
O.A. Boikanyo and G. Morosanu, Modified Rockafellar's algorithms, Math. Sci. Res. J. 13 (2009), no. 5, 101--122.
M. Bridson and A. Haeiger, Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences, Springer, Berlin, 1999.
H. Brezis and P.L. Lions, Produits infinis deresolvantes, Israel J. Math. 29 (1978), no. 4, 329--345.
K.S. Brown, Buildings, Springer, New York, 1989.
D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math. 33, Amer. Math. Soc. Providence, RI, 2001.
S. Dhompongsa and B. Panyanak, On Δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008) 2572--2579.
B. Djafari Rouhani and H. Khatibzadeh, On the proximal point algorithm, J. Optim.Theory Appl. 137 (2008), no. 2, 411--417.
R. Espinola and A. Fernandez-Leon, CAT(k)-spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009), no. 1, 410--427.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpan-sive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1984.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by S.M. Bates, Progr. Math. 152, Birkhäuser, Boston, MA, 1999.
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29 (1991), no. 2, 403--419.
M.T. Heydari and S. Ranjbar, Halpern-type proximal point algorithm in complete CAT(0) metric spaces, An. Ştiint. Univ. Ovidius Constanta Ser. Mat. 24 (2016), no. 3, 141--159.
J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math. ETH Zürich, Birkhäuser, Basel 1997.
H. Khatibzadeh, Some remarks on the proximal point algorithm, J. Optim. Theory Appl. 153 (2012), no. 3, 769--778.
H. Khatibzadeh and S. Ranjbar, A variational inequality in complete CAT(0) spaces, J. Fixed Point Theory Appl. 17 (2015), no. 3, 557--574.
H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math. Soc. 2016 (2016) DOI:10.1017/s1446788716000446.
W.A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl. 2004 (2004), no. 4, 309--316.
W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689--3696.
C. Li, G. Lopez and V. Martin-Marquez, Monotone vector fields and the proximal point algorithm, J. Lond. Math. Soc. 679 (2009), no. 3, 663--683.
T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179--182.
B. Martinet, Regularisation d'inequations variationnelles par approximations successives, Revue Francaise d'Informatique et de Recherche Operationnelle, 4 (1970) 154--158.
G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane (and D. Reidel publishing Company), Bucharest, 1988.
R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877--898.

### History

• Receive Date: 30 January 2016
• Revise Date: 21 February 2016
• Accept Date: 04 March 2016